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Grid integration of renewable energy sources via virtual synchronous generator Dong, Shuan 2019

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Grid Integration of Renewable Energy Sources via VirtualSynchronous GeneratorbyShuan DongB.Sc., Tianjin University, 2012M.Sc., China Electric Power Research Institute, 2015a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoral studies(Electrical and Computer Engineering)The University of British Columbia(Vancouver)November 2019c© Shuan Dong, 2019The following individuals certify that they have read, and recommend to the Faculty of Grad-uate and Postdoctoral Studies for acceptance, the dissertation entitled:Grid Integration of Renewable Energy Sources via Virtual Synchronous Gen-eratorsubmitted by Shuan Dong in partial fulfillment of the requirements for the degree of Doctorof Philosophy in Electrical and Computer Engineering.Examining Committee:Yu Christine Chen, Electrical and Computer EngineeringSupervisorJuri Jatskevich, Electrical and Computer EngineeringSupervisory Committee MemberJose´ Mart´ı, Electrical and Computer EngineeringSupervisory Committee MemberHermann Dommel, Electrical and Computer EngineeringSupervisory Committee MemberRyozo Nagamune, Mechanical EngineeringUniversity ExaminerShahriar Mirabbasi, Electrical and Computer EngineeringUniversity ExamineriiAbstractOngoing efforts toward environmentally sustainable electricity generation give rise to gradualdisplacement of synchronous generators by renewable energy resources (RESs). This paradigmshift reshapes power system dynamics and presents numerous challenges to reliable and efficientgrid operations. For example, our power system will have reduced inertia and be at higher riskof instability, since the RESs generally interconnect to the grid via power-electronic converterswith less or no inertia. Motivated by these challenges arising from RES integration, the con-cept of virtual synchronous generator (VSG) has been proposed to provide virtual inertia byemulating the SG dynamics in the RES controller. Among all VSG designs, synchronverter isa representative one with concise structure.However, this dissertation finds that conventional synchronverter designs lack in controldegrees of freedom, require trial-and-error tuning process, synchronize with the grid slowly,and suffer from output-power coupling. Also, their active-power transfer capacity has not beenstudied, especially under weak-grid conditions. In order to address these problems and integratemore RESs into our system, my dissertation has five major contributions ranging from controldesign to tuning method to operation characteristics. First, in order to improve the synchron-verter control degrees of freedom, I augment the synchronverter with a damping correctionloop, which freely adjusts its response speed without affecting the steady-state performance.In order to simplify the tuning process, I propose a tuning method that evaluates the feasi-ble pole-placement region and directly computes synchronverter parameters to achieve desireddynamics. My proposed tuning method completely avoids the trial-and-error tuning processand thus has overwhelming advantages over conventional tuning methods. Next, in order tosynchronize the synchronverter quickly to the grid and enable the flexible “plug-and-play” op-eration of RESs, this dissertation proposes a self-synchronizing synchronverter design with bothiiifast self-synchronization speed and easily tuneable parameters. Then, to further improve thetracking performance, I propose a design with reduced output-power coupling. Finally, in orderto integrate synchronverter-based RESs in weak grid, this dissertation analytically studies itsactive power transfer capacity and proposes two countermeasures to improve it. All my pro-posed designs and analyses are verified through extensive numerical or experimental studies.ivLay SummaryDriven by environmental considerations, renewable energy sources (RESs) are expected to dis-place a sizeable portion of conventional fossil fuel-based synchronous generators in our powersystem. However, RESs usually have less or no inertia and thus their integration may causesystem instability. In order to tackle this problem, I propose a controller design for RESs,which contributes virtual inertia to the power system and has freely adjustable response speed.Next, in order to facilitate its tuning process, I propose a method to analytically evaluate theparameter tuning range and directly compute controller parameters to achieve desired dynam-ics. Then, I improve the controller such that it quickly connects the renewables to the powersystem and has better tracking performance. These designs have simple structure and easilytuneable parameters. Moreover, I systematically evaluate the transfer limit of renewables withthe proposed design. All these significantly facilitate the renewable energy integration in powersystem.vPrefaceThis dissertation has been written and prepared by the author under supervision of Prof. YuChristine Chen. Most of the research results in this dissertation have been published in, sub-mitted to, or will be submitted to scientific journals and conference proceedings. In all of them,I was responsible for developing the ideas, deriving the mathematical models, performing thetheoretical analyses, and conducting simulation studies. My supervisor Prof. Yu Christine Chenprovides detailed guidance, highly valuable feedback, and extremely helpful comments at allstages of these research works, from developing the ideas to writing the papers. My collaboratorJingya Jiang from Beijing Jiaotong University helps conduct the experimental verification inChapter 5. Prof. Ryozo Nagamune from The University of British Columbia provides insightfuland helpful discussions on root locus analysis used in Chapter 3. All publications resulting fromthis doctoral dissertation are listed as follows:Chapter 2• S. Dong and Y. C. Chen, “Adjusting synchronverter dynamic response speed via dampingcorrection loop,” IEEE Trans. Energy Convers., vol. 32, no. 2, pp. 608–619, Jun. 2017.Chapter 3• S. Dong and Y. C. Chen, “A Method to Directly Compute Synchronverter Parametersfor Desired Dynamic Response,” IEEE Trans. Energy Convers., vol. 33, no. 2, pp.814-825, Jun. 2018.viChapter 4• S. Dong and Y. C. Chen, “Analysis of Feasible Synchronverter Pole-placement Regionto Facilitate Parameter Tuning,” under preparation.Chapter 5• S. Dong, J. Jiang, and Y. C. Chen, “Analysis of Synchronverter Self-synchronization Dy-namics to Facilitate Parameter Tuning,” IEEE Trans. Energy Convers., to be published.• S. Dong and Y. C. Chen, “A Fast Self-synchronizing Synchronverter Design with Eas-ily Tuneable Parameters,” in Proc. of IEEE Power & Energy Society General Meeting,Portland, OR, Aug. 2018.Chapter 6• S. Dong and Y. C. Chen, “Reducing Output Active- and Reactive-power Coupling in Vir-tual Synchronous Generators,” in Proc. of IEEE International Symposium on IndustrialElectronics (ISIE), Vancouver, BC, Jun. 2019.Chapter 7• S. Dong and Y. C. Chen, “Improving Active-power Transfer Capacity of Virtual Syn-chronous Generator in Weak Grid,” in Proc. of 20th IEEE Workshop on Control andModeling for Power Electronics (COMPEL), Toronto, ON, Jun. 2019.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxivAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxviDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxviii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.1 Lacking in Control Degrees of Freedom . . . . . . . . . . . . . . . . . . . 41.2.2 Inefficient Tuning Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.3 Slow Self-synchronization Speed . . . . . . . . . . . . . . . . . . . . . . . 71.2.4 Undesired Output-power Coupling . . . . . . . . . . . . . . . . . . . . . . 81.2.5 Unrevealed Active-power Transfer Capacity . . . . . . . . . . . . . . . . . 81.3 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9viii1.3.1 Objective I: Increasing Synchronverter Control Degrees of Freedom . . . . 91.3.2 Objective II: Proposing Efficient Synchronverter Tuning Method . . . . . 91.3.3 Objective III: Achieving Fast Self-synchronization . . . . . . . . . . . . . 101.3.4 Objective IV: Reducing Synchronverter Output-power Coupling . . . . . . 111.3.5 Objective V: Improving Synchronverter Active-power Transfer Capacity . 111.4 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Adjusting Synchronverter Dynamic Response Speed via Damping Correc-tion Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.1 Overview of Conventional Synchronverter . . . . . . . . . . . . . . . . . . 142.1.2 Response Speed of Conventional Synchronverter . . . . . . . . . . . . . . 192.2 Proposed Synchronverter Controller Design . . . . . . . . . . . . . . . . . . . . . 232.3 Small-signal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.1 Development of Synchronverer State-space Model . . . . . . . . . . . . . . 312.3.2 Verification of Linearized State-space Model . . . . . . . . . . . . . . . . . 332.3.3 Impact of Damping Correction Loop on Eigenvalues . . . . . . . . . . . . 342.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.4.1 During Normal Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.4.2 Realizing Self Synchronization . . . . . . . . . . . . . . . . . . . . . . . . 432.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 Directly Computing Synchronverter Parameters for Desired Dynamic Re-sponse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.1 Problems in Parameter Tuning of Synchronverter . . . . . . . . . . . . . . . . . . 463.2 Explanations for Problems in Parameter Tuning . . . . . . . . . . . . . . . . . . 493.2.1 Transfer Function Modeling of Synchronverter APL . . . . . . . . . . . . 493.2.2 Criterion for Different Eigenvalue Variation Patterns . . . . . . . . . . . . 513.3 Proposed Parameter Tuning Method . . . . . . . . . . . . . . . . . . . . . . . . . 573.3.1 Summary of Proposed Parameter Tuning Method . . . . . . . . . . . . . . 613.4 Case Studies in Single-synchronverter Infinite-bus System . . . . . . . . . . . . . 61ix3.4.1 Verifying the Reduced-order Model and the γ-Criterion . . . . . . . . . . 613.4.2 Validating the Proposed Parameter Computation Method . . . . . . . . . 643.5 Application of Proposed Parameter Computation Method in Power System . . . 653.5.1 Obtaining an Infinite-bus Equivalent for the External System . . . . . . . 673.5.2 Tuning Synchronverter Parameters with the Proposed Method . . . . . . 713.5.3 Verifying Synchronverter Dynamic Performance with Actual ExternalSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784 Analyzing Feasible Pole-placement Region of Synchronverter to FaciliateIts Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.1 Motivation of Studying Feasible Pole-placement Region . . . . . . . . . . . . . . 804.1.1 Overview of Direct Computation Method . . . . . . . . . . . . . . . . . . 814.1.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.2 Analysis of Feasible Synchronverter Pole-placement Region . . . . . . . . . . . . 844.2.1 Range of ω?n With Fixed ζ? ∈ (0, 1] . . . . . . . . . . . . . . . . . . . . . . 854.2.2 Feasible Pole-placement Region . . . . . . . . . . . . . . . . . . . . . . . . 904.2.3 Updated Parameter Tuning Procedures . . . . . . . . . . . . . . . . . . . 914.3 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.3.1 Verification of µ-Criterion that Specifies Range of ω?n . . . . . . . . . . . . 934.3.2 Verification of Feasible Pole-placement Region . . . . . . . . . . . . . . . 934.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955 Achieving Fast Self-synchronization Speed with Easily Tuneable Parameters 965.1 Self-synchronizing Synchronverter Design . . . . . . . . . . . . . . . . . . . . . . 985.1.1 Proposed Self-synchronizing Synchronverter Design . . . . . . . . . . . . . 985.1.2 Dynamic Response of Self Synchronization . . . . . . . . . . . . . . . . . 1035.2 Analysis of Self-synchronization Dynamics . . . . . . . . . . . . . . . . . . . . . . 1065.2.1 Phase-angle Self-synchronization Dynamics . . . . . . . . . . . . . . . . . 1065.2.2 Voltage-magnitude Self-synchronization Dynamics . . . . . . . . . . . . . 1095.2.3 Parameter Values to Achieve Fast Self Synchronization . . . . . . . . . . . 109x5.3 Simulation Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.3.1 Verification of Self-synchronization Analysis . . . . . . . . . . . . . . . . . 1135.3.2 Verification of Reduced Second-order APL Model . . . . . . . . . . . . . . 1185.4 Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216 Reducing Transient Active- and Reactive-power Coupling in Synchronverter1226.1 Proposed Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.2 Transfer-function Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.2.1 Transfer-function Model of the APL . . . . . . . . . . . . . . . . . . . . . 1256.2.2 Analysis of Output-power Coupling . . . . . . . . . . . . . . . . . . . . . 1266.2.3 APL Parameter Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.3 Simulation Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.3.1 Active- and Reactive-power Coupling . . . . . . . . . . . . . . . . . . . . 1316.3.2 Steady-state Frequency-droop Characteristics . . . . . . . . . . . . . . . . 1346.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347 Improving Active-power Transfer Capacity of Synchronverter in Weak Grid1367.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377.1.1 Synchronverter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377.1.2 Test System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1397.1.3 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417.2 Synchronverter Active-power Transfer Capacity . . . . . . . . . . . . . . . . . . . 1427.2.1 Analysis of Synchronverter Active-power Transfer Capacity . . . . . . . . 1427.2.2 Connection to Voltage Stability . . . . . . . . . . . . . . . . . . . . . . . . 1457.2.3 Improving Synchronverter Active-power Transfer Capacity . . . . . . . . . 1477.3 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1487.3.1 Activating Voltage-droop Control . . . . . . . . . . . . . . . . . . . . . . . 1497.3.2 Using Reactive-power Compensation Devices . . . . . . . . . . . . . . . . 1497.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150xi8 Research Contributions and Future Directions . . . . . . . . . . . . . . . . . 1518.1 Contribution Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1518.1.1 A Synchronverter Design With Freely Adjustable Response Speed . . . . 1518.1.2 A Tuning Method to Directly Computing Synchronverter Parameters . . 1528.1.3 A Fast Self-synchronizing Synchronverter Design . . . . . . . . . . . . . . 1538.1.4 A Synchronverter Design With Reduced Output-power Coupling . . . . . 1548.1.5 Improvement of Synchronverter Active-power Transfer Capacity in WeakGrids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1558.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1558.2.1 Unified Synchronverter Design for Various Grid Conditions . . . . . . . . 1558.2.2 Application of Proposed Synchronverter Designs in Power System . . . . 1568.2.3 Parameter Tuning Method of High-order System . . . . . . . . . . . . . . 1578.2.4 Studying Dynamics of Power-Electronics-Based Power System . . . . . . . 157Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159A Linearized State-space Model of Synchronverter Proposed in Chapter 2 . . 170B Parameters of Components in the Test System in Fig. 3.8 . . . . . . . . . . 175C Necessity of Adopting Virtual-Resistance in Self-synchronizing Synchron-verter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177C.1 Synchronverter with only virtual resistance R˜v. . . . . . . . . . . . . . . . . . . . 178C.2 Synchronverter with only virtual inductance L˜v. . . . . . . . . . . . . . . . . . . 182C.3 Proposed synchronverter with the virtual resistance Rv along with coordinatetransformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189D Derivation of Reduced Second-order APL Model in Chapter 5 . . . . . . . 192E Proof of Self-synchronization Capability of Synchronverter Proposed inChapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194xiiList of TablesTable 2.1 Parameters of Synchronverter-connected System in Fig. 2.1 . . . . . . . . . . . 19Table 2.2 Eigenvalue sensitivities to parameter Df . . . . . . . . . . . . . . . . . . . . . . 38Table 3.1 Direct Computation of APL Parameters Jg and Df . . . . . . . . . . . . . . . 63Table 3.2 Cases I–IV used to verify synchronverter dynamic performance with actualexternal system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Table 4.1 Parameters of Synchronverter-connected System in Fig. 4.1 . . . . . . . . . . . 83Table 5.1 Experimental Hardware and Controller Parameters in Fig. 5.12 . . . . . . . . 119Table 6.1 Parameter values used to verify proposed synchronverter design tuned to re-spond quickly (case I) and slowly (case II) in Section 6.3.1. . . . . . . . . . . . 131Table 7.1 Parameters of Synchronverter in Fig. 7.2 . . . . . . . . . . . . . . . . . . . . . 139Table B.1 Parameters of synchronous generator SG1 and SG2 in Fig. 3.8. . . . . . . . . . 175Table B.2 Parameters of modified Woodward governors used in SG1 and SG2 in Fig. 3.8. 175Table B.3 Parameters of excitation system used in SG1 and SG2 in Fig. 3.8. . . . . . . . 175Table B.4 Parameters of the synchronverter in Fig. 3.8. . . . . . . . . . . . . . . . . . . . 176Table B.5 Parameters of transformers T1, T2 and T3 in Fig. 3.8. . . . . . . . . . . . . . 176Table B.6 Parameters of lines Line45, Line46 and Line56 in Fig. 3.8. . . . . . . . . . . . 176Table B.7 Parameters of constant-impedance loads Load1, Load2 and Load3 in Fig. 3.8. 176Table C.1 Parameters of the synchronverter in cases (i)–(iii) during self synchronization. 178xiiiList of FiguresFigure 1.1 Illustrative diagram of self synchronization. For t < ts, the breaker is open,and the VSG automatically synchronizes its inner voltage eg(t) to the gridvoltage u∞(t) = ut(t) without PLLs. For t ≥ ts, eg(t) and ut(t) are syn-chronized, and the breaker may close without causing large start-up currentsafter grid connection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Figure 1.2 Coupling between VSG active- and reactive-power loops during normal op-eration, e.g., varying RPL regulation signal Q?t causes transient variations inAPL output Pt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Figure 2.1 The conventional synchronverter [1]. (a) synchronverter APL. (b) synchron-verter RPL. (c) synchronverter grid interface. . . . . . . . . . . . . . . . . . . 14Figure 2.2 Phasor diagram of the synchronverter-connected system in Fig. 2.1. . . . . . 18Figure 2.3 Transient response of APL when: (a) Jg increases from 0.2814 to 150 (Dp =1407); (b) Dp decreases from 1407 to 250 (Jg = 2.814). . . . . . . . . . . . . . 20Figure 2.4 Four realizations of proposed synchronverter controller with damping correc-tion loop. (a) Realization I. (b) Realization II. (c) Realization III. (d) Real-ization IV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Figure 2.5 Proposed synchronverter with damping correction loop and its grid interface. 31Figure 2.6 Verification of the linearized small-signal state-space model. . . . . . . . . . . 34Figure 2.7 Dominant modes of improved synchronverter without the damping correc-tion loop. (a) Jg increases from 0.2814 to 150 (Df = 0, τf = 0.01, Pt =0.6 MW, Qt = 0 MVar). (b) Dp decreases from 1407 to 250 (Df = 0, τf =0.01, Pt = 0.6 MW, Qt = 0 MVar). . . . . . . . . . . . . . . . . . . . . . . . 35xivFigure 2.8 Dominant modes of improved synchronverter with the damping correctionloop. (a)Df decreases from 0 to−3 (τf = 0.01, Pt = 0.6 MW, Qt = 0 MVar).(b) τf increases from 0.007 to 0.05 (Df = −2.76, Pt = 0.6 MW, Qt =0 MVar). (c) Df decreases from 0 to −4.1 when τf respectively takes 0.01,0.001, and 0.0001 (Pt = 0.6 MW, Qt = 0 MVar). (d) Df decreases from0.2 to −4.1 when Jg respectively takes 0.02, 2.814, 100 (τf = 0.01, Pt =0.6 MW, Qt = 0 MVar). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Figure 2.9 Dominant modes of improved synchronverter when validating the robustnessof the proposed design. (a) Pt increases from 0 to 1 MW (Df = −2.76, τf =0.01, Qt = 0 MVar). (b) Qg increases from 0 to 0.8 MVar (Df = −2.76, τf =0.01, Pt = 0 MW). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Figure 2.10 Transient response of APL when: (a) Df decreases from 0 to −2.76 (Jg =2.814). (b) Jg increases from 2.814 to 200 (ζ2 ≈ 0.8). . . . . . . . . . . . . . . 40Figure 2.11 Comparison between damping correction loop (method A) and transientdroop function (method B). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Figure 2.12 Impact of the damping correction loop on dc-side current idc and ac-sidepower Pt when the APL response is fast (Jg = 2.814, Df = −2.76) andslow (Jg = 200, Df = 0.42). We can find that the damping correction loopinfluences idc and Pt nearly in the same way. . . . . . . . . . . . . . . . . . . 41Figure 2.13 Self-synchronization process of improved synchronverter with the dampingcorrection loop when f∞ = 60.1 Hz. . . . . . . . . . . . . . . . . . . . . . . . 42Figure 3.1 Synchronverter augmented with damping correction loop. The synchron-verter parameters are usually tuned via an iterative method, which requiresrepeated computation of system eigenvalues and onerous trial-and-error ef-fort. The parameter tuning method proposed in this chapter avoids theseshortcomings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47xvFigure 3.2 Parameter Df influences APL dominant mode (represented by eigenvalues λ2and λ3) differently depending on the operating condition. This results inmore trial-and-error effort to tune synchronverter parameters via the itera-tive method based on the linearized state-space model in (2.66) and (2.67).(a) With Dp = 1407, Df increases from −6.5 to 1. (b) With Dp = 0, Dfincreases from −1.2 to 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 3.3 Different root loci patterns of 1 +KG(s) = 0 in the s-plane with (a) γ > 1,(b) γ = 1, and (c) 0 < γ < 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Figure 3.4 Image of function g(x) = x3 + bx2 + d. . . . . . . . . . . . . . . . . . . . . . . 55Figure 3.5 (a) Desired APL pole locations in the s-plane used to directly compute pa-rameters Jg and Df . (b) Desired APL time-domain response correspondingto the desired APL dominant poles λ?2 and λ?3. . . . . . . . . . . . . . . . . . 57Figure 3.6 Verification of the reduced third-order APL model and the proposed γ-criterion in the s-plane, i.e., varying Df influences the APL damping ra-tio differently when γ takes values in different ranges. Particularly, onlywhen γ ≥ 1 can the APL damping ratio be adjusted freely in the range (0, 1).(a) γ = 3.58, −6.2 ≤ Df ≤ 0.7 (Jg = 2.814, Dp = 1407.0, τf = 0.01).(b) γ = 1.00, −1.6 ≤ Df ≤ 0.5 (Jg = 2.814, Dp = 190.25, τf = 0.01).(c) γ = 0.60, −0.9 ≤ Df ≤ 0.7 (Jg = 2.814, Dp = 0, τf = 0.01). . . . . . . . . 62Figure 3.7 Verification of the proposed direct computation method achieving (a) accu-rate APL pole placement in the s-plane. (b) desired APL dynamic responsein the time domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Figure 3.8 6-bus test system used to verify the proposed parameter tuning method. . . . 65Figure 3.9 Modified Woodward governor used in SG1 and SG2 to achieve primary fre-quency regulation [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Figure 3.10 Excitation system used in SG1 and SG2 to control their terminal voltages [3]. 66xviFigure 3.11 Obtaining the infinite-bus equivalent for the external system in the six-bustest system used to verify the proposed parameter tuning method. (a) Con-verting all component impedance values of the 6-bus test system in Fig. 3.8with respect to the 6.6 kV voltage level. (b) Eliminating nodes 4-6 from thetest system via kron reduction. (c) Merging nodes 2 and 3 and obtaining theinfinite-bus equivalent for the external system. . . . . . . . . . . . . . . . . . 68Figure 3.12 Verification of the proposed direct computation method to achieve desiredfast dynamic response in actual grid conditions (ω?n1 = 30 rad/s and ζ?1 =0.707). (a)–(e) Time-domain response comparison between the actual andequivalent systems when Hsg1 = Hsg2 = 8.0 s. (f)–(j) Time-domain re-sponse comparison between the actual and equivalent systems when Hsg1 =Hsg2 = 3.0 s. (a)(b)(f)(g) synchronverter active-power output Pt in the ac-tual and equivalent systems. (c)(h) active power outputs of SG1 and SG2 Psg1and Psg2 in the actual system. (d)(i) frequency of SG1 and SG2 fsg1 and fsg2in the actual system. (e)(j) input torque of SG1 and SG2 Tsg,m1 and Tsg,m2in the actual system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Figure 3.13 Verification of the proposed direct computation method to achieve desiredslow dynamic response in actual grid conditions (ω?n1 = 10 rad/s and ζ?1 =0.707). (a)–(e) Time-domain response comparison between the actual andequivalent systems when Hsg1 = Hsg2 = 8.0 s. (f)–(j) Time-domain re-sponse comparison between the actual and equivalent systems when Hsg1 =Hsg2 = 3.0 s. (a)(b)(f)(g) synchronverter active-power output Pt in the ac-tual and equivalent systems. (c)(h) active power outputs of SG1 and SG2 Psg1and Psg2 in the actual system. (d)(i) frequency of SG1 and SG2 fsg1 and fsg2in the actual system. (e)(j) input torque of SG1 and SG2 Tsg,m1 and Tsg,m2in the actual system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75xviiFigure 4.1 Synchronverter augmented with damping correction loop proposed in Chap-ter 2. My previous work in Chapter 3 proposes a method to directly computecontrol parameters and place the APL dominant poles at desired locations.However, trial-and-error work is needed when specifying the desired domi-nant poles, since it may be impossible to achieve pole placement at somelocations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Figure 4.2 Desired APL pole locations in the s-plane used to directly compute parame-ters Jg and Df . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Figure 4.3 Impacts of ω?n and ζ? on the pole-placement results and the APL dynamics.We find that for some specified ω?n and ζ?, though we place s2 and s3 atthe desired APL dominant pole locations (4.7) and (4.8), s1 may insteadbe the APL dominant pole and we may not achieve desired APL dynamicsin (4.9). (a) APL pole locations in the s-plane when ω?n takes, respectively,55 and 100 rad/s (ζ? = 0.707). (b) Actual and desired APL step responseswhen ω?n takes, respectively, 55 and 100 rad/s (ζ?=0.707). . . . . . . . . . . . 84Figure 4.4 Range of ω?n which ensures that s2 and s3 are the APL dominant poles (i.e.,s1 < Re(s2)). (a) when 0 < µ < cosϕ? (0 < Dp < N cos2 ϕ?). (b) when µ ≥cosϕ? (Dp ≥ N cos2 ϕ?). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Figure 4.5 Different patterns of feasible pole-placement region for APL dominant poles s2and s3 in the s-plane when (a) µ = 0 (Dp = 0), (b) 0 < µ < 1 (0 < Dp < N),and (c) µ ≥ 1 (Dp > N). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Figure 4.6 Verification of µ-criterion in (4.28) that specifies range of ω?n for certain ζ?(ζ? = cosϕ? = 0.707) when (a) µ = 0 (Dp = 0), (b) 0 < µ = 0.618 <ζ? (0 < Dp = 75 < N cos2 ϕ?), and (c) µ = 0.782 ≥ ζ? (Dp = 120 ≥N cos2 ϕ?). Note that in Figs. 4.6(a)–(c), I use pink shadow to mark therange of ω?n computed from (4.28), and by choosing ω?n within the markedrange, we ensure that s2 and s3 represent the APL dominant poles andachieve successful pole placement when tuning synchronverter parameters. . . 92xviiiFigure 4.7 Verification of feasible pole-placement region developed based on the µ-criterion when (a) µ = 0 (Dp = 0), (b) 0 < µ = 0.677 < 1 (0 < Dp =90 < N), and (c) µ = 1.001 > 1 (Dp = 200 > N). In this figure, I highlightthe feasible pole-placement region with green or yellow colours. By plac-ing the APL poles s2 and s3 within the feasible pole-placement region, wesatisfy (4.13) and ensure that s2 and s3 represent the APL dominant poles. . 94Figure 5.1 Proposed self-synchronizing synchronverter, which has few parameters thatrequire tuning. Highlighted in red colour are aspects of particular relevanceto the proposed design. (a) Grid interface. (b) Power computation block.(c) Active-power loop. (d) Reactive-power loop. . . . . . . . . . . . . . . . . 97Figure 5.2 Equivalent representation of proposed synchronverter design in Fig. 5.1 dur-ing self synchronization (Switch 1 in Fig. 5.1(b) is in position 2). (a) Equiv-alent grid interface corresponding to Figs. 5.1(a) and 5.1(b), in which Rvacts as virtual reactance jRv due to the algebraic coordinate transformationin (5.5). (b) Active- and reactive-power feedback signals. (c)(d) APL andRPL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Figure 5.3 Impacts ofDf on phase-angle self-synchronization dynamics. We find that: (i)increasing Df accelerates the APL response speed and enables θg∞(t) to con-verge to θ◦g∞ more quickly, but there is an upper bound to phase-angle self-synchronization speed, (ii) reduced phase-angle self-synchronization speed re-sults in slower voltage-magnitude self synchronization. (a)(b) Self-synchronizationdynamics with θg∞(0) = 3.14 rad. (c)(d) Self-synchronization dynamicswith θg∞(0) = −3.14 rad. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Figure 5.4 Impacts ofKg on voltage-magnitude self-synchronization dynamics with θg∞(0) =3.14 ∈ (0, pi) rad. We find that decreasing Kg accelerates the RPL responsespeed so that ψf (t) converges to 1 p.u. more quickly, but sufficiently small Kgresults in transient overshoots in ψf (t). . . . . . . . . . . . . . . . . . . . . . 105xixFigure 5.5 Phase portrait of the reduced-order APL model consisting of (5.12) and (5.20).Trajectories marked as (i) and (ii) approach but do not cross the boundarydelineated by ωg∞ = −ω∞ (traces marked as (iii)), so dθg∞dt > −ω∞ duringphase-angle self synchronization. (a) Df = 4.38. (b) Df = 20.0. . . . . . . . . 107Figure 5.6 Root loci patterns of the linearized discrete-time APL model in the complexplane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112Figure 5.7 Self-synchronization simulation results using the proposed controller design.These verify my analyses of self-synchronization dynamics, which leveragetwo suitable reduced-order models to study phase-angle and voltage-magnitudedynamics independently, as detailed in Section 5.2. . . . . . . . . . . . . . . . 114Figure 5.8 Self-synchronization simulation results of the proposed controller design. As-suming that the processor sampling time is 50 µs, Df is computed with η = 0.6.116Figure 5.9 Grid interface of the LCL-filter-based synchronverter. . . . . . . . . . . . . . 116Figure 5.10 Self-synchronization simulation results of the proposed controller design withan LCL filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117Figure 5.11 Verification of the reduced second-order APL model (Model A) via compar-isons with the full-order self-synchronizing synchronverter model in (5.12)–(5.17) (Model B) and Model B with (5.13) replaced by (D.2) (Model C).(a) θg∞(t) dynamics. (b) Phase portraits (ωg∞-θg∞ plots). . . . . . . . . . . 118Figure 5.12 Schematic diagram of self-synchronizing synchronverter experimental setup. . 119Figure 5.13 Experimental results of the self-synchronization dynamics using the proposedcontroller design when (a)–(d) θg∞(0) = −3.14 rad, (e)–(h) θg∞(0) = 0,and (i)–(l) θg∞(0) = 3.14 rad/s. . . . . . . . . . . . . . . . . . . . . . . . . . . 120xxFigure 6.1 Proposed synchronverter design that combines the damping correction loopand the transient droop function. Specifically, by setting Dm = 0, this figurerepresents the synchronverter with the damping correction loop only, and bysetting Df = 0, this figure represents the synchronverter with the transientdroop function only). This controller design is able to reduce the coupling be-tween active- and reactive-power loops regardless of the tuned synchronverterresponse speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Figure 6.2 Block diagram for small-signal model of the APL (omitting LPF1 and LPF2).The block marked in red is associated with the damping correction loop, andthose in blue are related to the transient droop function. . . . . . . . . . . . . 125Figure 6.3 Block diagram for small-signal model of the APL (including LPF1 and LPF2marked in purple colour). Blocks marked in red are associated with thedamping correction loop, and the one in blue is related to the transientdroop function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130Figure 6.4 Comparison of dynamic response of proposed synchronverter design (method A)with synchronverter augmented with only the damping correction loop (method B)and one with only the transient droop function (method C). Indeed, method Ahas the least coupling with (a) fast and (b) slow APL response speed. . . . . 133Figure 6.5 Impact of β on active- and reactive-power coupling. By tuning parameterssuch that β < 0, active- and reactive-power coupling is reduced when com-pared with setting β = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133Figure 6.6 Active- and reactive-power coupling in synchronverters augmented with ei-ther damping correction loop (method B) or transient droop function (method C)are nearly identical with ω?n =Bζ?ADp. Indeed, the relative values of ω?n andBζ?ADpdetermine whether the damping correction loop or the transient droop func-tion results in lower coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . 134Figure 6.7 Steady-state frequency-droop characteristics are maintained under both (i) fastand (ii) slow APL response speeds. . . . . . . . . . . . . . . . . . . . . . . . . 134xxiFigure 7.1 Synchronverter augmented with damping correction loop. (a) Active-powerloop. (b) Reactive-power loop. . . . . . . . . . . . . . . . . . . . . . . . . . . 137Figure 7.2 Microgrid test system used to demonstrate the active-power transfer capacityof the synchronverter under weak-grid conditions. . . . . . . . . . . . . . . . . 139Figure 7.3 Without voltage-droop controller: system dynamics caused by increasing thesynchronverter active-power reference P ?t from 0 to 0.5 (case I) and 0.7 MW(case II), respectively. (a)(b) Case I. (c)(d) Case II. . . . . . . . . . . . . . . 140Figure 7.4 Equivalent circuit of the system in Fig. 7.2 as seen from the synchronverter. . 141Figure 7.5 Synchronverter active-power transfer capacity P t(Qt) for rated capacity SN(grid impedance Xt) in the ranges of (a)(d) 0 < SN ≤ U2∞4Xt(0 < Xt ≤ U2∞4SN),(b)(e) U2∞4Xt< SN ≤ U2∞2Xt( U2∞4SN< Xt ≤ U2∞2SN), and (c)(f) SN >U2∞2Xt(Xt >U2∞2SN). 144Figure 7.6 Synchronverter PV curves parameterized by α with Qt = αPt. . . . . . . . . 146Figure 7.7 With voltage-droop controller: system dynamics resulting from increasingthe synchronverter active-power reference P ?t from 0 to 0.7 MW. . . . . . . . 149Figure 7.8 With reactive-power compensation devices: system dynamics resulting fromincreasing the synchronverter active-power reference P ?t from 0 to 0.7 MW. . 150Figure C.1 Case (i)—synchronverter equipped with only virtual resistance R˜v ((a1)–(d1)) and its equivalent representation during self synchronization ((a2)–(d2)).179Figure C.2 Time-domain simulation of self synchronization with only virtual resistance R˜v.180Figure C.3 Eigenvalues of the synchronverter with only virtual resistance R˜v > 0 incase (i). When changing R˜v from 1 to 799 Ω, we find that one pair of eigen-values, i.e., λ1 5 and λ1 6, are always on the right half-plane in the complexplane. This explains the unsuccessful self-synchronization simulation resultsin case (i), as observed in Fig. C.2. . . . . . . . . . . . . . . . . . . . . . . . . 182Figure C.4 Case (ii)—synchronverter equipped with only virtual inductance L˜v ((a1)–(d1)) and its equivalent representation during self synchronization ((a2)–(d2)).183Figure C.5 Time-domain simulation of self synchronization with only virtual inductance L˜v.184xxiiFigure C.6 Phasor diagram of proposed synchronverter with only virtual inductance L˜vin Fig. C.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184Figure C.7 Eigenvalues of the synchronverter with only virtual inductance L˜v > 0 incase (ii). When changing L˜v from 1 to 100 mH, we find that one pair ofeigenvalues, i.e., λ2 7 and λ2 8, are always on the right half-plane in the com-plex plane. This explains the unsuccessful self-synchronization simulationresults in case (ii), as observed in Fig. C.5. . . . . . . . . . . . . . . . . . . . 188Figure C.8 Case (iii)—synchronverter with virtual resistance R˜v together with a coordi-nate transformation ((a1)–(d1)) and its equivalent representation during selfsynchronization ((a2)–(d2)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190Figure C.9 Self-synchronization process with virtual resistance Rv together with the co-ordinate transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191Figure E.1 Lyapunov function V (θg∞, ωg∞) of the second-order APL model in (5.12)and (5.20). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195xxiiiGlossaryAPL active-power loopDFIG doubly fed induction generatorFACTS flexible AC transmission systemHVDC high-voltage direct-currentIGBT insulated-gate bipolar transistorLPF low-pass filterLVRT low-voltage ride-throughMIMO multi-input multi-outputPCC point of common couplingPLL phase-locked loopPMSG permanent magnet synchronous generatorPV photovoltaicRES renewable energy resoureRMS Root-Mean-SquaredRoCoF rate of change of frequencyRPL reactive-power resoureSG synchronous generatorxxivVSC voltage source converterVSG virtual synchronous generatorxxvAcknowledgmentsFirst, I would like to express my most sincere thanks to my PhD supervisor Prof. Yu ChristineChen. Without Prof. Chen’s offer four years ago, I may not have the opportunity to studyin this beautiful campus, enjoy everyday’s research work, and have so many good memories.During my PhD journey, Prof. Chen always encouraged me whenever I got stuck in the research,and always trusted me whenever I tried different ideas. This dissertation cannot be completedwithout Prof. Chen’s continuous support, patient guidance, and valuable comments. I believewhat I have learned from Prof. Chen, whether those on academic research or on real life, wouldbe a treasure of my lifetime.Many thanks also go to Prof. Ryozo Nagamune and Prof. Juri Jatskevich. Prof. Nagamunepatiently taught me many interesting proofs in control engineering with his magical propellingpencil. Prof. Jatskevich is always full of wisdom, and I learned quite a lot whenever askinghim for suggestions. Also, I would like to thank my other dissertation committee members anduniversity examiners, Prof. Jose´ Mart´ı, Prof. Hermann Dommel, and Prof. Shahriar Mirabbasi,for their valuable suggestions that helped me improve my dissertation.Also, I wish to thank my Master supervisor Prof. Yongning Chi from China Electric PowerResearch Institute (CEPRI) for his trust and help. Five years have passed, I still rememberthe day when he called me to his office and gave me his credit card to support my PhDapplication. Moreover, I would like to thank Prof. Jiabing Hu from Huazhong University ofScience and Technology and Prof. Hua Geng from Tsinghua University for their encouragement,and Prof. Xiaoxin Zhou from CEPRI and Profs. Chengshan Wang and Yuan Zeng from TianjinUniversity for their recommendations when I applied for Master and PhD positions before.I really appreciate the caring and support of my friends here. When I first came to Van-couver, Mrs. Hui Jia, Mr. Lei Zhang and Duoduo picked me up from the airport in a rainyxxviday, and after that, we lived in Acadia Park for half a year. For a time I wanted to solve theproblem in Chapter 2, but returned home exhausted with little progress every mid-night. Theykept the light in the living room on for me everyday, and I still remember the warm feelingswhen seeing the light from outside.I would also like to thank my friends and colleagues in UBC Electric Power and EnergySystems Group who helped me a lot and made the life here so memorable (in alphabeticalorder): Abdullah Al-Digs, Arash Tavighi, Bo Chen, Hua Chang, Javier Tarazona, Jiayue Xu,Rubinder Nagi, Wonbae Choi, Xiaotong Wang, Xing Liang, Yajian Tong, Yingwei Huang,Zemeng Wang, Zhi Qu, Zhibang Liang, and those who are not listed here. Also, I would liketo thank my friends Tiancong Shao and Jingya Jiang from Beijing Jiaotong University for allthe helpful discussions on daily life and scientific research.Table tennis always cheered me up whenever I got stuck in the research, and I really appre-ciate the days of playing table tennis with my friends (in alphabetical order): Alison, Becky,Chendi Wang, Dima, Dian Sheng, Guanpeng Li, Ian, Jixing Li, Jiaxi Gao, Jing, Junyuan Zheng,Ke Dai, Liang Zou, Lidong Wang, Marco Wong, Mary Shen, Mr. Ding, Mr. Shao, Peter Liu,Phil, Prof. Xiaoliang Jin, Prof. Yu, Yanan Sun, Yang Li, Zemeng Wang, Ziang Yan, Ziyang Jin,and those who are not in the list. I wish you all the best in the future.Moreover, I would like to thank my parents for their unconditional love and support thathave always been with me. When I was a kid, I often sat in our yard and watched how my fatherrepaired his old bike, and sometime my mother patiently showed me how to solve elementaryequations. All these ignited my interests to learn more about science and engineering later, andin my mind, they provided me the best family education.Especially, I would like to thank my wife Yu Jing for her continuous support for my PhDresearch work. Hope that our life will get better and better in the future.xxviiDedicationTo my familyxxviiiChapter 1Introduction1.1 BackgroundAs a result of skyrocketing prices, finite reserves, and environmental concerns of fossil fu-els, fossil fuel-based generators are expected to be gradually displaced by renewable energysources (RESs), such as wind and solar. However, due to the way in which RESs are integratedinto the grid, high RES penetration level poses numerous technical challenges to existing gridoperations. First, since the electricity generated by most RESs is either with variable ampli-tude/frequency or in dc, which is not directly compatible with the synchronous ac grid, mostRESs are coupled with the grid through power-electronic converters, e.g., voltage source con-verter (VSC) [4]. Since power-electronic converters fully or partially decouple the generatorfrom the grid, the large-scale displacement of thermal or hydro power reduces the total systeminertia1 [7]. Subsequently, the system would have larger frequency deviation and rate of changeof frequency (RoCoF) when generation-load imbalances or faults occur [8]. Moreover, sincemany RESs are distributed in remote areas with weak grids, the system is prone to instabilityif the power-electronic interfaces adopt conventional control methods, such as vector currentcontrol [9] and direct power control [10], which depend on phase-locked loops (PLLs) [11, 12].Additionally, the intermittent, variable, and uncertain nature of RESs leads to severe voltage1In physics, inertia usually refers to the resistance of any physical object to any change in its velocity, andthus, it is called vis inertiae or force of inactivity in Newton’s Principia [5]. Specifically, in our power engineeringfield, the system inertia is proportional to the instantaneous power reserve that prevents any significant gridfrequency change and stabilizes our power system following disturbances, e.g., the kinetic energy in the rotors ofthe synchronous generators and motors [6].1and frequency fluctuations and thus presents notable challenges to the system operation [8].Finally, depending on the controllers of their power-electronic interfaces, the renewable energygenerator units may exhibit entirely different dynamic characteristics from the conventionalsynchronous generators (SGs) used in the thermal and hydro power plants. This also bringsnew challenges to existing power system operation and control.To tackle these problems, the virtual synchronous generator (VSG) concept [1, 4, 13–56]provides a method for mimicking the SG and providing the effect of inertia support usingpower-electronic devices. First, by contributing inertia to the grid, the VSG helps to reducefrequency excursions when loads vary or faults occur. Second, the operation of VSGs does notrely on the phase angle of the local grid voltage, which is measured from PLLs that are foundin conventional VSC control methods. In this way, it avoids the PLL-related instability whenintegrating RES into a weak grid. Moreover, via its active- and reactive-power control loops,the VSG realizes frequency and voltage droop controls in order to improve rotor angle stabilityand system voltage stability by using only local information [57]. Finally, since the VSG canemulate the SG dynamic behaviour, it significantly facilitates the power system operation.Note that the VSG is not limited by the physical design of an actual SG [14], and its embeddedparameters can be tuned freely and can even be changed online [30].There are many different realizations of the VSG, e.g., power synchronization control [13],synchronverter [1, 17, 22, 41, 47, 51, 52, 55], synchronous generator emulation control [21], andsynchronous power controller [15]. My research in this dissertation mainly centres around thesynchronverter control scheme, since it not only inherits all advantages of the VSG, but alsohas two additional merits as follows. The synchronverter has a simpler controller structurecompared with other VSG realizations. For example, unlike some VSG designs requiring PLLsto measure local grid frequencies, the synchronverter reuses its own rotor speed as the inputsignal of its governor part and thus completely removes the PLL during its normal operation.Also, in contrast to some VSG designs that are controlled as a current source and only deliverpower to an energized grid, the synchronverter is able to support the grid voltage and operatein an island grid.Based on the synchronverter design in [1], extensive research has been conducted to furtherimprove the synchronverter performance and make it suitable for RES integration. For exam-2ple, to share inertia between different non-synchronous ac grids and facilitate RES integration,[22] and [41] propose synchronverter-based high-voltage direct-current (HVDC) transmissionsystem. To improve the synchronverter low-voltage ride-through (LVRT) capability, [51] de-signs a mode switching controller, which protects synchronverter from large transient inrushfault current. To reduce the complexity and computation burden of the controller, [55] proposesself-synchronized synchronverters, which can synchronize to the grid without dedicated synchro-nization units like PLLs. To further improve the synchronverter stability and performance, [47]proposes five modifications including adopting virtual impedance method in the synchronverteralgorithm. To analyze the stability of synchronverter-dominated power system, [52] presents abifurcation theory-based analysis method, which accurately analyzes and predicts the impactof control parameters on the system stability.1.2 MotivationAlthough a lot of research has been done on the synchronverter, this dissertation finds that sev-eral important aspects of the synchronverter during its normal operation and start-up phase,from the controller design to the tuning method to the operation characteristics, still havenot been thoroughly investigated, and this handicaps the RES integration via synchronverter2.First, conventional synchronverter designs do not have freely adjustable response speed duringnormal operation, and also, their tuning processes require a lot trial-and-error efforts. Next,conventional synchronverter designs synchronize with the grid slowly before physical connectionto the grid, and we need a fast self-synchronizing synchronverter design to enable the “plug-and-play” integration of RESs. Then, in terms of output characteristics, conventional synchronverterdesigns are multi-input multi-output (MIMO) system with severe output-power coupling duringnormal operation, which significantly affects the power quality. Last, the active-power transfercapability of the synchronverter during normal operation has not been studied, especially un-der weak grid conditions. This hinders the development of synchronverter-based RESs, mostof which are located in remote areas and connected to weak grids. Below, I introduce theshortcomings in previous researches on synchronverter in detail.2It is noted that these research gaps also exist for most other VSG designs, and thus replacing synchronverterwith a different VSG design will not address all these challenges.31.2.1 Lacking in Control Degrees of FreedomAlthough the synchronverter is suitable for integrating renewable energy into the grid, one of itsmajor shortcomings is that during normal operation, the dynamic response speed of its active-power loop (APL) cannot be adjusted freely without affecting the steady-state frequency droopcharacteristic, i.e., the frequency droop coefficient must be modified [22]. Such a modificationis undesirable, since the droop coefficient value is fixed by local grid standards [14]. Thus,conventional synchronverter designs lacks in the control degrees of freedom, which could beindicated by the number of the relevant tuneable control parameters [58]. If the synchronverteris used in renewable energy unit and its APL response speed during normal operation cannot betuned freely, this would limit the ability of the synchronverter to achieve fast maximum power-point tracking and to provide timely frequency regulation. This shortcoming is also evident inother VSG control schemes [14, 18].Owing to the importance of unrestricted adjustment of the APL dynamic response speedduring normal operation, numerous approaches have been proposed to solve this problem inthe literature. In studies on frequency droop control, several methods have been proposed, see,e.g., [27, 42, 59]. If used directly in synchronverters, however, the transient droop function [42]and the adaptive transient droop function [59] methods induce the undesirable side effect of cou-pling between the APL and the reactive-power loop (RPL), and the parameters in the modifieddroop control law method [27] cannot be analytically tuned to adjust the APL response speed.In studies on VSGs, the alternating inertia control [30] and the virtual impedance method [45]aim to suppress frequency and power oscillations, and as a bonus, also modify the APL responsespeed. However, the controller in [30] cannot be analytically tuned, and, by adding a virtualimpedance, [45] can only reduce the APL response speed. On the other hand, the distributedfrequency control [16], the differential algorithm [54], and the configurable natural droop con-troller [15] directly aim to modify the response speed of the APL. However, [16] requires gridfrequency measurements from a PLL, which deteriorates this method’s effectiveness, especiallyin weak grid conditions. The differential algorithm in [54] increases the APL response speed byadding a differential term, which may amplify noise in measured signals and result in numericalinstability. Finally, the method proposed in [15] induces a zero in the transfer function from4the active- power reference value to the active-power output, which may be positive when theAPL responds quickly, and thus may cause undesirable non-minimum phase behaviour.1.2.2 Inefficient Tuning MethodAnother important problem on the synchronverter is the need for an accurate and efficientmethod to tune relevant controller parameters for normal operations. Relevant VSG parametertuning methods can be roughly categorized into offline approaches, which tune parametersprior to online operation, and online ones, in which parameters are adjusted in real time duringonline operation. Among offline approaches [1, 14–20, 22], parameters are tuned based on eitherempirical formulas ([1]), or small-signal analysis with the linearized system transfer function([14–16]) or state-space model ([17–20, 22]). Empirical formulas used in [1] do not preciselyplace the APL dominant poles, so they are unable to tune parameters to achieve the exactdesired time-domain transient behaviour. The methods in [14] and [15] (which use transfer-function-based models) neglect the effects of low-pass filters (LPFs) that smooth out measuredsignals, which may cause inaccurate parameters. Though the method in [16], which is alsobased on transfer-function analysis, accounts for the LPFs, potentially tedious trial-and-errorwork is needed during the tuning process. This is also true for the methods in [17–19], whichtune parameters based on eigenvalues of the linearized system state-space model. The methodin [20] tunes parameters using eigenvalue parametric sensitivities rather than through trialand error, but it aims only to stabilize the system instead of achieving exact pole placement.While the method in [22] places poles at prescribed locations, the solution to a computationallyburdensome nonlinear optimization problem is required to obtain parameter values. Amongonline approaches, [23] tunes parameters using the linearized system transfer function withoutconsidering LPFs, and thus may result in inaccurate parameters, as in [14] and [15]; [24] runsa self-tuning algorithm and searches for parameter values for the VSG during online operation,which adds nontrivial computational burden.Besides the tuning methods, our incomprehension of the feasible pole-placement region 33Here, feasible pole-placement region represents the region in the s-plane where we can place the synchron-verter APL poles freely and ensure that they represent the APL dominant mode. Since the system dynamicsare mainly determined by its dominant poles, we cannot achieve desirable APL dynamics after tuning process ifthe placed poles are not the dominant ones. Indeed, the feasible pole-placement region represents the set of theachievable system dynamics via the parameter tuning.5Figure 1.1: Illustrative diagram of self synchronization. For t < ts, the breaker is open, andthe VSG automatically synchronizes its inner voltage eg(t) to the grid voltage u∞(t) = ut(t)without PLLs. For t ≥ ts, eg(t) and ut(t) are synchronized, and the breaker may close withoutcausing large start-up currents after grid connection.also complicates other VSG tuning methods. Without this feasible pole-placement region, wemight have to repeatedly choose the desirable synchronverter poles in the s-plane, computethe synchronverter parameters based on these poles, and check the system poles until that thechosen desirable poles are the dominant ones. For example, when tuning VSG with methodsin [19] and [60], we still need to repeatedly specify the natural frequency and the damping ratiofor desired dominant poles and tune the VSG parameters with small-signal analysis beforeachieving successful pole-placement. If knowing the feasible pole-placement ahead, we canavoid this trial-and-error process and significantly simplify the VSG tuning process. In viewof this problem, there are two mitigation measures in previous studies [14, 20, 22, 61]. Onetechnical route is solving the VSG parameters from an optimization problem that minimizes thedistance between desired and actual poles in the s-plane [22]. However, this method might notensure that the objective function can be minimized to an acceptable value such that we achievethe desired dynamic response. The other technical route is to delimit a range for the desireddominant poles instead of setting their desired values preciously [14, 20, 61]. For example,[20] moves all the system poles to the left half s-plane to stabilize the whole system, and [61]further sets a range for the system damping ratio to improve system dynamic performance(equivalently, [14] tunes the VSG parameters to satisfy a phase margin requirement). Thesemethods save some trial-and-error efforts, but with the cost that we might not achieve poleplacement and attain desirable system dynamics with full accuracy and precision.61.2.3 Slow Self-synchronization SpeedBefore being physically connected to the grid, I find that most VSG designs, including synchron-verter, synchronize their inner voltage to the grid voltage slowly, and this affects the flexible“plug-and-play” operation of VSG-based RESs. Here, we first note that a highly desirableproperty for the VSG to have is the so-called self-synchronization capability, which refers tothe ability for the VSG to automatically synchronize its inner voltage to the grid-side voltagewithout PLLs before closing the breaker, as shown in Fig. 1.1 [55]. This capability helps toavoid potentially large start-up currents when physically connecting the VSG to the grid, andin turn, protects the power-electronic devices in the VSG. Moreover, if the VSG is not endowedwith the self-synchronization capability, then a PLL is required to measure the grid-side volt-age phase angle in order to synchronize the VSG voltage to that of the grid before physicalconnection [13, 19]. The self-synchronizing VSG completely obviates the need for a PLL inthe controller and significantly simplifies the overall design. Specifically, compelling argumentsfor bypassing the PLL include past observations that PLL-based designs are associated withgreater computational burden [55, 62, 63], unnecessary controller redundancy [64, 65], and morecomplex parameter tuning [55, 66, 67].Due to the importance of achieving self synchronization, numerous self-synchronizing VSGdesigns have been proposed in the literature [17, 55, 68, 69]. However, I find that their pa-rameters are not easily tuneable to achieve fast self-synchronization speed. This is mainlybecause conventional self-synchronization designs typically adopt a virtual impedance, which isused to generate active- and reactive-power feedback signals during self synchronization. How-ever, the virtual impedance requires simultaneous tuning of two parameters, i.e., resistance andreactance, which may be difficult. For example, [68] requires tuning of two sets of virtual-impedance values, a large one during self synchronization to reduce start-up currents and asmall one for normal operation afterwards. Furthermore, in the method proposed in [70], thevirtual-impedance value requires the solution of a system of inequality constraints constructedfrom a set of performance requirements. The designs in [17, 55, 69] use the virtual impedanceduring self synchronization and then bypass it immediately after grid connection. Improperlytuned impedance parameters may lead to slow or even unsuccessful self synchronization [71].7Figure 1.2: Coupling between VSG active- and reactive-power loops during normal operation,e.g., varying RPL regulation signal Q?t causes transient variations in APL output Pt.1.2.4 Undesired Output-power CouplingOne shortcoming in synchronverter as well as other VSG designs is that its APL and RPL are notcompletely decoupled during normal operation. For example, as depicted in Fig. 1.2, adjustingthe RPL regulation signal Q?t may not only affect the reactive-power output Qt (as expected),but also cause transient variations in the VSG active-power output Pt, which is undesirable.Ideally, Pt would track the APL input signal P?t in Fig. 1.2. Transient active-power variationsmay then cause unwanted grid-voltage frequency deviations, consequently affecting the powerquality. The output-power coupling in Fig. 1.2 may result from high line-resistance-to-reactanceratio or large phase-angle difference between the converter output voltage and the grid-sidevoltage. In order to address the output-power coupling resulting from large line-resistance-to-reactance ratio, the coordinate transformation method [66, 72], the virtual negative resistormethod [73], and the virtual impedance method [74] have been proposed to reshape the gridimpedance and reduce the associated output-power coupling. In order to reduce the APL-RPL coupling caused by large phase-angle difference, the cross feedforward compensation [75],the linear control theory-based approach [23], and the current compensation method [76] havebeen proposed. However, these either cannot freely adjust the VSG response speed, or theysignificantly complicate the controller structure.1.2.5 Unrevealed Active-power Transfer CapacityAn aspect of the synchronverter (or other VSG designs) that has not been thoroughly investi-gated in the literature is the active-power transfer capacity of synchronverter-controlled RESsoperating under weak-grid conditions. If the actual active power delivered from the RES to theload centre exceeds the transfer capacity, the system would not converge to a viable power-flowsolution, and this leads to so-called voltage instability [77]. Conventionally, voltage stability is8often compromised by heavy loading conditions, and the impacts of grid voltage variations onthe voltage stability has been studied in [78]. In my setting, however, loss of stability stems fromgreater synchronverter active-power output than that can be delivered to the rest of the sys-tem. Till now, research efforts are lacking on the analysis of the synchronverter active-transfercapacity as well as practical countermeasures to improve it.1.3 Research ObjectivesAs identified in Section 1.2, conventional synchronverter designs lack in control degrees offreedom, require trial-and-error tuning process, synchronize with the grid slowly, suffer fromoutput-power coupling, and have unrevealed active-power transfer capacity during its normaloperation or start-up process. Thus, this dissertation proposes five research objectives, targetingat addressing the shortcomings of previous research on synchronverter and facilitating the RESintegration via synchronverter. The five research objectives of my dissertation are listed asfollows.1.3.1 Objective I: Increasing Synchronverter Control Degrees of FreedomAccording to Section 1.2.1, conventional synchronverter design lacks in control degrees of free-dom and cannot adjust its APL dynamic response speed freely without affecting its steady-statefrequency-droop characteristics during normal operation. In order to address this problem, myobjective I aims to propose a synchronverter design which is able to adjust its APL responsespeed freely without affecting the steady-state performance during normal operation. Also, Iwill identify the root cause of the conventional synchronverters’ shortcoming in Section 1.2.1.Finally, I plan to verify the effectiveness of the proposed synchronverter design with small-signalanalysis and time-domain simulations.1.3.2 Objective II: Proposing Efficient Synchronverter Tuning MethodAccording to 1.2.2, we are in urgent need of an accurate and efficient synchronverter parametertuning method for its normal operation, which avoids the trial-and-error process. In order toachieve this goal, my object II consists of two sub-objectives, developing pole-placement methodand deriving feasible pole-placement region.9Developing Pole-placement MethodIn this sub-objective, I plan to develop a pole-placement method to achieve desired synchron-verter dynamic response during normal operation via the following steps. First, I plan to developa reduced-order synchronverter APL model, which captures the synchronverter dynamics withhigh accuracy while being analytically tractable. Next, based on the APL model, I formulate aset of equations that connect the desired APL dominant poles and the parameters that requiretuning. Then, I try to solve the parameters from these equations and achieve pole placement.Finally, I verify the obtained parameters via both small-signal analysis and extensive simu-lation studies. We note that the proposed pole-placement method will completely avoid thetrial-and-error process when placing the APL poles.Deriving Feasible Pole-placement RegionIn this sub-objective, I plan to analytically derive or numerically compute the feasible pole-placement region during normal operation for the synchronverter design. Within this region,we are able to place the APL dominant poles freely and achieve desired dynamics during normaloperation. I will verify the derived region via small-signal analysis and numerical studies. Itis worth noting that the feasible pole-placement region completely avoids the of repeatedlyspecifying the APL poles, compute control parameters and checking whether the placed APLdominant poles represent the APL dominant mode.1.3.3 Objective III: Achieving Fast Self-synchronizationConsidering the shortcoming in Section 1.2.3, my objective III will study how to equip oursynchronverter with self-synchronization capability during its start-up process, and at the sametime, ensure that the controller achieve fast self-synchronization speed and has easily tuneableparameters. To reach this goal, I plan to redesign the self-synchronization controller such that ithas less parameters that require tuning. With the proposed controller, I will further analyze thefull-order nonlinear system model. In so doing, I hope to estimate the synchronization time,prove the self-synchronization capability, and recommend parameter setting to facilitate theparameter tuning of the self-synchronization design. Finally, I will verify the proposed designas well as my analysis via extensive simulation studies or experimental results.101.3.4 Objective IV: Reducing Synchronverter Output-power CouplingTargeting at addressing the shortcoming in Section 1.2.4, I plan to add an auxiliary powerdecoupling loop in my proposed synchronverter design in Section 1.3.1. In this way, I decouplethe active- and reactive-power control and improve the controller tracking performance duringnormal operation. I will provide analytical justification for the proposed design via transfer-function analysis. Moreover, I will seek for closed-form expressions or numerical computationmethods for the control parameters of the proposed controller scheme, and thus simplify theparameter tuning process. Finally, I will validate my design and analysis via extensive time-domain simulations.1.3.5 Objective V: Improving Synchronverter Active-power TransferCapacityAs mentioned in Section 1.2.5, the active-power transfer capacity of synchronverter-controlledRESs during normal operation has not been studied so far, especially under weak-grid con-ditions. Also, voltage stability acts as one limitation for the active-power transfer capacity.Bearing these in mind, my objective V plans to study the active-power transfer capacity of thesynchronverter during normal operation under weak grid conditions and propose countermea-sures to improve its transfer capacity. First, I will build a microgrid test system which connectsa synchronverter via a long transmission line. With this system, I will identify the active-powertransfer capacity of synchronverter during normal operation through both numerical studiesand voltage stability analysis. Also, I will explore the connections between the voltage stabilityin this setting and that in conventional power system. Then based on these analysis, I willprovide countermeasures to improve synchronverter active-power transfer capability in weakgrid.1.4 Dissertation OutlineThe remainder of this dissertation is organized as follows. First, in Chapter 2, I propose asynchronverter design with freely adjustable response speed during normal operation. Then,Chapters 3 and 4 jointly propose a tuning method which directly computes the synchronverterparameters to achieve desired dynamics and also, provide the feasible pole-placement region to11guide the tuning process. In Chapter 5, I propose a self-synchronizing synchronverter designwith fast self-synchronization speed and easily tuneable parameters, and in Chapter 6, I proposea synchronverter design with reduced output-power coupling during normal operation. Next,in Chapter 7, I study the synchronverter active-power transfer capability and also providecountermeasures to improve it. Finally, Chapter 8 provides concluding remarks and compellingfuture directions.12Chapter 2Adjusting Synchronverter Dynamic ResponseSpeed via Damping Correction LoopIn this chapter, I propose to augment the conventional synchronverter control scheme withan auxiliary loop to freely adjust the dynamic response speed during normal operation. Thisloop is dubbed as a damping correction loop since its form and function are reminiscent ofthe damping component in the classical SG model. Central to the proposed auxiliary loopis the creation of an additional tuneable parameter that allows for unrestricted adjustmentof the system damping ratio without affecting the steady-state frequency droop characteristic,which is an improvement over the conventional synchronverter design. In the proposed method,relevant parameters are analytically tuned and active- and reactive-power coupling effects arereduced when operating with increased response speed.This chapter is organized as follows. Section 2.1 provides an overview of the conventionalsynchronverter and identifies its shortcoming. Section 2.2 describes the proposed synchronverterwith the added damping correction. In Sections 2.3 and 2.4, the performance of the proposedsynchronveretr design is validated through small-signal analysis and time-domain simulations.Finally, concluding remarks and directions for future research are provided in Section 2.5.13Figure 2.1: The conventional synchronverter [1]. (a) synchronverter APL. (b) synchronverterRPL. (c) synchronverter grid interface.2.1 Preliminaries2.1.1 Overview of Conventional SynchronverterThe synchronverter is a power-electronic converter with an SG mathematical model embeddedinto its controller, so that it emulates SG rotor dynamics and contributes inertia to the grid.The synchronverter consists of active- and reactive-power control loops, among which the APLregulates the active-power output by controlling the virtual rotor speed, thereby adjusting therotor angle. However, I find that the response speed of the APL cannot be adjusted freely, sinceto preserve the synchronverter’s steady-state frequency droop characteristic, only the inertiaconstant can be adjusted in the APL. In this section, I provide an overview of the conventionalsynchronverter and motivate the need to improve the control aimed at unrestricted adjustmentof the APL response speed.For the SG emulated by the synchronverter control system, assume that (i) it has a roundrotor with no damper windings, (ii) there is no flux saturation, (iii) the number of pole-pairsis 1, and (iv) stator windings are star connected with no neutral line. Figure 2.1 shows a14synchronverter-connected system, which consists of the active-power loop (APL) (Fig. 2.1(a)),the reactive-power loop (RPL) (Fig. 2.1(b)), and the interface to the grid (Fig. 2.1(c)). Notethat the synchronverter dc-bus voltage udc is assumed to be constant, and this assumption ismade throughout this dissertation. Below, with reference to Fig. 2.1, I explain each componentin detail.1) Active-power Loop: As depicted in Fig. 2.1(a), APL emulates SG rotor dynamics and real-izes self synchronization without a PLL. Let ωg denote the rotating speed of the synchronverter-emulated SG. In order to describe the electromechanical behaviour of the SG, I adopt thestandard swing equation, as follows: [79]Jgdωgdt= Tm − Te −Dp(ωg − ω?g), (2.1)where Jg is the inertia constant, Tm is the mechanical input torque, Te is the electromagnetictorque, and ω?g is the reference value of ωg. In (2.1), the term −Dp(ωg − ω?g) represents asimplified governor with no time delay, so that the synchronverter APL can realize frequencydroop control. The frequency droop coefficient, Dp, is determined byDp =∆Tm∆ωg, (2.2)where ∆ωg = ωg − ω?g is the angular speed deviation and ∆Tm is the amount of input torquechange required by local grid code [14]. I assume that the torque would change by 100% if theangular speed changes by 0.5% [1].Let P ?g denote the reference value of the synchronverter active-power output Pg; also letωN denote the rated angular speed value. Then, in (2.1), the mechanical input torque iscomputed as Tm = P?g /ωN . Additionally, let ig = [iga, igb, igc]T denote the synchronverterstator current 1; and let ψf denote the excitation flux obtained from the synchronverter RPL.Then, the electromagnetic torque in (2.1) is evaluated as [1]Te = ψf iTg sin θ˜g, (2.3)1In this dissertation, variables in bold denote matrix quantities (including vectors) and those not in bold referto scalars.15wheresin θ˜g =[sin θg sin(θg − 2pi3)sin(θg +2pi3)]T(2.4)and the virtual rotor angle θg is computed according toθg =∫ t0ωg dt. (2.5)2) Reactive-power loop: As shown in Fig. 2.1(b), depending on the states of Switches 1 and 2,the synchronverter RPL regulates its Qg or the line-to-line RMS value Ut of the terminal voltageut by adjusting the excitation flux ψf . Let S1 and S2 represent the states of Switches 1 and 2(Si = 1 if Switch i is ON, and Si = 0 if Switch i is OFF, i = 1, 2). Then, the dynamics of ψfare described byKgdψfdt= S1(Q?g −Qg) + S2√23Dq(U?t − Ut), (2.6)where Kg is the control parameter that adjusts the RPL response speed, and Q?g and U?t arereference values for Qg and Ut. The voltage droop coefficient, Dq, is expressed asDq =∆Qg∆Ut, (2.7)where ∆Ut = Ut−U?t and ∆Qg is the amount of reactive-power variation required for a commen-surate change in Ut, as set by the local grid code [14]. Here, I assume that Qg increases 100%if Ut drops 5% [1]. In (2.6), Qg obeys [1]Qg = −ωgψf iTg cos θ˜g, (2.8)in whichcos θ˜g =[cos θg cos(θg − 2pi3)cos(θg +2pi3)]T. (2.9)We note that Qg also includes the reactive power consumed by the output L-type filter of thesynchronverter.Depending on the states of Switches 1 and 2, the RPL operates under different control modeswhen used in different application scenarios [55]. For example, in the main grid, with S1 = 116and S2 = 0, Q-mode supplies the desired reactive power based on its reference Q?g received fromhigher level control centre. On the other hand, in microgrid settings, with S1 = S2 = 1, QD-mode is preferred to realize voltage droop control and improve voltage stability. Specifically,the synchronverter injects more reactive power automatically when there is a voltage sag orabsorbs more reactive power when these is a voltage swell. In this chapter, Q-mode is adoptedfor analysis and simulation, but note that this choice has no significant impact on my results.3) Grid Interface: The final component of the system depicted in Fig. 2.1(c) is the syn-chronverter interface with the grid, which is described here to facilitate my analysis later. Thesynchronverter inner voltage eg and its corresponding line-to-line RMS value are, respectively,expressed aseg = ωgψf sin θ˜g, (2.10)Eg =√32ωgψf , (2.11)where θg is the phase angle of eg. As depicted in Fig. 2.1(c), the synchronverter has an outputL-type filter with resistance Rs and reactance Xs, and at the point of common coupling (PCC),it is connected to the grid through a transmission line with resistance Re and reactance Xe. Thischapter assumes that the line is predominantly inductive, i.e., Xe  Re, and that dynamics ofpassive components, e.g., inductors, are negligible. Denote the phase angle difference betweeneg and the grid voltage u∞ by θg∞ (since Pg > 0, θg∞ ∈ [0, pi/2]), which obeysdθg∞dt=dθgdt− dθ∞dt= ωg − ω∞, (2.12)where ω∞ = 2pif∞ is the angular speed of u∞, and θ∞ is the phase angle of u∞. Furthermore,denote, by Pt and Qt, the active and reactive power injected into the grid from the PCC,respectively, so thatPt ≈ Pg =√32ωgψfU∞ sin θg∞Xt=: fp(ωg, θg∞, ψf ), (2.13)Te =Pgωg≈ PtωN, (2.14)17Figure 2.2: Phasor diagram of the synchronverter-connected system in Fig. 2.1.Qt =XeX2tE2g −XsX2tU2∞ +Xs −XeX2tEgU∞ cos θg∞ =: fq(ωg, θg∞, ψf ), (2.15)where U∞ is the line-to-line RMS value of u∞, and Xt = Xs +Xe. According to the geometricrelationship highlighted in Fig. 2.2, the RMS value Ut of terminal voltage ut isUt =√X2eX2tE2g +X2sX2tU2∞ +2XeXsX2tEgU∞ cos θg∞ =: fu(ωg, θg∞, ψf ). (2.16)Remark 1 (Derivation of expressions for Pt, Qt, and Ut in (2.13), (2.15) and (2.16)). In thisremark, I present the derivation process of (2.13), (2.15) and (2.16). First, according to Fig. 2.2,the synchronverter current phasor satisfiesIg∠ϕg =Eg∠θg − U∞∠θ∞jXt, (2.17)and thus, the terminal voltage phasor can be computed as followsUt∠θt = U∞∠θ∞ + jXeIg∠ϕg =XeXtEg∠θg +XsXtU∞∠θ∞. (2.18)With (2.17) and (2.18), we can obtain the synchronverter active-power output Pt, its reactive-power output Qt, and the terminal voltage RMS value Ut as followsPt = Re {Ut∠θt · (Ig∠ϕg)∗}= fp(ωg, θg∞, ψf ), (2.19)Qt = Im {Ut∠θt · (Ig∠ϕg)∗}= fq(ωg, θg∞, ψf ), (2.20)18Table 2.1: Parameters of Synchronverter-connected System in Fig. 2.1Parameters Values Parameters Values Parameters ValuesRs 0.741 Ω Ls 20 mH S1, S2 1, 0Re 0.0 Ω Le 38.5 mH DC-link voltage 13 kVJg 2.814 kg·m2 Kg 27980 Var·radV rated voltage 6.6 kVrmsDp 1407N·m·srad Dq 3711 Var/V rated frequency 60 HzU∞ 6.6 kVrms ωN , ω?g 376.99 rad/s rated capacity 1 MVAUt =∣∣∣∣XeXtEg∠θg + XsXtU∞∠θ∞∣∣∣∣= fu(ωg, θg∞, ψf ), (2.21)where Re{·}, Im{·}, and ‖{·}‖ denote the real part, the imaginative part, and the magnitudeof {·}, respectively. 2.1.2 Response Speed of Conventional SynchronverterAs described above, the synchronverter APL emulates SG rotor dynamics and realizes frequencydroop control. The dynamic behaviour of the APL, as described by (2.1), can be tuned byvarying the inertia Jg as well as the droop coefficient Dp. Next, via a numerical example, weshow that only varying Jg does not provide adequate adjustment of the APL response speed.Furthermore, while varying Dp results in modified response speeds, the steady-state droopcharacteristic is also affected, which is undesirable.Example 1 (Conventional APL Response Speed). This example uses the system shown inFig. 2.1 in conjunction with the relevant parameter values reported in Table 2.1, which aretuned according to [1]. In this example, at t = 0.0 s, the active- and reactive-power referencevalues are set to 0, i.e., P ?g = 0 MW and Q?g = 0 MVar, and the grid frequency f∞ = 60 Hz.Then, at t = 1.0 s, P ?g increases to 0.6 MW while Q?g remains unchanged. Finally, at t = 3.0 s,P ?g and Q?g remain unchanged, while the grid frequency f∞ suddenly increases to 60.1 Hz.As evident in Fig. 2.3(a), even though Jg is varied over the wide range from 0.2814 to 150, noappreciable change in the APL response speed is observed after the events introduced at t = 1.0 sand 3.0 s. On the other hand, as shown in Fig. 2.3(b), decreasing Dp increases the dynamic19Figure 2.3: Transient response of APL when: (a) Jg increases from 0.2814 to 150 (Dp = 1407);(b) Dp decreases from 1407 to 250 (Jg = 2.814).response speed after the event introduced at t = 1.0 s. However, in order to achieve the desiredsteady-state frequency droop characteristic, Dp should be fixed at 1407. The phenomena observed in Example 1 can be explained as follows. Consider small vari-ations in APL input variables, P ?g , ω∞, ω?g , and ψf , denoted by ∆P ?g , ∆ω∞, ∆ω?g , and ∆ψf ,respectively. Further denote by ∆Pg the variations in the APL output, Pg, resulting from smallvariations in the input variables. With the above notation in place, linearize (2.1) and (2.13)around the equilibrium point (denoted by superscript ◦) and take the Laplace transformationof the resulting small-signal model to obtain the following transfer function (Please refer toRemark 2 for detailed derivation process):∆Pg = G11(s)∆P?g +G12(s)∆ω∞ +G13(s)∆ψf , (2.22)where variations in ω?g are neglected by setting ∆ω?g = 0. In (2.22), G11(s) describes dynamicsin ∆Pg with respect to ∆P?g , G12(s) reflects the dynamics in ∆Pg with respect to ∆ω∞, andbecause Qg is adjusted by ψf , G13(s) highlights the influence of the RPL on APL dynamics.20The transfer functions G11(s), G12(s), and G13(s) share the same poles and are given byG11(s) =ω2n1s2 + 2ζ1ωn1s+ ω2n1, (2.23)G12(s) =−M · (s+ 2ζ1ωn1)s2 + 2ζ1ωn1s+ ω2n1, (2.24)G13(s) =N · (s2 + 2ζ1ωn1s)s2 + 2ζ1ωn1s+ ω2n1, (2.25)respectively, where M and N are expressed asM =√32ωNψ◦fU∞ cos θ◦g∞Xt, (2.26)N =√32ωNU∞ sin θ◦g∞Xt, (2.27)respectively. In (2.23)–(2.25), the damping ratio ζ1 and the natural frequency ωn1 are given byζ1 =(√Xt2√6ψ◦fU∞ cos θ◦g∞)· Dp√Jg, (2.28)ωn1 =(√√32ψ◦fU∞ cos θ◦g∞Xt)· 1√Jg, (2.29)respectively. Note that, in (2.28) and (2.29), both ζ1 and ωn1 are inversely proportional to√Jg. Hence, reducing Jg results in larger ζ1, which causes the system to be over damped andto respond slowly. Conversely, increasing Jg results in smaller ωn1, which limits the systembandwidth and still makes the APL respond slowly. Thus, as shown in Fig. 2.3(a), tuningonly Jg cannot satisfy both ζ1 and ωn1 design requirements and consequently cannot adjustthe APL response speed effectively. On the other hand, as shown in Fig. 2.3(b) and deducedfrom (2.28) and (2.29), the response speed can be adjusted by varying Dp, as decreasing Dpreduces ζ1 but does not influence ωn1. In this case, however, the steady-state frequency droopcharacteristic is not preserved, i.e., Pg decreases less when f∞ increases to 60.1 Hz. This isundesirable as the droop coefficient Dp is set to a particular value based on local grid code toenhance stability [14].In summary, the response speed of the conventional synchronverter APL cannot be ade-quately adjusted without affecting the steady-state droop characteristic. As described above,21the root cause is that tuning only one parameter Jg cannot simultaneously satisfy both ζ1 andωn1 design requirements. In the next section, I aim to adjust the response speed of the APLwithout affecting the desired droop characteristic.Remark 2 (Derivation of Transfer Functions (2.22)–(2.25)). We begin with the swing equationof the original synchronverter, i.e., (2.1), in which the mechanical input torque Tm satisfiesTm =P ?gωN, (2.30)and according to (2.13) and (2.14), the electromagnetic torque Te is expressed asTe =Pgωg=fp(ωg, θg∞, ψf )ωg=√32ψfU∞ sin θg∞Xt. (2.31)Substituting these expressions for Tm and Te into (2.1), we get thatJgdωgdt=P ?gωN−√32ψfU∞ sin θg∞Xt−Dp(ωg − ω?g). (2.32)As for ωg, it can be obtained from (2.12) asωg =dθg∞dt+ ω∞. (2.33)Then, take the derivative of (2.33) to getdωgdt=d2θg∞dt2+dω∞dt. (2.34)Next, we substitute (2.33) and (2.34) into (2.32) to getJg(d2θg∞dt2+dω∞dt)=P ?gωN−√32ψfU∞ sin θg∞Xt−Dp((dθg∞dt+ ω∞)− ω?g). (2.35)Then, as described in Section II-B, consider small variations in the APL input variables P ?g ,ω∞, and ψf (ω?g remains unchanged), and denote these small variations as ∆P ?g , ∆ω∞, and∆ψf , respectively. Linearizing (2.35) around the equilibrium point (denoted by superscript ◦),22we have thatJg(d2∆θg∞dt2+d∆ω∞dt)=∆P ?gωN−√32U∞Xt(sin θ◦g∞∆ψf + ψ◦f cos θ◦g∞∆θg∞)−Dp(d∆θg∞dt+ ∆ω∞), (2.36)where ∆θg∞ denotes small variations in θg∞ arising from ∆P ?g , ∆ω∞, and ∆ψf . Next, bytaking the Laplace transformation of (2.36), we get (while mildly abuse notation by using thesame symbols for both time- and s-domain variables)Jg(s2∆θg∞ + s∆ω∞)=1ωN∆P ?g −√32U∞Xt(sin θ◦g∞∆ψf + ψ◦f cos θ◦g∞∆θg∞)−Dp(s∆θg∞ + ∆ω∞). (2.37)Solving (2.37) for ∆θg∞, we get∆θg∞ =∆P ?gωN− (Jgs+Dp)∆ω∞ −√32U∞ sin θ◦g∞Xt∆ψfJgs2 +Dps+√32ψ◦fU∞ cos θ◦g∞Xt. (2.38)Next, consider small variations in the APL output Pg expressed in (2.13), resulting from smallvariations in the input variables, and linearize (2.13) with respect to ωg, θg∞, and ψf by takingappropriate partial derivatives to arrive at a small-signal model. It turns out that∂Pg∂ωg|x◦ ismuch smaller than∂Pg∂ψf|x◦ and ∂Pg∂θg∞ |x◦ , so we neglect the∂Pg∂ωg|x◦∆ωg term in ∆Pg. Assumingthat ω◦g = ωN , the small-signal output variable ∆Pg can be expressed as∆Pg =√32ωNU∞Xt(sin θ◦g∞∆ψf + ψ◦f cos θ◦g∞∆θg∞). (2.39)Finally, by substituting (2.38) into (2.39) and defining M , N , ζ1 and ωn1 as in (2.26)–(2.29),the transfer functions (2.22)–(2.25) are obtained, as desired. 2.2 Proposed Synchronverter Controller DesignTargeted at unrestricted adjustment of the APL dynamic response speed, this section proposesa damping correction loop to be added into the existing APL. With the augmented controller23in place, I show that the response speed of the improved synchronverter can be adjusted freelywithout affecting its steady-state frequency droop characteristic. We note that my proposeddamping correction loop has four kinds of realizations, i.e., realizations I–IV, as depicted inFigs. 2.4(a)–(d). In this chapter, I use the realization I in Fig. 2.4(a) as an example to introducethe proposed synchronverter design. For the introduction of other three realizations, pleaserefer to Remark 3. The proposed controller design includes both the APL and the RPL, and isinterfaced with the grid as in Fig. 2.1(c).In order to focus on the core idea behind the proposed damping correction loop, for now,neglect all first-order low-pass filters (LPFs) in Fig. 2.4. In other words, even though Tef andψff are obtained by filtering Te and ψf signals with LPFs, for now assume that Tef ≈ Teand ψff ≈ ψf (we will return to these filters in a later discussion). The inputs to the dampingcorrection loop are Tef and ψff , and its output is Dfddt(Tefψff), where Df [V· s2/rad] is a tuneableparameter. After adding this loop into the existing APL, (2.1) becomesJgdωgdt= Tm − Tef −Dp(ωg − ω?g)−Dfddt(Tefψff). (2.40)With the proposed damping correction loop in place, we make two key observations. First,note that the steady-state value of the additional term Dfddt(Tefψff)in (2.40) is exactly zero.As a direct consequence, Dp remains as the original droop coefficient, and the desired steady-state frequency droop characteristic is preserved. Second, during the transient period prior toreaching steady state, the additional term Dfddt(Tefψff)acts to modify the APL damping. This isevident by substituting (2.13) into (2.14) and further the resultant expression into Dfddt(Tefψff)to obtainDfddt(Tefψff)≈ Df ddt(Teψf)= Dfddt(√32U∞ sin θg∞Xt)=(Df√32U∞ cos θg∞Xt)· (ωg − ω∞), (2.41)where Tef ≈ Te and ψff ≈ ψf are assumed in the first equality.24Figure 2.4: Four realizations of proposed synchronverter controller with damping correctionloop. (a) Realization I. (b) Realization II. (c) Realization III. (d) Realization IV.To illustrate how the damping correction loop influences the APL response speed, analo-gous to the analysis conducted in Section 2.1.2, consider small variations in the same inputvariables ∆P ?g , ∆ω∞, and ∆ψf . Here, I define the system output as Pt (instead of Pg inthe original synchronverter) as this is the actual active power injected into the grid. Denoteby ∆Pt variations in Pt resulting from small variations ∆P?g , ∆ω∞, and ∆ψf . Then, by lineariz-ing (2.13) and (2.40) around the equilibrium point and applying the Laplace transformation on25the resulting small-signal model, we obtain the following transfer function:∆Pt = G21(s)∆P?g +G22(s)∆ω∞ +G23(s)∆ψf , (2.42)whereG21(s) =ω2n2s2 + 2ζ2ωn2s+ ω2n2, (2.43)G22(s) =−M · (s+ 2ζ1ωn2)s2 + 2ζ2ωn2s+ ω2n2, (2.44)G23(s) =N · (s2 + 2ζ2ωn2s)s2 + 2ζ2ωn2s+ ω2n2, (2.45)M and N are given by (2.26) and (2.27), and the damping ratio ζ2 and the natural frequencyωn2 are expressed asζ2 =(√Xt2√6ψofU∞ cos θog∞)·Dp +Df√32U∞ cos θog∞Xt√Jg, (2.46)ωn2 = ωn1 =(√√32ψ◦fU∞ cos θ◦g∞Xt)· 1√Jg, (2.47)respectively. In the augmented APL, according to (2.47), the natural frequency ωn2 can betuned by Jg, as described in Section 2.2. At the same time, the damping ratio ζ2 can beindependently adjusted by tuning Df . For example, with θ◦g∞ ∈ [0, pi/2], ζ2 = ζ1 if Df = 0,ζ2 > ζ1 if Df > 0, and ζ2 < ζ1 if Df < 0. Thus, by introducing an additional degree of freedomto the original APL controller, the proposed damping correction loop allows ζ2 and ωn2 designrequirements to be satisfied simultaneously.Remark 3 (Other Realization of Damping Correction Loop). BesidesDfddt(Tefψff)in Fig. 2.4(a),the proposed damping correction loop also has other three realization forms as shown inFigs. 2.4(b)-(d). All these realizations are equivalent in terms of their functions, and I brieflyexplain this by introducing realizations II–IV as follows.Realization II: The second realization of the damping correction loop is marked in the reddashed line box in Fig. 2.4. This realization uses the synchronverter output currents ig and itsrotor angle θg as inputs. Here, I show that realization II indeed has same output as realization I.26First, according to (2.10), we havesin θ˜g =1ωgψfeg. (2.48)Also, bear in mind that the active power Pt, which satisfies (2.13), can be computed from [80]Pt = iTg eg. (2.49)With (2.48) and (2.49), we haveiTg sin θ˜g =iTg egωgψf=Ptωgψf=√32U∞ sin θg∞Xt(2.50)Thus, by omitting its LPF, the output signal of the realization II of the damping correctionloop is indeedDfddt(iTg sin θ˜g)= Dfddt(√32U∞ sin θg∞Xt)=(Df√32U∞ cos θg∞Xt)· (ωg − ω∞), (2.51)which is same as the expression in (2.41). We note that one minor difference between realiza-tions I and II is that realization II does not need the filtered signal ψff together with its limiteras shown in Fig. 2.4(a).Realizations III and IV: As shown in Figs. 2.4(c) and (d), the third and fourth realizations ofthe damping correction loop are, respectively, similar to the first and second ones in Figs. 2.4(a)and (b), with the exception that realizations III and IV avoid the differential operator, whichmight amplifier noises and lead to instability. This is achieved by appending the outputs ofboth realizations III and IV to the output side of the integrator 1Jgs instead of the input side,as suggested in [81].Remark 4 (Connection to Transient Droop Function [42]). While myriad other schemes havebeen proposed in the literature, I comment on the connection between my method and the so-27called “transient droop function” method since it boils down to augmenting the swing equationin (2.1) with a derivative term and is thus most closely related to the method proposed in thischapter. Specifically, the method in [42] is equivalent to augmenting (2.1) so that it becomesJgdωgdt= Tm − Tef −Dp(ωg − ω?g)−DmdPtdt. (2.52)Via a similar exercise as the one done to obtain (2.22), we get∆Pt = G31(s)∆P?g +G32(s)∆ω∞ +G33(s)∆ψf . (2.53)It is straightforward to show that setting Dm = Df/(ωNψ◦f ) results in G31(s) = G21(s),G32(s) = G22(s), andG33(s) =N · (s2 + 2ζ1ωn2s)s2 + 2ζ2ωn2s+ ω2n2. (2.54)Since G21(s) = G31(s) and G22(s) = G32(s), both the proposed damping correction loop andthe transient droop function lead to the same dynamic performance in ∆Pt with respect to∆P ?g and ∆ω∞. However, with increased APL dynamic response speed, the transient droopfunction strengthens the coupling between the APL and the RPL. To see this, consider thecase in which Jg is reduced to increase the bandwidth of the APL, in an effort to increasethe synchronverter response speed. According to (2.28), however, this causes ζ1 to increase,creating an over-damped system. In order to increase the response speed, the core idea inboth the proposed damping correction loop and the transient droop function is to reduce thedamping ratio independently, by tuning Df in the former method and Dm in the latter. Recallthat G23(s) and G33(s) isolate the impact of the RPL on the APL in the two methods, and theonly difference between G23(s) and G33(s) is in the coefficients multiplying s in the numerators,namely, 2Nζ2ωn2 in (2.45) and 2Nζ1ωn2 in (2.54). As ζ1 > ζ2 and N > 0 for θ◦g∞ ∈ [0, pi/2], wehave that 2Nζ1ωn2 > 2Nζ2ωn2. Consequently, if ψf is increased to supply more reactive power,for the same ∆ψf > 0, G33(s)∆ψf would imply a larger temporary rise in ∆Pt than G23(s)∆ψf .In this way, the transient droop function strengthens the coupling between the APL and theRPL. So comparatively, for fast APL response, the proposed damping correction loop achieves28the desired dynamic performance with reduced active- and reactive-power coupling. Remark 5 (Realizing Self Synchronization). For the synchronverter, synchronization is theability to closely track its internal voltage eg to ut, and hence to the grid voltage u∞, priorto closing the breaker in Fig. 2.1(c) [55]. Synchronization helps to minimize start-up currentswhen the VSC becomes physically connected to the grid. To avoid the need for a dedicatedsynchronization unit and to simplify the controller, a desirable synchronverter feature is tohave self-synchronization capability, i.e., the ability to synchronize the VSC with the gridbefore connecting them without a PLL [55]. In order to achieve self synchronization, thesynchronverter must be able to (i) operate in P -mode in which Pt tracks P?g without steady-state error and Q-mode with both P ?g and Q?g set to zero, and (ii) via a virtual impedancebranch, compute virtual currents and feed them back into the controller [55]. These featuresensure that eg ≈ ut before the VSC is physically connected to the grid, and thus enablethe synchronverter to realize synchronization and reduce start-up currents immediately afterbreaker closure. The proposed controller design is equipped to achieve the features mentionedabove and, in turn, self synchronization. First, with reference to Fig. 2.1(c), before closing thebreaker, the synchronverter can operate in P -mode with the damping torque supplied only bythe damping correction loop (i.e., set Dp = 0). Also, as shown in Fig. 2.1, the RPL can operatein Q-mode with switches S1 = 1 and S2 = 0. The current ig that feeds back into the controllerin Fig. 2.4 is computed by having the difference (eg − ut) pass through a virtual impedancebranch (Lvs + Rv)−1, where Lv and Rv are tuned by trial and error. Finally, both active-and reactive-power reference values P ?g and Q?g are set to zero during the self-synchronizationprocess prior to breaker closure. In this way, my design satisfies both features (i) and (ii) aboveand thus achieves self synchronization. In addition to the damping correction loop, there are two minor differences between theconventional and proposed synchronverter designs. First, in the RPL, first-order LPFs areimplemented to smooth ψf , Te, Qt, and Ut. The smoothed signals are ψff , Tef , Qtf , and Utf ,respectively, whereτfdψffdt= −ψff + ψf , (2.55)29τfdTefdt= −Tef + Te, (2.56)τfdQtfdt= −Qtf +Qt, (2.57)τfdUtfdt= −Utf + Ut, (2.58)with Te = Pt/ωN , and Pt, Qt, and Ut are, respectively, given by (2.13), (2.15), and (2.16). Notethat all filter time constants are assumed to be equal, and their impacts are studied in Section2.3. Furthermore, Qtf (instead of Qg in the original synchronverter) is used as the feedbacksignal, as it reflects the actual reactive power (after being filtered) injected into the grid. Thus,the dynamics of ψf obeyKgdψfdt= S1(Q?g −Qtf ) + S2√23Dq(U?t − Utf ). (2.59)During normal operation, parameters in the proposed synchronverter controller are tuned asfollows. Statuses of Switches 1 and 2 are chosen based on the application scenario as discussed inSection II-A. The droop coefficients, Dp and Dq, are set by the local grid code. The parameterKg can be chosen based on the method described in [1]. Finally, τf , Jg, and Df can be tunedusing small-signal analysis, which is detailed next.2.3 Small-signal AnalysisIn Section 2.2, the synchronverter with the proposed damping correction loop is described, butthe impacts of the first-order LPFs are neglected. Also, we note that small-signal analysisis usually needed to tune the synchronverter parameters. Thus, this section conducts thesmall-signal analysis on the proposed synchronverter design. Below, I first include all LPFsand develop the full small-signal state-space model of the synchronverter-connected systemlinearized around the equilibrium point. After that, I verify the accuracy of the developedstate-space model by comparing its time domain response with the simulation results of thenonlinear synchronverter-connected system. Finally, by conducting small-signal analysis withmy state-space model, I show the efficacy and robustness of my proposed controller design.30Figure 2.5: Proposed synchronverter with damping correction loop and its grid interface.2.3.1 Development of Synchronverer State-space ModelIn this Subsection, I develop the linearized state-space model for the synchronverter-connectedsystem, which consists of Figs.2.1(c) and 2.4. No that to this end, the system dynamics arefully described by (2.12), (2.40), and (2.55)–(2.59). First, we collect the system state variablesand specify the input and output vectors in Figs. 2.1(c) and 2.4. As the state-space model takesthe form x˙(t) = f(x(t),u(t)), in which x(t) is the state vector, u(t) is the input vector, wehave x(t) =∫ t0 f(x(t),u(t))dt, and the state vector x is likely composed of integrator outputs.Also, the state variables remain constant in steady state, i.e., f(x(t),u(t)) = 0. With these inour mind, we choose the state vector x(t) as followsx = [ωg, θg∞, ψf , ψff , Tef , Qtf , Utf ]T. (2.60)Note that in (2.60), I choose θg∞ rather than θg because θg∞ is constant in steady state whileθg varies as its derivative ωg is not zero. In terms of the state-space model input vector u, first,the four reference values P ?g , Q?g, U?t , and ω?g in the controller need to be included, then I alsoconsider the grid frequency ω∞ as an input since we are also interested in the system response31when ω∞ varies. Thus, the input vector u isu = [P ?g , Q?g, U?t , ω?g , ω∞]T. (2.61)Furthermore, we recognize Pt and Qt as system output variables, resulting the output vectory = [Pt, Qt]T. (2.62)Next, with the nonlinear system state vector x, the input vector u, and the output vec-tor y in place, I described the nonlinear state-space model. To show the modeling processmore clearly, by combining Figs. 2.1(c) and 2.4, the synchronverter-connected system is de-picted in Fig. 2.5. By substituting (2.13), (2.15), and (2.16) into (2.56)–(2.58), and puttingtogether (2.40), (2.12), (2.59), (2.55)–(2.58), (2.13), and (2.15), we obtain the full-order non-linear state-space model, as follows:ddtωg =1JgωNP ?g −1JgTef − DfJgψff(fp(ωg, θg∞, ψf )ωN− Tef ψfψff)− DpJg(ωg − ω?g) =: F1 1,ddtθg∞ = ωg − ω∞ =: F1 2,ddtψf =S1K(Q?g −Qtf ) +√23S2DqK(U?t − Utf ) =: F1 3,ddtψff = − 1τfψff +1τfψf =: F1 4,ddtTef = − 1τfTef +1τffp(ωg, θg∞, ψf )ωN=: F1 5,ddtQtf = − 1τfQtf +1τffq(ωg, θg∞, ψf ) =: F1 6,ddtUtf = − 1τfUtf +1τffu(ωg, θg∞, ψf ) =: F1 7, (2.63)Pt = fp(ωg, θg∞, ψf )),Qt = fq(ωg, θg∞, ψf )), (2.64)in which (2.63) and (2.64), respectively, represent the state and output equation. Denote the32equilibrium point of the nonlinear system in (2.63) asxo = [ωog , θog∞, ψof , ψoff , Toef , Qotf , Uotf ]T, (2.65)we linearize (2.63) and (2.64) around x◦ to obtain the following small-signal state-space model:d∆xdt= A∆x+B∆u, (2.66)∆y = C∆x+D∆u. (2.67)where ∆(·) denotes small perturbations in variable (·), and the state vector ∆x and input vector∆u are∆x = [∆ωg,∆θg∞,∆ψf ,∆ψff ,∆Tef ,∆Qtf ,∆Utf ]T, (2.68)∆u = [∆P ?g ,∆Q?g,∆U?t ,∆ω?g ,∆ω∞]T, (2.69)respectively. Please see Appendix A for the state matrix A, the input matrix B, the outputmatrix C, and the direct transition matrix D. We note that the matrix A has 7 eigenvalues,denoted by λk, k = 1, ..., 7. As the LPFs in (2.55)–(2.58) are considered in (2.66), this modelaccurately describes the small-signal dynamic behaviour of the synchronverter. With the modelin (2.66) and (2.67), we are able to conduct eigenvalue sensitivity analysis to validate theeffectiveness of the proposed controller design and highlight its robustness against operating-point changes.2.3.2 Verification of Linearized State-space ModelHere, I verify the linearized small-signal state-space model, which consists (2.66) and (2.67),against the nonlinear synchronverter-connected system modelled in PSCAD/EMTDC. In orderto achieve this, I conduct time-domain simulations and compare the dynamic response of thetwo models. Both models use parameter values reported in Table 2.1, Df = −2.76 V ·m2/rad,and τf = 0.01 s. The linearized model is obtained by linearizing the nonlinear system aroundthe equilibrium point corresponding to Pt = 0.6 MW and Qt = 0.0 MVar.Simulation results are displayed in Fig. 2.6, in which cases (i) (blue trace) and (ii) (red trace),33Figure 2.6: Verification of the linearized small-signal state-space model.respectively, represent the active-power output Pt obtained from the nonlinear and linearizedmodels. Beginning at t = 0.5 s, the active-power reference value P ?g = 0.6 MW, the reactive-power reference value Q?t = 0 MVar, the grid frequency f∞ = 60 Hz, and the synchronverteroperates in steady state. Then, at t = 1.0 s, P ?g decreases from 0.6 MW to 0 MW. Afterthis disturbance, from Fig. 2.6, we note that the dynamic response of the nonlinear modelhas a slightly larger overshoot than that of the linearized one. This is reasonable because thesynchronverter operating point is moved away from the linearization point, and according to myanalysis in Section 2.3.3 later, a smaller Pt corresponds to a smaller damping ratio ζ2. Basedon this, we expect that the nonlinear model would have a slightly larger overshoot when P ?gbecomes 0 MW. For the remainder of the simulation, at t = 2.0 s, P ?g increases from 0 MW to0.6 MW, and at t = 3.0 s, f∞ increases from 60 Hz to 60.1 Hz, and then decreases from 60.1 Hzto 60 Hz at t = 4.0 s. Following each of these disturbances, the dynamic response resultingfrom the nonlinear and linearized models are well matched.Based on the simulation results shown in Fig. 2.6, we conclude that the dynamic behaviourof the nonlinear model is sufficiently mimicked by the linearized model composed of (2.66)and (2.67), thus verifying its A matrix to be used to tune design parameters as done in Sec-tion 2.3.3.2.3.3 Impact of Damping Correction Loop on EigenvaluesAfter verifying the linearized state-space model in (2.66) and (2.67), I show that the dampingcorrection loop is able to adjust the APL damping ratio via the small-signal analysis on thismodel. Using participation-factor analysis, I find that the state ∆Tef , which reflects the APL34Figure 2.7: Dominant modes of improved synchronverter without the damping correction loop.(a) Jg increases from 0.2814 to 150 (Df = 0, τf = 0.01, Pt = 0.6 MW, Qt = 0 MVar). (b) Dpdecreases from 1407 to 250 (Df = 0, τf = 0.01, Pt = 0.6 MW, Qt = 0 MVar).dynamics, is closely related to eigenvalues λ2 and λ3, so I focus on these in the analysis below.Without Damping Correction LoopLet Df = 0, so that the dynamic performance of the synchronverter without the dampingcorrection loop can be examined. Figure 2.7(a) shows that when Jg increases, the two realpoles λ2 and λ3 meet and split into complex-conjugate poles, i.e., the damping ratio ζ2 of theAPL decreases. However, their magnitudes |λ2| and |λ3| decrease, i.e., increasing Jg reduces thenatural frequency ωn2. Thus, only tuning Jg cannot place this pair of eigenvalues further awayfrom the imaginary axis and cannot adjust the APL response speed freely. Figure. 2.7(b) showsthat varying Dp changes the APL damping ratio, but has negligible influence |λ2| and |λ3|. Asthe original APL is over damped, reducing Dp makes λ3 move towards the left and increasesthe APL response speed, but this affects the steady-state frequency droop characteristic asexplained in Section II.35Figure 2.8: Dominant modes of improved synchronverter with the damping correction loop.(a) Df decreases from 0 to −3 (τf = 0.01, Pt = 0.6 MW, Qt = 0 MVar). (b) τf increasesfrom 0.007 to 0.05 (Df = −2.76, Pt = 0.6 MW, Qt = 0 MVar). (c) Df decreases from 0to −4.1 when τf respectively takes 0.01, 0.001, and 0.0001 (Pt = 0.6 MW, Qt = 0 MVar). (d)Df decreases from 0.2 to −4.1 when Jg respectively takes 0.02, 2.814, 100 (τf = 0.01, Pt =0.6 MW, Qt = 0 MVar).36With Damping Correction LoopHere, allow Df to be an additional tuneable parameter. Figure 2.8(a) shows that, by decreasingDf , the two real poles λ2 and λ3 meet and become two complex-conjugate poles, and there-after |λ2| and |λ3| remain unchanged. Thus, as (2.46) predicts, tuning Df adjusts the dampingratio ζ2 without affecting the natural frequency ωn2. Next, Fig. 2.8(b) shows that increasing τfnot only reduces the damping ratio ζ2, but also makes the natural frequency ωn2 smaller.Figures 2.8(c) and 2.8(d) show two families of root loci when Df varies with different τfand Jg. We observe that if τf or Jg becomes smaller, the natural frequency ωn2 would increase.It is worth noting that there are mutual influences between τf and Jg. For example, if τf takesthe value of 0.01 s, as shown in Fig. 2.8(d), a larger Jg would reduce ωn2 significantly, whilea smaller Jg would only increase ωn2 slightly. Also, there is a tradeoff between the first-orderLPFs’ filtering ability and the APL dynamic performance, i.e., while reducing τf increases ωn2,it also reduces the LPFs’ noise rejection ability.The analysis above shows that, with the proposed damping correction loop, and by simulta-neously tuning Df , Jg and τf , eigenvalues λ2 and λ3 can be placed more freely, and the degreesof freedom in the controller are increased. We also note that the damping correction loop hassmaller impacts on other eigenvalues, as explained in Remark (6) below.Remark 6 (Eigenvalue Sensitivities to ParameterDf ). In this remark, I systematically evaluatethe impact of the proposed damping correction loop on all eigenvalues of the linearized systemand conclude that indeed, this loop mainly influences λ2 and λ3, to which, as stated in thisSection, ∆Tef is most closely related. To assess these influences, we first note that studying theimpact of the damping correction loop on the eigenvalues is equivalent to studying the impactof varying the parameter Df . The influence of the parameter Df on the eigenvalue λk can bequantitatively determined by computing the sensitivity of each eigenvalue to Df , as follows:[79]∂λk∂Df=qTk[∂A∂Df]pkqTk pk, (2.70)where qk is the left eigenvector of the state matrix A (see Appendix A) corresponding to λk(i.e., qTkA = λkqTk ), and pk is the right eigenvector ofA corresponding to λk (i.e., Apk = λkpk).For the sensitivity analysis, I assume that the synchronverter-connected system parameters37Table 2.2: Eigenvalue sensitivities to parameter Df .Value of λk Sensitivity of λk to parameter Df (i.e.,∂λk∂Df) Impact of increasing Dfλ1 −4.9433 + j0.0000 0.13161− j0.0000 Re(λ1)↗λ2 −14.556 + j10.723 −11.840− j15.304 Re(λ2)↘, Im(λ2)↘λ3 −14.556− j10.723 −11.840 + j15.304 Re(λ3)↘, Im(λ3)↗λ4 −100.00 + j0.0000 −0.0000 + j0.0000λ5 −94.800 + j0.0000 0.0059503 + j0.0000 Re(λ5)↗λ6 −100.00 + j0.0000 0.0000 + j0.0000λ7 −541.72 + j0.0000 12.884 + j0.0000 Re(λ7)↗are as in Table 2.1, Df = −2.76 V · s2/rad, and τf = 0.01 s. Linearizing the system around theoperating point Pt = 0.6 MW and Qt = 0.0 MVar, and using (2.70), I compute the sensitivityof each eigenvalue to the parameter Df , the results from which are summarized in Table 2.2(Re(λk) and Im(λk) represent the real and imaginary parts of λk, respectively). We find thatother than λ2 and λ3, the damping correction loop also influences λ1, λ5, and λ7. Specifically,λ1, λ5, and λ7 would increase as Df increases. However, from Table 2.2, we note that themagnitudes of sensitivities∣∣Re( ∂λ2∂Df )∣∣ = 11.840 and ∣∣Im( ∂λ2∂Df )∣∣ = 15.304 are comparable to themagnitude of the eigenvalue itself∣∣λ2∣∣ = 18.079, and this is also true for λ3. On the otherhand, for λ1, λ5, and λ7,∣∣Re( ∂λk∂Df )∣∣  ∣∣λk∣∣ and ∣∣Im( ∂λk∂Df )∣∣  ∣∣λk∣∣, k = 1, 5, 7. For example,∣∣Re( ∂λ7∂Df )∣∣ = 12.884  ∣∣λ7∣∣ = 541.72. Based on this reasoning, I conclude that the dampingcorrection loop influences λ2 and λ3 more than λ1, λ5, and λ7.It is worth noting that although the eigenvalue sensitivity analysis above only evaluatesthe impact of Df on λk with one set of parameters, it can be repeated under other operatingconditions, with Df taking other values. The repeated analysis reveals similar trends in thatλ2 and λ3 are influenced more than other eigenvalues. Detailed results from this are omittedhere to avoid repetition. Robustness of Proposed DesignFigure 2.9(a) shows that when the synchronverter active-power output Pt increases from 0 MWto 1 MW, the damping ratio ζ2 increases accordingly. On the other hand, Fig. 2.9(b) shows thatan increase in the reactive-power output Qt from 0 MVar to 0.8 MVar only causes ζ2 to decreaseslightly. In this way, the proposed controller is robust against variations in the operating point.38Figure 2.9: Dominant modes of improved synchronverter when validating the robustness of theproposed design. (a) Pt increases from 0 to 1 MW (Df = −2.76, τf = 0.01, Qt = 0 MVar).(b) Qg increases from 0 to 0.8 MVar (Df = −2.76, τf = 0.01, Pt = 0 MW).2.4 Simulation ResultsIn this section, via time-domain simulations, I not only validate the effectiveness of the pro-posed damping correction loop during normal operation, but also verify that the proposeddesign realizes self synchronization. The simulated system consists of the proposed synchron-verter controller shown in Fig. 2.4 and its interface to the grid shown in Fig. 2.1(c). Thesynchronverter-connected system is modelled in PSCAD/EMTDC, with τf = 0.01 s and otherparameter values reported in Table 2.1, unless otherwise noted. Note that the synchronverteris implemented with detailed model, which consists of insulated-gate bipolar transistor (IGBT)switches.2.4.1 During Normal OperationIn this case study, at t = 1.0 s, P ?g increases from 0 MW to 0.6 MW, and at t = 3.0 s, thegrid frequency f∞ increases from 60.0 Hz to 60.1 Hz. As shown in Fig. 2.10(a), decreasing Dfreduces the damping ratio of the synchronverter APL as (2.46) predicts. Since the originalAPL is over damped, a smaller Df causes the APL to respond more quickly. By setting39Figure 2.10: Transient response of APL when: (a) Df decreases from 0 to −2.76 (Jg = 2.814).(b) Jg increases from 2.814 to 200 (ζ2 ≈ 0.8).Figure 2.11: Comparison between damping correction loop (method A) and transient droopfunction (method B).Df = −2.76 V· s2/rad, the APL damping ratio ζ2 ≈ 0.8, and satisfactory transient responsespeed is obtained, as illustrated in Fig. 2.10(a). With ζ2 held at 0.8 by adjusting Df , increasingJg causes the synchronverter APL to respond more slowly, as shown in Fig. 2.10(b). This is dueto the decreased bandwidth as predicted by (2.29) and (2.47). We note that the steady-statefrequency droop characteristic is preserved in both Figs. 2.10(a) and 2.10(b).Remark 7 (Comparison with Transient Droop Function [42]). To ensure a fair comparison,the transient droop function (method B) is tuned so that it has the same dynamic response as40Figure 2.12: Impact of the damping correction loop on dc-side current idc and ac-side power Ptwhen the APL response is fast (Jg = 2.814, Df = −2.76) and slow (Jg = 200, Df = 0.42). Wecan find that the damping correction loop influences idc and Pt nearly in the same way.the proposed damping correction loop method (method A) with respect to variations in P ?g .As shown in Fig. 2.11, following the increase in P ?g from 0 MW to 0.6 MW at t = 1.0 s, traces(a1) and (b1), corresponding to the dynamic response of Pt in methods A and B, respectively,are identical. At t = 3.0 s, Q?g increases from 0 MVar to 0.4 MVar. After this event, traces(a2) and (b2) in Fig. 2.11, corresponding to the dynamic response of Qt in methods A andB, respectively, are identical, since the same value of Kg is chosen for both methods. Qt isincreased by making ψf larger, and Pt is influenced by ψf (note that this influence is discussedin detail in Remark 1), so at t = 3.0 s, Pt traces in both methods A and B rise temporarily.As shown in Fig. 2.11, the transient induced in method A is much smaller than that in methodB. We conclude that, with high APL response speeds, the proposed damping correction loopmethod significantly diminishes the coupling between the APL and the RPL. Remark 8 (Impact on dc-side Current). The proposed damping correction loop influences thedynamics of the ac-side output power, Pt, as observed in Fig. 2.10. As a side effect, the dynamicsof the dc-side current idc are also affected, because the dc side of the synchronverter transfers41Figure 2.13: Self-synchronization process of improved synchronverter with the damping correc-tion loop when f∞ = 60.1 Hz.power to the ac side, as shown in Fig. 2.1(c). To examine the effect of the damping correctionloop on idc, we consider two cases: (i) APL responds quickly with Jg = 2.814 kg ·m2 and Df =−2.76 V· s2/rad, and (ii) APL responds slowly with Jg = 200 kg ·m2 and Df = 0.42 V· s2/rad.In Fig. 2.12, we adopt per-unit quantities for P ?g , Pt, and idc to promote ease of comparison.42During the simulations, at t = 1.0 s, P ?g increases from 0 to 0.6 p.u. (i.e., 0.6 MW). We canfind that trace (a) in Fig. 2.12(a) and trace (a) in Fig. 2.12(b), respectively correspondingto idc and Pt when the APL responds quickly, are nearly identical. Similarly, trace (b) inFig. 2.12(a) and trace (b) in Fig. 2.12(b), corresponding to the case of slow APL response,are nearly identical. These observations are explained as follows. Referring to Fig. 2.1(c),suppose losses in the VSC as well as the L-type filter resistance Rs are neglected, we havethat Pt ≈ udcidc. And since the dc-side voltage udc is fixed at its rated value (or in per-unitquantities, udc = 1 p.u.), then indeed, Pt ≈ idc in per-unit quantities, as shown in Fig. 2.12.From the above, we conclude that the impact of the proposed damping correction loop on theac-side output power and the dc-side current are nearly identical in per-unit quantities (andthey are directly proportional to each other in actual quantities with fixed udc). 2.4.2 Realizing Self SynchronizationVia time-domain simulations, we validate the proposed self-synchronization method describedin Remark 5. In this case study, the grid frequency f∞ = 60.1 Hz, τf = 0.01 s, the virtualimpedance branch is set to (0.05s+ 10)−1 (tuned by trial and error). All parameters except Dpare reported in Table 2.1. Prior to physically connecting the VSC to the grid, i.e., closing thebreaker in Fig. 2.1(c), the synchronverter is synchronized to the grid. Then, the breaker isclosed and normal operation begins. With reference to simulation results shown in Fig. 2.13,at t = 0.0 s, the synchronverter operates in P -mode (with Dp = 0 N ·m · s/rad and Df =1.00 V · s2/rad) and Q-mode (with S1 = 1 and S2 = 0), where P ?g = 0 MW and Q?g =0 MVar. As an example, Fig. 2.13(b) shows that the synchronverter a-phase inner voltageega converges to the grid voltage uta (note that ut = u∞ before the breaker is closed) within0.15 s. When the breaker is closed at t = 0.8 s and normal operation (as depicted in Fig. 2.4)begins, self synchronization is achieved with ega ≈ uta, as shown in Fig. 2.13(c). Moreover,Fig. 2.13(d) shows that there is no significant surge current after closing the breaker. (Notethat in Fig. 2.13(c), the harmonics in uta after t = 0.8 s are caused by fast switching devices,and an LCL-type filter can be included to filter them out. As this issue is not the focus of thischapter, we refrain from dwelling on it further.) At t = 1.0 s, the active-power reference valueP ?g increases from 0.0 to 0.6 MW. Since the frequency droop controller is inactive (i.e., Dp = 0),43the active-power output Pt tracks P?g to 0.6 MW even though the grid frequency f∞ = 60.1 Hz.Finally, at t = 2.0 s, the frequency droop controller is activated with Dp = 1407 N ·m · s/rad,and the damping correction loop is reset with Df = −2.76 V · s2/rad. As shown in Fig. 2.13(e),the synchronverter soon decreases its active-power output automatically to regulate the gridfrequency. To close this discussion, we note that the results shown in Fig. 2.13 are typicaland, in general, my proposed design achieves self synchronization regardless of the initial gridfrequency.2.5 SummaryThis chapter identifies a shortcoming in the synchronverter in that its APL lacks in controldegrees of freedom, and thus is unable to adjust its dynamic response speed freely duringnormal operation. To mitigate this problem, I propose to add a damping correction loop to thesynchronverter APL, which allows the response speed to be adjusted freely without affectingthe steady-state frequency droop characteristic. Moreover, the coupling between the APL andthe RPL is reduced when the APL is tuned to respond quickly.44Chapter 3Directly Computing Synchronverter Param-eters for Desired Dynamic ResponseWith the proposed synchronverter design with freely adjustable response speed in Chapter 2,this chapter further proposes a method to directly compute its controller parameter values fordesired transient and steady-state response during normal operation. The proposed approach isgrounded in a reduced third-order system model that captures pertinent dynamic characteristicsof the synchronverter active-power loop (APL), particularly those of the dominant mode. Thisreduced-order model helps to identify and explain a shortcoming in a previous parameter tuningmethod based on the small-signal analysis. Central to the proposed parameter computationmethod is to express APL parameters of the original system as closed-form functions of the polesof the reduced-order system. Since the reduced-order model retains dominant-mode dynamicbehaviours of the original system, APL parameters can be directly computed according tospecified APL dominant mode.In this chapter, Section 3.1 identifies a shortcoming in tuning the synchronverter parameterswith small-signal analysis. In Section 3.2, I develop and adopt a criterion to explain the high-lighted shortcoming. Then, in order to avoid the shortcoming in conventional tuning method,in Section 3.3, I further propose the direct parameter computation method. Through extensivecase studies, Sections 3.4 and 3.5 validate the proposed criterion and the direct computationmethod. Finally, Section 3.6 provides concluding remarks and directions for future research.453.1 Problems in Parameter Tuning of SynchronverterRecall that in Chapter 2, I append the damping correction loop to the synchronverter so thatits response speed can be adjusted without violating the frequency regulation requirement. Totune its APL parameters, we usually adopt an iterative method based on small-signal analysis asdone in Section 2.3, in which system eigenvalues are checked repeatedly with different parametervalues in an effort to place the APL dominant poles at desired locations. However, I find thatthe damping correction loop parameter influences the APL dominant model differently underdifferent conditions, which introduces uncertainty to the iterative tuning process. Below, wemotivate the need to improve the parameter tuning method that is based on the small-signalanalysis in Section 2.3.Based on the discussion in Section 2.3, we note that the synchronverter APL dynamics aremainly influenced by parameters Jg, Dp, Df , and τf , which is also evident from (2.40) and (2.56).Bear in mind that among these parameters, Dp is set according to the local frequency regulationrequirements, and τf is set to ensure that the LPFs have desired filtering abilities. As presentedin Section 2.3, remaining parameters Jg and Df are tuned by performing small-signal analysison a linearized state-space model (2.66) and (2.67). That is, we repetitively compute the systemeigenvalues and do trial and error with different parameter values until the synchronverter hassatisfactory dynamic performance. We refer to this parameter tuning method as iterative tuningmethod.Among the eigenvalues of the state matrix A in (2.40), denoted by λk (k = 1, ..., 7), λ2 =−α + jβ and λ3 = −α − jβ represent the APL dominant mode, which is tuned by varyingparamters Jg and Df . Via a numerical example below, I show that under two different operatingconditions, Df influences the APL dominant mode in distinct ways.Example 2 (Impact ofDf on the APL Dominant Mode). In this example, for the synchronverter-connected system in Fig. 3.1, which combines the proposed controller in Fig. 2.1(c) and thegrid interface in Fig. 2.4, I adjust the APL dynamic response, which is mainly governed bythe APL dominant mode, by varying parameter Df . To highlight the impact of varying Df onthe dominant mode under different operating conditions, I consider cases corresponding to twovalues of Dp, which may differ depending on different grid codes at different areas [14]. Values46Figure 3.1: Synchronverter augmented with damping correction loop. The synchronverterparameters are usually tuned via an iterative method, which requires repeated computation ofsystem eigenvalues and onerous trial-and-error effort. The parameter tuning method proposedin this chapter avoids these shortcomings.for other system parameters are adopted as reported in Table 2.1. As shown in Fig. 3.2(a),if Dp = 1407 N ·m · s/rad, increasing Df from −4.0 to −2.5 V · s2/rad causes the dampingratio ζ of the APL dominant mode, represented by λ2 and λ3, to increase from 0 to 1. On theother hand, the natural frequency ωn of the APL dominant mode is minimally affected. Thiscase is in accordance with expectations, since by omitting LPF dynamics, we have the followingexpressions according to (2.46) and (2.47),ζ =α√α2 + β2∝ 1√Jg(Dp +Df√32U∞ cos θog∞Xt), (3.1)ωn =√α2 + β2 ∝ 1√Jg. (3.2)Particularly, Df adjusts ζ freely without impacting ωn. However, if Dp = 0 N ·m · s/rad, as47Figure 3.2: Parameter Df influences APL dominant mode (represented by eigenvalues λ2and λ3) differently depending on the operating condition. This results in more trial-and-error effort to tune synchronverter parameters via the iterative method based on the linearizedstate-space model in (2.66) and (2.67). (a) With Dp = 1407, Df increases from −6.5 to 1.(b) With Dp = 0, Df increases from −1.2 to 2.shown in Fig. 3.2(b), Df does not tune ζ freely in the range (0, 1), and it also influences ωnsignificantly. This case does not match the relationships in (3.1) and (3.2). Problem Statement: As revealed in Example 2, varying the parameter Df influences theAPL dominant mode differently under different operating conditions. Note that Jg influencesboth ζ and ωn, as shown in (3.1) and (3.2). Thus, in the first case with Dp = 1407 N ·m · s/rad,since Df mainly influences ζ, we can tune Jg and Df independently according to the desired ωnand ζ via the iterative tuning process. In the second case with Dp = 0, however, ωn and ζ areboth influenced by Df and cannot be adjusted independently using pararameters Jg and Df .In this case, (3.1) and (3.2) contradict with the practical results, and further investigation isneeded to explain the eigenvalue variation pattern in Fig. 3.2(b). Since the pattern in which Dfaffects eigenvalues is uncertain, the iterative tuning method based on (2.66) andd (2.67) lead tomore trial-and-error work to achieve desired APL dominant-mode behaviour. Next, I developa criterion to differentiate and predict the eigenvalue variation patterns due to changes in Dffor different values of Dp.483.2 Explanations for Problems in Parameter TuningThis section develops a criterion to explain and predict the impact of changes in parameter Dfon eigenvalue variation patterns, specifically those of eigenvalues corresponding to the APLdominant mode. Here, instead of relying on the full linearized model in (2.66) and (2.67), Idevelop a third-order linearized APL model that captures pertinent system dynamic and steady-state behaviours. Such a reduced-order model is satisfactory when Xt  Rt, which leads todecoupled APL and RPL dynamics [14]. In this case, the roots of the characteristic equation ofthe third-order APL model accurately approximate the eigenvalues λ2 and λ3, which representthe APL dominant mode obtained from the original linearized model in (2.66) and (2.67).Below, I first describe the third-order linearized APL model, and then based on this model,develop the criterion and explain the phenomenon in Example 2.3.2.1 Transfer Function Modeling of Synchronverter APLInspired by [42], here, I find that the key to ensuring the reduced-order model captures pertinentcharacteristics of the full-order model is to include LPF dynamics for the signal Te in additionto the rotor-angle dynamics. Since the APL transfer-function model in (2.42)–(2.47) neglectsthe LPF dynamics, it does not model the phenomenon uncovered in Example 2, and so cannotbe used for accurate parameter tuning. With this in mind, we start from (2.33) and (2.34) inlast chapter, i.e.,ωg =dθg∞dt+ ω∞,dωgdt=d2θg∞dt2+dω∞dt. (3.3)Then, omitting the RPL dynamics in (2.55) and (2.57)–(2.59), setting ψff = ψf = ψ◦f , andassuming that ω?g remains unchanged (∆ω?g = 0), we substitute (3.3) into the swing equa-tion (2.40) and linearize the resultant around the equilibrium point x◦ to get the followingsmall-signal model:Jg(d2∆θg∞dt2+d∆ω∞dt)=∆P ?tωN−∆Tef −Dp(d∆θg∞dt+ ∆ω∞)− Dfψ◦fd∆Tefdt. (3.4)49Next, in order to account for the LPF dynamics for Te, we substitute (2.13) into (2.56), andthen linearize (2.56) around x◦ to getτfd∆Tefdt= −∆Tef +√32ψ◦fU∞ cos θ◦g∞Xt∆θg∞. (3.5)By taking the Laplace transformation of (3.4) and (3.5), and solving them for ∆θg∞, weget∆θg∞ =(τfs+ 1) (∆P?t − ωN (Jgs+Dp)∆ω∞)τfJgωN · (s3 + bs2 +Ks+ d) , (3.6)whereb =Jg + τfDpτfJg, (3.7)K =1τfJg(Dp +Df√32U∞ cos θ◦g∞Xt), (3.8)d =√32ψ◦fU∞ cos θ◦g∞τfJgXt, (3.9)Finally, linearize (2.13) and substitute (3.6) into the resultant (2.13), we have the third-orderAPL model as follows:∆Pt =d · (τfs+ 1) · (∆P ?t − ωN (Jgs+Dp)∆ω∞)s3 + bs2 +Ks+ d=: G1(s)∆P?t +G2(s)∆ω∞, (3.10)The characteristic equation of the model in (3.10) iss3 + bs2 +Ks+ d = 0. (3.11)Note that if the LPF dynamics were neglected as in (2.42)–(2.47), we would obtain a second-order model with lower accuracy. Next, I conduct root locus analysis for (3.11), which helps toestablish a criterion to predict the pattern in which Df affects the roots of (3.11), and in turn,the APL dominant mode of the full model in (2.66) and (2.67).503.2.2 Criterion for Different Eigenvalue Variation PatternsBy performing root locus analysis on the characteristic equation in (3.11), I evaluate the in-fluence of varying Df on the APL dominant mode under different operating conditions. Theeigenvalue variation patterns resulting from varying Df can be categorized into three differenttypes, depending on the number of breakaway points (0, 1, or 2) in the root loci.The root locus analysis begins by rewriting (3.11) as1 +KG(s) = 0, (3.12)whereG(s) =ss3 + bs2 + d. (3.13)According to (3.8), K is a linear function of Df , so variations in Df and K produce the sametrends on the root loci of (3.12). Moreover, according to (3.11), K > 0 is a necessary conditionfor the system in (3.10) to be stable. Thus, I increase K from 0 to +∞ (by varying Df ), anddetermine the root loci [82]. Please refer to Remark 9 for details of the root locus analysis,which is summarized as Fig. 3.3.Based on the analysis in Remark 9, I find that as Df varies, the root loci of 1 +KG(s) = 0have three different types of patterns depending on the value ofγ :=133√√23Xtψ◦fU∞ cos θ◦g∞(3√Jgτ2f+Dp 3√τfJ2g). (3.14)These cases are summarized as follows:(i) if γ > 1, as shown in Fig. 3.3(a), the root loci of 1 + KG(s) = 0 have two breakawaypoints δ1 and δ2,(ii) if γ = 1, as shown in Fig. 3.3(b), the root loci of 1 + KG(s) = 0 have one breakawaypoint δ1 = δ2,(iii) if 0 < γ < 1, as shown in Fig. 3.3(c), the root loci of 1 + KG(s) = 0 have no breakawaypoints.51Figure 3.3: Different root loci patterns of 1+KG(s) = 0 in the s-plane with (a) γ > 1, (b) γ = 1,and (c) 0 < γ < 1.Note that only if γ ≥ 1 can the damping ratio of the APL dominant mode be tuned freelyin the range (0, 1) by varying only Df . In the remainder of the dissertation, I refer to theseconclusions on γ as the γ-criterion. As Df varies, the two branches of the root loci beginningat p2 and p3 correspond to the trajectories of λ2 and λ3 in the eigenvalue variation patterns,52which can be verified using (2.66) and (2.67). Thus, we can readily use the γ-criterion developedfrom the third-order model to predict the eigenvalue variation patterns of λ2 and λ3 due tochanges in Df .Remark 9 (Proof of Proposed γ-Criterion via Root Locus Analysis). In this remark, I provethe proposed γ-criterion by conducting the root locus analysis, as follows:(1) Start and end points: The root loci of (3.12) have three branches. They start from theopen-loop poles of G(s) in (3.13), i.e., the three roots of the equations3 + bs2 + d = 0. (3.15)One branch ends at the open-loop zero of G(s), i.e., z1 = 0, and the other two branchesend at ∞. We notice that since b > 0 and d > 0, the discriminant ∆1 of the third-orderpolynomial in (3.15) is [83]∆1 = −4b3d− 27d2 < 0. (3.16)Thus, (3.15) has one real root, denoted by p1, and a pair of complex-conjugate poles,denoted by p2 and p3 (p2p3 ∈ R and p2p3 ≥ 0). According to Vieta’s formulas [84],−b = p1 + p2 + p3, (3.17)0 = p1p2 + p2p3 + p1p3, (3.18)−d = p1p2p3. (3.19)From (3.19), p1 = −d/(p2p3) < 0, and from (3.18), Re(p2) = Re(p3) = (p2 + p3)/2 =−(p2p3)/(2p1) > 0. Thus, p1 is a negative real root, and the real parts of p2 and p3 are bothpositive. Note that the two branches starting from p2 and p3 provide good approximationsfor the trajectories of λ2 and λ3 obtained from (2.66).(2) Asymptotes of the root loci: Two of the branches are asymptotic to the lines emanatingfrom the point σa = (p1 + p2 + p3 − z1) /2 = −b/2, with angles ϕa = ±pi/2.53(3) Root locus segments on the real axis: The interval (p1, 0) is part of the root loci, sincethere is an odd number of poles and zeros, i.e., p2, p3, and z1 to its right side in the s-plane.(4) Angles of departure/arrival: Denote angles of departure from poles p1, p2, and p3as θp1, θp2, and θp3, respectively, then by applying the angle criterion, we have θp1 = 0,θp2 ∈ (pi/2, pi), and θp3 ∈ (−pi,−pi/2). Further denote the angle of arrival at the zero z1as ϕz1, we get ϕz1 = pi.(5) Intersection of the root loci with the imaginary axis: By substituting s = jωinto (3.11), and solving it for ω and K, we know that the root loci of (3.12) intersects theimaginary axis at s = jω = ±j√d/b when K = d/b.(6) Breakaway points: These are determined by finding the roots ofdKds=dds(− 1G(s))=dds(−s3 + bs2 + ds)= 0, (3.20)which boils down to2s3 + bs2 − d = 0. (3.21)The roots of (3.21) are denoted by, respectively, δ1, δ2, and δ3 with assumption that Re(δ1) ≤Re(δ2) ≤ Re(δ3). According to Vieta’s formulas [84]− b2= δ1 + δ2 + δ3, (3.22)0 = δ1δ2 + δ2δ3 + δ1δ3, (3.23)d2= δ1δ2δ3. (3.24)Define (which is equivalent to (3.14))γ :=b3 3√d, (3.25)then the discriminant ∆2 of (3.21) is [83]∆2 = 4b3d− 108d2 = 108d2 (γ3 − 1) , (3.26)54Figure 3.4: Image of function g(x) = x3 + bx2 + d.and the sign of ∆2 is determined by the relationship between γ and 1.(i) If γ > 1, then ∆2 > 0, and (3.21) has three distinct real roots δ1, δ2, and δ3 (δ1 <δ2 < δ3). In this case, I can further show thatp1 < δ1 < δ2 < 0 < δ3, (3.27)i.e., (3.21) has two negative real roots δ1 and δ2 in the range (p1, 0), and one positivereal root δ3 /∈ (p1, 0). Since only the interval (p1, 0) is part of the root loci on the realaxis, the root loci of (3.12) have two distinct breakaway points δ1 and δ2.We note that (3.27) can be determined by proving that δ1 < δ2 < 0 < δ3 and p1 < δ1.First, according to (3.22), at least one root of (3.21) is negative. Further con-sider (3.24), we know that (3.21) has two negative roots and one positive root, i.e.,δ1 < δ2 < 0 < δ3. Then, in order to prove that p1 < δ1, I define functiong(x) = x3 + bx2 + d, (3.28)in which x denotes the function variable. Since p1 is the negative real root of (3.15),we have g(p1) = 0, and the image of g(x) is shown in Fig. 3.4. Since δ1 is the rootsof (3.21), we know that 2δ31 + bδ21 − d = 0. With this, we haveg(δ1) = δ31 + bδ21 + d=12(2δ31 + bδ21 − d)+12(3d+ bδ21)> 0. (3.29)55According to the image of the function g(x) in Fig. 3.4, we have p1 < δ1.(ii) If γ = 1, then ∆2 = 0, and (3.21) has a pair of repeated roots and a real root. In thiscase, I can further show thatp1 < δ1 = δ2 < 0 < δ3, (3.30)i.e., (3.21) has repeated negative roots δ1 and δ2 in the range (p1, 0), and a positivereal root δ3 /∈ (p1, 0). Since only the interval (p1, 0) is part of the root loci on the realaxis, the root loci of (3.12) have one breakaway point δ1 = δ2.(iii) If 0 < γ < 1, then ∆2 < 0, and (3.21) has a pair of complex conjugate roots and areal root. In this case, I can further show thatRe(δ1) = Re(δ2) < 0 < δ3, (3.31)i.e., δ3 /∈ (p1, 0), and thus the root loci of (3.12) have no breakaway points. .With this I complete the proof of the proposed γ-criterion in (3.14). Example 3 (Differentiating Eigenvalue Variation Patterns). The proposed γ-criterion can beused to explain the different eigenvalue variation patterns resulting from varying Df in Ex-ample 2. In the first case where Dp = 1047 N ·m · s/rad, we have that γ = 3.58 > 1 (i.e.,Fig. 3.3(a)). Indeed, as shown in Fig. 3.2(a), varying only Df adjusts the damping ratio of theAPL dominant mode freely in the range (0, 1). On the other hand, in the case where Dp = 0,we have that γ = 0.60 ∈ (0, 1) (i.e., Fig. 3.3(c)). Accordingly, as shown in Fig. 3.2(b), tuningonly Df cannot adjust the damping ratio of the APL dominant mode freely in the range (0, 1).Moreover, as predicted in Fig. 3.3(c), the root loci go to ∞ along the asymptotes as K (or Df )increases. Thus, in this case, variations in Df also significantly influence ωn. With the proposed γ-criterion in place, we are able to predict the effects of Df variationson eigenvalues of (2.66) corresponding to the APL dominant mode under different operatingconditions. In the next section, in order to avoid the shortcoming of the iterative tuning methodbased on (2.66), I propose a method to compute APL parameters Jg and Df directly to satisfy56Figure 3.5: (a) Desired APL pole locations in the s-plane used to directly compute parameters Jgand Df . (b) Desired APL time-domain response corresponding to the desired APL dominantpoles λ?2 and λ?3.prescribed damping ratio and natural frequency requirements.3.3 Proposed Parameter Tuning MethodIn this section, I propose a direct computation method to obtain the APL parameters Jg and Dffor given APL dominant mode requirements, regardless of the value of γ. Using this method,we avoid the trial-and-error process in tuning the APL parameters, which plagued the iterativemethod in Section 3.1. First, denote ζ? and ω?n as, respectively, the required damping ratio andnatural frequency for the APL dominant mode. Also denote ϕ? = arccos ζ? ∈ (0, pi/2). Then,as depicted in Fig. 3.5(a), the desired APL dominant pole locations areλ?2 = −ω?n cosϕ? + jω?n sinϕ?, (3.32)λ?3 = −ω?n cosϕ? − jω?n sinϕ?. (3.33)57In other words, my goal is to achieve the APL time-domain response corresponding to thefollowing transfer function when varying the active-power reference value P ?t∆Pt(s)∆P ?t (s)=ω?n2s2 + 2ζ?ω?ns+ ω?n2 =: G?1(s). (3.34)Using the inverse Laplace transformation, the desired APL dynamic response is indeed∆Pt(t) = L−1{G?1(s)∆P ?t (s)}= ∆P ?t(1− e−ζ?ω?nt√1− ζ?2sin(ω?n√1− ζ?2 t+ ϕ?)), (3.35)which has settling time ts ≈ 4/(ζ?ω?n), as shown in Fig. 3.5(b) [82]. The proposed directcomputation method leverages the fact that the poles of the third-order system in (3.10) closelyapproximate the APL dominant mode, which is represented by λ2 and λ3 (eigenvalues of thefull linearized system in (2.66) and (2.67)). In this way, I translate the problem of achievingdesired pole locations for the full system in (2.66) and (2.67) to one for the reduced third-ordersystem in (3.10). To this end, denote poles of (3.10) (or the roots of (3.11)) as s1, s2, and s3.By tuning Jg and Df , I wish to place two complex-conjugate roots of the characteristic equationin (3.11) at the locations of λ?2 and λ?3 in the s-plane, so we set s2 = λ?2 and s3 = λ?3. Furtherset s1 = −α1 (α1 > 0), which is an unspecified real-valued root of (3.11). Then, according toVieta’s formulas [84], for (3.11), we have−b = s1 + s2 + s3 = −α1 − 2ω?n cosϕ?, (3.36)K = s1s2 + s2s3 + s1s3 = 2α1ω?n cosϕ? + ω?n2, (3.37)−d = s1s2s3 = −α1ω?n2, (3.38)where the second equality in each of (3.36)–(3.38) results by substituting s2 = λ?2 and s3 = λ?3from (3.32) and (3.33), as well as s1 = −α1. Recall that b, K, and d are functions of Jg and Dfaccording to (3.7)–(3.9), so α1, Jg, and Df are the three unknown variables in (3.36)–(3.38) for58which to be solved. We first obtain from (3.38) thatα1 =dω?n2 . (3.39)Then, by substituting (3.39) into (3.36) and (3.37), and then solving them for Jg and Df , weget the following closed-form expressions for tuneable controller parameters Jg and Df :Jg =√32ψ◦fU∞ cos θ◦g∞ − τfDpXtω?n2ω?n2Xt(1− 2τfω?nζ?), (3.40)Df =2ψ◦fζ?ω?n+τfψ◦f1− 2τfω?nζ?−√23XtDpU∞ cos θ◦g∞(1 +τf2ω?n21− 2τfω?nζ?), (3.41)which achieve the desired pole locations as defined in (3.32) and (3.33). Note that Jg and Dfcan be recomputed easily for different operating points. In fact, through extensive simulations,I find that these parameters do not vary significantly over the range of normal operating points.After obtaining the APL parameters Jg and Df with (3.40) and (3.41), we verify that s1 <Re(s2). This condition ensures that s2 and s3 represent the APL dominant mode as shownin Fig. 3.5. If s1 ≥ Re(s2), we need to choose a different ω?n and recompute Jg and Df . Thecondition on ωn will be analyzed in detail in next chapter.Remark 10 (Effective Inertia and Damping Constants). Due to the effects of the dampingcorrection loop in the modified APL swing equation in (2.40) and the LPF dynamics, Jg alonedoes not fully capture the synchronverter inertia characteristic. In this remark, I characterizethe effective synchronverter inertia and damping. With Jg andDf chosen as in (3.40) and (3.41),respectively, and assuming that ∆ω∞ = 0, (3.6) can be expressed as∆θg∞ =(τfs+ 1) ∆P?tτfJgωN · (s− s1)(s− s2)(s− s3)=(τfs+ 1) ∆P?tτfJgωNα1(1α1s+ 1)(s2 + 2ζ?ω?ns+ ω?n2) . (3.42)In practice, α1  1, and moreover, the dynamics associated with the pole s1 = −α1 are fasterthan those associated with poles s2 and s3, so here, I approximate1α1s + 1 ≈ 1 in (3.42) [82].Also, it turns out that by choosing τf to satisfy noise rejection requirements, 0 < τf  1 and59the zero −1/τf is sufficiently far away from the APL dominant poles, so its effect on the time-domain response is small, and I approximate τfs+1 ≈ 1 in (3.42) [82]. With these assumptionsin place, inverse Laplace transformation of (3.42) yieldsd2∆θg∞dt2=∆P ?tτfJgωNα1− 2ζ?ω?nd∆θg∞dt− ω?n2∆θg∞. (3.43)Next, to incorporate ∆Te into (3.43), linearize (2.31) with respect to θg∞ to get∆Te =√32ψ◦fU∞ cos θ◦g∞Xt∆θg∞. (3.44)Also, since ∆ω∞ = 0, we linearize (3.3) and obtain∆ωg =d∆θg∞dt,d∆ωgdt=d2∆θg∞dt2. (3.45)Solving for ∆θg∞ in (3.44) and substituting the resultant along with (3.45) and (3.39) into (3.43),we get thatJeffd∆ωgdt= ∆Tm −∆Te −Deff ∆ωg, (3.46)where ∆Tm = ∆P?t /ωN , the effective inertia constant isJeff =√32ψ◦fU∞ cos θ◦g∞Xt·(1ω?n)2, (3.47)and the effective damping constant isDeff =√6ψ◦fU∞ cos θ◦g∞ ·(ζ?ω?n). (3.48)Note that, according to (3.47), Jeff ∝ (1/ω?n)2 and Jeff is independent of the desired APLdamping ratio ζ?. Thus, in my proposed method, by specifying the natural frequency ω?n forthe APL dominant poles, we conveniently achieve the desired inertia characteristic. We alsonote that the effective inertia Jeff may be larger, smaller, or the same as the inertia constant Jg,depending on the system parameters, operating point, and desired time-domain performance.Once the desired Jeff is satisfied, the desired effective damping constant Deff can be achieved60by specifying ζ?, in accordance with (3.48). 3.3.1 Summary of Proposed Parameter Tuning MethodBased on the discussion above, I propose to tune the synchronverter APL parameters Dp, τf ,Jg, and Df as follows. First, Dp is specified to satisfy local grid frequency-droop requirements,which may differ in different geographical locations. Next, we choose τf based on the LPF noiserejection requirements. Then, according to the desired ω?n and ζ? of the APL dominant mode,we compute Jg and Df using (3.40) and (3.41). Finally, with the chosen parameter values, weverify that s1 < Re(s2). Note that the RPL parameters can still be chosen based on the methodproposed in [1]. In the next section, I validate the proposed γ-criterion in Section 3.2 and thedirect computation method in this section.3.4 Case Studies in Single-synchronverter Infinite-bus SystemIn this section, I first verify that two complex-valued roots of the characteristic equation for thethird-order model closely approximate eigenvalues λ2 and λ3 of the full linearized model in (2.66)and (2.67), which represent the APL dominant mode. I also show that the proposed γ-criterionaccurately predicts the eigenvalue variation patterns resulting from varying Df . Then, viaseveral case studies, I verify that the proposed direct computation method is highly effective inobtaining the APL parameters and placing the eigenvalues λ2 and λ3 at their respective desiredlocations. The system under study is shown in Fig. 3.1, with the parameter values reported inTable 2.1, unless otherwise specified.3.4.1 Verifying the Reduced-order Model and the γ-CriterionIn this case study, I vary Df with Dp taking values of 1407.0, 190.25, and 0 N ·m · s/rad,and correspondingly, Figs. 3.6(a)–(c) show the trajectories of the APL characteristic equationroots si (i = 1, 2, 3) and the state matrix A eigenvalues λk (k = 1, ..., 7). In Fig. 3.6, si and λkare, respectively, marked with ◦ and ×, and the root loci of 1 + KG(s) = 0 are indicatedby the solid lines. Note that γ > 1 when Dp > 190.25 N ·m · s/rad, γ = 1 when Dp =190.25 N ·m · s/rad, and γ < 1 when Dp < 190.25 N ·m · s/rad,.Based on visual inspection of Fig. 3.6, we make two observations. First, the roots s2 and s361Figure 3.6: Verification of the reduced third-order APL model and the proposed γ-criterion inthe s-plane, i.e., varying Df influences the APL damping ratio differently when γ takes valuesin different ranges. Particularly, only when γ ≥ 1 can the APL damping ratio be adjustedfreely in the range (0, 1). (a) γ = 3.58, −6.2 ≤ Df ≤ 0.7 (Jg = 2.814, Dp = 1407.0, τf = 0.01).(b) γ = 1.00, −1.6 ≤ Df ≤ 0.5 (Jg = 2.814, Dp = 190.25, τf = 0.01). (c) γ = 0.60,−0.9 ≤ Df ≤ 0.7 (Jg = 2.814, Dp = 0, τf = 0.01).62Table 3.1: Direct Computation of APL Parameters Jg and DfDesired λ?2 and λ?3 Jg Df Actual λ2 and λ3 Error 1 −9.239± j3.827 57.86 2.221 −9.380± j4.076 2.86%2 −7.071± j7.071 54.94 1.602 −7.194± j7.057 1.24%3 −3.827± j9.239 51.08 0.6781 −3.952± j9.188 1.36%4 −18.48± j7.654 16.44 0.9433 −18.31± j7.801 1.11%5 −14.14± j14.14 14.45 0.6154 −14.27± j13.99 0.982%6 −7.654± j18.48 12.24 0.1334 −7.929± j18.41 1.42%7 −27.72± j11.48 7.965 0.5269 −27.34± j11.24 1.49%8 −21.21± j21.21 6.166 0.2770 −21.57± j20.82 1.78%9 −11.48± j27.72 4.608 −0.06764 −12.08± j27.71 1.98%Figure 3.7: Verification of the proposed direct computation method achieving (a) accurate APLpole placement in the s-plane. (b) desired APL dynamic response in the time domain.of (3.11) and the eigenvalues λ2 and λ3 of the state matrix A in (2.66) are well matched. Thus,the reduced third-order APL model in (3.10) is sufficiently accurate, and its characteristicequation roots s2 and s3 provide satisfactory approximations for λ2 and λ3, which representthe APL dominant mode. Moreover, with γ ≥ 1 (as shown in Figs. 3.6(a)(b)), the trajectoriesof s2 and s3 (and those of λ2 and λ3) have at least one breakaway point in the left half s-plane.Otherwise, when 0 < γ < 1 (as shown in Fig. 3.6(c)), the trajectories of s2 and s3 (and also λ2and λ3) have no breakaway points in the left half-plane. In other words, varying Df adjuststhe damping ratio of the APL dominant mode freely in the range (0, 1) only when γ ≥ 1. Thisagrees well with the root locus analysis results summarized in Fig. 3.3. Thus, I conclude thatthe proposed γ-criterion indeed accurately predicts the eigenvalue variation patterns arisingfrom varying Df .633.4.2 Validating the Proposed Parameter Computation MethodIn this case study, via comparison with the iterative tuning method in Section 3.1, I demonstratethe accuracy and effectiveness of the proposed direct computation method. The comparisonsare summarized in Table 3.1, in which (i) the natural frequency ω?n of the desired APL dominantmode in rows 1–3, rows 4–6, and rows 7–9 are, respectively, 10, 20, and 30; and (ii) the dampingratio ζ? of the desired APL dominant mode in rows (3m+1), rows (3m+2), and rows (3m+3)are, respectively, 0.924, 0.707, and 0.383 (m = 0, 1, 2). Based on these desired characteristics,I compute the APL parameters Jg and Df using (3.40) and (3.41). Next, we substitute theresulting Jg and Df into the state matrix A, and compute the eigenvalues λ2 and λ3. I alsoobtain the relative error  between the desired λ?2 and the actual λ2 according to =‖λ?2 − λ2‖‖λ?2‖× 100%. (3.49)The resulting Jg, Df , λ2, λ3, and  for all cases are reported in Table 3.1, and both the desired λ?2and the actual λ2 are plotted in Fig. 3.7(a) for comparison. By graphically comparing λ?2 and λ2,we find that they are well matched. Moreover, their relative errors  are all less than 3%. Thisis also true for λ?3 and λ3. Thus, we conclude that, with the proposed method, we can directlycompute Jg and Df to effectively place the eigenvalues λ2 and λ3, which represent the APLdominant mode, at their desired locations λ?2 and λ?3 with minimal error.Via time-domain simulations, I show that the minor errors between actual and desiredeigenvalues do not cause noticeable differences in the APL dynamic response. We take casescorresponding to rows 2 and 8 in Table 3.1 as examples. Using the iterative tuning method,we find that when Jg = 55.66 kg ·m2 and Df = 1.597 V · s2/rad, λ2 and λ3 are exactlyequal to desired λ?2 and λ?3 in row 2 (Case B1). Via a similar method, when Jg = 6.081 kg ·m2and Df = 0.2627 V · s2/rad, λ2 and λ3 are exactly equal to desired λ?2 and λ?3 in row 8 (Case B2).I model the synchronverter-connected system as shown in Fig. 3.1 in the PSCAD/EMTDC.At t = 0.1 s, the active-power reference value P ?t increases from 0 MW to 0.6 MW. Figure 3.7(b)shows the time-domain simulation results when I adopt 4 different set of parameters Jg and Df .Cases A1 and A2 use the APL parameters computed from (3.40) and (3.41) in rows 2 and 8,respectively, of Table 3.1. Cases B1 and B2 adopt the APL parameters obtained with the64Figure 3.8: 6-bus test system used to verify the proposed parameter tuning method.iterative tuning method. As shown in Fig. 3.7(b), traces (a1) and (b1), which correspond toCases A1 and B1, respectively, are nearly identical. This is also true for traces (a2) and (b2),which correspond to Cases A2 and B2. Thus, we conclude that the APL parameters obtainedfrom the proposed direct computation method achieve desired APL dynamic response.In this section, I verify the proposed γ-criteria and the proposed parameter computationmethod. The γ-criteria explains the different eigenvalue patterns when varying Df with high ac-curacy, and the direct computation method directly computes Jg and Df that satisfy prescribeddamping ratio and natural frequency characteristics, which leads to desired APL time-domainresponse. We note that the direct computation method is verified with a single-synchronverterinfinite-bus system, which does not consider the dynamics of the external grid. In the nextsection, I apply my proposed direct computation method in actual power system and furthervalidate its efficacy.3.5 Application of Proposed Parameter Computation Methodin Power SystemTo further validate the proposed method, I implement it to tune parameters of a synchronverterconnected to a six-bus system with one-line diagram shown in Fig. 3.8. In this system. asynchronverter and two SGs (SG1 and SG2) supply power, via three transformers (T1, T2, and65Figure 3.9: Modified Woodward governor used in SG1 and SG2 to achieve primary frequencyregulation [2].Figure 3.10: Excitation system used in SG1 and SG2 to control their terminal voltages [3].T3, respectively), to three constant-impedance loads (Load1, Load2, and Load3) in the system.Generators SG1 and SG2 are each equipped with a Woodward governor (see Fig. 3.9) to achieveprimary frequency control and a standard excitation system (see Fig. 3.10) to regulate terminalvoltages. The high-voltage sides of transformers T1, T2, and T3 are, respectively, connected tobuses 4–6. Three transmission lines (Line45, Line46, and Line56) connect buses 4–6, forming aring structure. Parameter values for all components in this system are reported in Appendix B.Note that I omit the capacitance values of Line45, Line46, and Line56, since they have littleinfluence on the simulation results here.With the test system in place, I tune and verify parameters of the synchronverter via thefollowing procedure:(1) Network reduction: Obtain an infinite-bus equivalent of the external system (highlightedin green in Fig. 3.8) as seen from the synchronverter using network reduction, and obtaina single-synchronverter infinite-bus system.(2) Parameter computation: Compute synchronverter parameters with the method pro-posed in Section III-B of the revised manuscript based on the equivalent external system66obtained from the first step.(3) Simulation verification: Verify the synchronverter dynamic response against the actualexternal system.In the remainder of this section, I present the detailed parameter tuning procedure.3.5.1 Obtaining an Infinite-bus Equivalent for the External SystemAs shown in Fig. 3.11(a), I represent SG1 and SG2, respectively, with voltage sources usg1and usg2, since their terminal voltages are regulated by their excitation systems. I also labelthe 6 buses in the six-bus system as nodes 1–6 as shown in Fig. 3.11(a). Then I convert allcomponent impedance values with respect to the 6.6-kV voltage level as follows:Zs = Rs + jXs = 0.741 + j7.54 Ω, (3.50)XT1 = 0.100× (6.60× 103)21.50× 106 = 2.904 Ω, (3.51)XT2 = 0.100× (6.60× 103)26.00× 106 = 0.726 Ω, (3.52)XT3 = 0.100× (6.60× 103)26.00× 106 = 0.726 Ω, (3.53)Z45 = (0.150 + j1.47)×(6.60× 10313.8× 103)2= 0.0343 + j0.336 Ω, (3.54)Z46 = (0.100 + j0.980)×(6.60× 10313.8× 103)2= 0.0229 + j0.224 Ω, (3.55)Z56 = (0.100 + j0.980)×(6.60× 10313.8× 103)2= 0.0229 + j0.224 Ω, (3.56)ZL1 =(6.60× 103)2(3.00− j0.600)× 106 = 13.962 + j2.792 Ω, (3.57)ZL2 =(6.60× 103)2(3.00− j0.600)× 106 = 13.962 + j2.792 Ω, (3.58)ZL3 =(6.60× 103)2(4.00− j0.500)× 106 = 10.723 + j1.340 Ω. (3.59)Let vectors I and U collect, respectively, the current-injection and nodal-voltage phasors atnodes 1–6 in the external system marked in green in Fig. 3.11(a), i.e.,I =[I1, I2, I3, I4, I5, I6]T, (3.60)67Figure 3.11: Obtaining the infinite-bus equivalent for the external system in the six-bus testsystem used to verify the proposed parameter tuning method. (a) Converting all componentimpedance values of the 6-bus test system in Fig. 3.8 with respect to the 6.6 kV voltage level.(b) Eliminating nodes 4-6 from the test system via kron reduction. (c) Merging nodes 2 and 3and obtaining the infinite-bus equivalent for the external system.U =[U1, U2, U3, U4, U5, U6]T. (3.61)68Note that U1, U2, and U3 are, respectively, the phasor representations of terminal voltages ofthe synchronverter, SG1, and SG2, the time-domain signals of which are, respectively, ut, usg1,and usg2. Further let Y denote the nodal admittance matrix of the six-bus system, then wehaveI = Y U , (3.62)i.e.,I1I2I3I4I5I6=y11 y12 y13 y14 y15 y16y21 y22 y23 y24 y25 y26y31 y32 y33 y34 y35 y36y41 y42 y43 y44 y45 y46y51 y52 y53 y54 y55 y56y61 y62 y63 y64 y65 y66U1U2U3U4U5U6=: YA YBYC YDU1U2U3U4U5U6, (3.63)wherey11 =1jXT1= −j0.344 S, (3.64)y14 = y41 = − 1jXT1= j0.344 S, (3.65)y22 =1jXT2= −j1.38 S, (3.66)y25 = y52 = − 1jXT2= j1.38 S, (3.67)y33 =1jXT3= −j1.38 S, (3.68)y36 = y63 = − 1jXT3= j1.38 S, (3.69)y44 =1jXT1+1Z45+1Z46+1ZL1= 0.820− j7.72 S, (3.70)y45 = y54 = − 1Z45= −0.300 + j2.94 S, (3.71)y46 = y64 = − 1Z46= −0.450 + j4.41 S, (3.72)y55 =1jXT2+1Z45+1Z56+1ZL2= 0.820− j8.75 S, (3.73)y56 = y65 = − 1Z56= −0.450 + j4.41 S, (3.74)69y66 =1jXT3+1Z46+1Z56+1ZL3= 0.993− j10.2 S, (3.75)and all remaining entries in the matrix Y are zeros. Note that the current injections at nodes 4–6are zeros, i.e., I4 = I5 = I6 = 0, so via Kron reduction [85], we get from (3.63) thatI1I2I3 = (YA − YBYD−1YC)U1U2U3 =: YkronU1U2U3 . (3.76)With this, as shown in Fig. 3.11(b), we eliminate nodes 4–6 from the external system. Next, Ifurther merge nodes 2 and 3, as done in [86], by assuming that the ratio between the SG2 andSG1 terminal voltages is complex number bu, i.e.,U3U2= bu. (3.77)Note that, following the method described in [86], we obtain that bu = 1.0 from the power flowsolution. Then we substitute U3 = buU2 into (3.76). Also, we define the equivalent currentinjection Ie associated with SG1 and SG2 asIe = I2 + b∗uI3, (3.78)where b∗u is the complex conjugate of bu. Using (3.78) to eliminate I2 and I3 from (3.76), weget that I1Ie = YU1U2 . (3.79)The entry Y (1, 2) (or Y (2, 1)) represents the equivalent admittance between nodes 1 and 2 inFig. 3.11(c)1. Let Ze denote the equivalent external grid impedance between nodes 1 and 2, asdepicted in Fig. 3.11(c), then we haveZe = − 1Y (1, 2). (3.80)1We refer to the entry in the m-th row and n-th column of matrix Y as Y (m,n).70Note that in (3.80), I neglect the node-to-ground admittances at nodes 1 and 2, i.e., Y (1, 1) +Y (1, 2) and Y (1, 2) +Y (2, 2), since their magnitudes are much smaller than those of the node-to-node admittance Y (1, 2) or Y (2, 1) between nodes 1 and 2. The resulting equivalent systemshown in Fig. 3.11(c) is highlighted in Fig. 3.11(c), where infinite-bus voltage is U∞ = Usg1 =1.03 p.u. In addition, as we consider a predominantly inductive network, I retain only thereactance Xe and neglect Re. In our test system, the equivalent reactance in the infinite-busequivalent for the external system isXe = Im (Ze) = Im(− 1Y (1, 2))= 3.45 Ω, (3.81)where Im(·) denotes the imaginary part of (·). So the total reactance is Xt = Xs+Xe = 11.0 Ω.3.5.2 Tuning Synchronverter Parameters with the Proposed MethodWith the infinite-bus equivalent (Fig. 3.11(c)) for the external system marked in green in Fig. 3.8in place, I tune the synchronverter parameters with my proposed method. For illustrationpurposes, I aim to achieve fast and slow APL time-domain dynamic response speeds, whichcorrespond to ω?n1 = 30 rad/s, ζ?1 = 0.707 and ω?n2 = 10 rad/s, ζ?2 = 0.707, respectively.In both cases, the sychronverter active- and reactive-power outputs are, respectively, Pt =0.6 MW and Qt = 0 MVar. With this assumption, I compute the equilibrium point x◦ forthe synchronverter-connected system in Fig. 3.11(c), the entries of which are ω◦g = 377 rad/s,θ◦g∞ = 0.142 rad, ψ◦f = ψ◦ff = 14.8 Wb, T◦ef = 1.59 × 103 N ·m, Qtf = 0.00 Var, and Utf =6.79 kV. The parameters Kg and Dq in the RPL are, respectively, set to be 27980 Var · rad/Vand 0 Var/V based on the tuning method in [1].I tune the APL parameters Dp, τf , Jg, and Df with my proposed method to ensure thatthe synchronverter responds (i) quickly and (ii) slowly. In both cases, I set Dp = 0 so that theAPL dynamics can be compared conveniently between the equivalent and actual systems bychanging the active-power reference value P ?t (bear in mind that in practice, Dp is determinedaccording to local grid codes). Also, I set τf = 0.01 s to reject measurement noise in feedbacksignals, e.g., the active-power output Pt. Then, I compute the values of parameters Jg and Dfwith (3.40) and (3.41).71(1) Synchronverter tuned to respond quickly (ω?n1 = 30 rad/s, ζ?1 = 0.707). In this case,parameters Jg and Df are evaluated to beJg1 =√32ψ◦fU∞ cos θ◦g∞ − τfDpXtω?n12ω?n12Xt(1− 2τfω?n1ζ?1 )= 21.4 kg ·m2, (3.82)Df1 =2ψ◦fζ?1ω?n1+τfψ◦f1− 2τfω?n1ζ?1−√23XtDpU∞ cos θ◦g∞(1 +τf2ω?n121− 2τfω?n1ζ?1)= 0.953 V · s/rad, (3.83)respectively. By substituting Jg = Jg1 and Df = Df1 back into the following APL charac-teristic equation (3.11), we find that its three roots ares1 = −57.6, s2 = −21.2 + j21.2, s3 = −21.2− j21.2. (3.84)Indeed, s1 < Re(s2) = Re(s3), i.e., s2 and s3 correspond to the APL dominant mode. Thus,with parameter values Jg1 and Df1, desired ω?n1 = 30 rad/s and ζ?1 = 0.707 are achieved.(2) Synchronverter tuned to respond slowly (ω?n2 = 10 rad/s, ζ?2 = 0.707). In this case,parameters Jg and Df are evaluated to beJg2 =√32ψ◦fU∞ cos θ◦g∞ − τfDpXtω?n22ω?n22Xt(1− 2τfω?n2ζ?2 )= 129 kg ·m2, (3.85)Df2 =2ψ◦fζ?2ω?n2+τfψ◦f1− 2τfω?n2ζ?1−√23XtDpU∞ cos θ◦g∞(1 +τf2ω?n221− 2τfω?n2ζ?2)= 2.26 V · s/rad, (3.86)respectively. By substituting Jg = Jg2 and Df = Df2 back into the APL characteristicequation in (3.11), we find that its three roots ares1 = −85.9, s2 = −7.07 + j7.07, s3 = −7.07− j7.07. (3.87)72Table 3.2: Cases I–IV used to verify synchronverter dynamic performance with actual externalsystem.Desired APLresponse speedω?n (rad/s) ζ? Hsg1 (s) Hsg2 (s)System inertialevelCase I fast 30.0 0.707 8.00 8.00 highCase II fast 30.0 0.707 3.00 3.00 lowCase III slow 10.0 0.707 8.00 8.00 highCase IV slow 10.0 0.707 3.00 3.00 lowIndeed, s1 < Re(s2) = Re(s3), i.e., s2 and s3 corresponds to the APL dominant mode.Thus, with parameter values Jg2 and Df2, desired ω?n2 = 10 rad/s and ζ?2 = 0.707 areachieved.Note that the direct computation of Jg and Df does not depend on the inertia constantof SG1 and SG2, i.e., Hsg1 and Hsg2. However, next, I verify my proposed tuning method toachieve desired dynamic performance for different values of Hsg1 and Hsg2.3.5.3 Verifying Synchronverter Dynamic Performance with ActualExternal SystemI simulate both the actual system in Fig. 3.8 with parameter values reported in Appendix Band the equivalent system in Fig. 3.11(c) in PSCAD/EMTDC. In Section 3.4.2, we observe onlyminor differences between the desired time-domain response and the time-domain results in theequivalent system. Thus, in order to promote readability of plots, here we interpret the resultsfrom the equivalent system as the desired response, and verify my proposed direct computationmethod by comparing time-domain response of synchronverter active-power output Pt in theactual and equivalent systems.We consider four cases as summarized in Table 3.2 to compare synchronverter dynamicperformance in the actual and equivalent external systems. In cases I and II, the synchronverteris tuned to respond quickly, but the system has high inertia in case I and low inertia in case II.In cases III and IV, the synchronverter responds slowly, but the system inertia level is high incase III and low in case IV. In cases I–IV, I increase the synchronverter active-power referencevalue P ?t from 0.0 MW to 0.6 MW at t = 1.0 s.(1) Synchronverter tuned to respond quickly (cases I and II). Simulation results for73Figure 3.12: Verification of the proposed direct computation method to achieve desired fastdynamic response in actual grid conditions (ω?n1 = 30 rad/s and ζ?1 = 0.707). (a)–(e) Time-domain response comparison between the actual and equivalent systems when Hsg1 = Hsg2 =8.0 s. (f)–(j) Time-domain response comparison between the actual and equivalent systemswhen Hsg1 = Hsg2 = 3.0 s. (a)(b)(f)(g) synchronverter active-power output Pt in the actualand equivalent systems. (c)(h) active power outputs of SG1 and SG2 Psg1 and Psg2 in theactual system. (d)(i) frequency of SG1 and SG2 fsg1 and fsg2 in the actual system. (e)(j) inputtorque of SG1 and SG2 Tsg,m1 and Tsg,m2 in the actual system.cases I and II are, respectively, shown in Figs. 3.12(a)–(e) and (f)–(j). We observe thatin both high- and low-inertia cases, there is little discrepancy between the synchronverteractive-power output Pt response in the actual (traces labelled as (i) in Figs. 3.12(a)(b)(f)(g))74Figure 3.13: Verification of the proposed direct computation method to achieve desired slowdynamic response in actual grid conditions (ω?n1 = 10 rad/s and ζ?1 = 0.707). (a)–(e) Time-domain response comparison between the actual and equivalent systems when Hsg1 = Hsg2 =8.0 s. (f)–(j) Time-domain response comparison between the actual and equivalent systemswhen Hsg1 = Hsg2 = 3.0 s. (a)(b)(f)(g) synchronverter active-power output Pt in the actualand equivalent systems. (c)(h) active power outputs of SG1 and SG2 Psg1 and Psg2 in theactual system. (d)(i) frequency of SG1 and SG2 fsg1 and fsg2 in the actual system. (e)(j) inputtorque of SG1 and SG2 Tsg,m1 and Tsg,m2 in the actual system.and equivalent (traces labelled as (ii) in Figs. 3.12(a)(b)(f)(g)) systems. This verifies ourproposed method to accurately tune the synchronverter parameters and achieve desired fasttime-domain response in cases with high and low levels of system inertia.75In addition, by inspecting Figs. 3.12(a)(b)(f)(g), we make the following observations:(i) The actual Pt in actual system (traces (i) in Figs. 3.12(a)(b)(f)(g)) are slightly be-low the desired one (traces (ii) in Figs. 3.12(a)(b)(f)(g)) during the transient periodafter t = 1.0 s. As the synchronverter begins to power the load at t = 1.0 s, theactive-power output of SG1 and SG2 decrease, as shown in Figs. 3.12(c)(h), whichoccurs more quickly than the governor action, as shown in Figs. 3.12(e)(j). Due tothe inertia response of SG1 and SG2, both rotating speeds fsg1 and fsg2 increase af-ter t = 1.0 s, as shown in Figs. 3.12(d)(i). These cause corresponding increases inthe SG1 and SG2 inner-voltage phase angles, which is equivalent to increasing θ∞in Fig. 3.1(c). According to (2.13) and (2.14), the synchronverter electromagnetictorque Te ∝ sin θg∞ ≈ θg − θ∞, so increasing θ∞ tends to decrease Te as well as thesynchronverter active-power output Pt. But in the equivalent system, I replace SG1and SG2 with infinite bus, which has fixed frequency. Thus, Pt in actual system isslightly smaller than that in the equivalent system, as shown in Figs. 3.12(a)(b)(f)(h).(ii) In the lower-inertia case shown in Figs. 3.12(f)(g), there is slightly larger discrepancybetween the actual and desired Pt. With lower system inertia, as shown in Fig. 3.12(i),the rate of change of frequency is higher, and fsg1 and fsg2 increase more quicklyafter t = 1.0 s. This is equivalent to increasing θ∞ more quickly in Fig. 3.1(c) after t =1.0 s. According to Te ∝ sin θg∞ ≈ θg − θ∞, increasing θ∞ more quickly tendsto decrease Te as well as Pt more obviously. This causes the actual and equivalentsystem dynamic response to differ more in the low-inertia case.(2) Synchronverter tuned to respond slowly (cases III and IV). Simulation resultsfor cases III and IV are, respectively, shown in Figs. 3.13(a)–(e) and (f)–(j). Comparedwith cases I and II in Fig. 3.12, there are larger deviations between the synchronverteractive-power output responses in the actual (traces (i) in Figs. 3.13(a)(b)(f)(g)) and equiv-alent (traces (ii) in Figs. 3.13(a)(b)(f)(g)) systems. In cases III and IV, the synchronverteris tuned to respond slowly, and its APL dynamics act in similar time scales as the SG1 andSG2 rotor dynamics. In contrast, the fast APL dynamics in Cases I and II are essentiallydecoupled from the comparatively slower SG rotor dynamics. Thus, after increasing the76synchronverter APL output Pt at t = 1.0 s, the resultant SG1 and SG2 frequency varia-tions, as shown in Fig. 3.13(d)(i), cause larger deviations between the actual and desired Ptin cases III and IV than in cases I and II. Even though there are larger deviations in thetrajectories, we highlight that the desired response speed is achieved following the increasein the APL output reference P ?t at t = 1.0 s. Thus, we verify the efficacy of the proposedparameter tuning method in cases where slower response speed is desirable.In addition, by inspecting Fig. 3.13(f) in case IV, we observe that the synchronverter active-power response has an undershoot during t = 1.0–1.7 s but an overshoot after 1.7 s. Due tothe increase in Pt at t = 1.0 s, the SG1 and SG2 active-power outputs Psg1 and Psg2 drop,as shown in Fig. 3.13(h). This causes the SG1 and SG2 frequencies, i.e., fsg1 and fsg2, toincrease after t = 1.0 s. But due to the subsequent governor action as shown in Fig. 3.13(j),which reduces the SG1 and SG2 input torques Tm,sg1 and Tm,sg2, fsg1 and fsg2 begin todecrease at t = 1.7 s after the initial increase. Recall that according to (2.13) and (2.14), thesynchronverter electromagnetic torque Te ∝ sin θg∞ ≈ θg−θ∞. So during t ∈ [1.0, 1.7] s, theincreases in fsg1 and fsg2 equivalently result in an increase in θ∞ and tend to decrease Te.On the other hand, after t = 1.7 s, the decreases in fsg1 and fsg2 correspond to decreasesin θ∞, and tend to increase Te. This explains the undershoot followed by overshoot inCase IV. In fact, this phenomenon also exists in cases I–III. But in cases I–III, either thesynchronverter responds quickly or the system has high inertia level, i.e., the time scale ofsynchronverter APL dynamics is much faster than that of the SG rotor dynamics, so thisphenomenon is not as obvious as that in case IV.Via the three steps above, I verify my proposed tuning method in a network system. Wenote that my method has two limitations:(1) I replace SGs and the external system network with an infinite bus with constant frequency,so my method does not capture how the synchronverter interacts with SGs in the desiredAPL time-domain response. On the other hand, we note that my method does not requireany information regarding system inertia level, which facilitates its usage in various settings.(2) My method assumes that the external grid is predominantly inductive, which may not holdfor low-voltage distribution systems.773.6 SummaryIn this chapter, I identify a shortcoming in tuning the synchronverter APL parameters fornormal operation with the conventionally used iterative method based on small-signal analysis.Specifically, varying Df influences the APL dominant mode differently under different operationconditions, and so more trial-and-error work is needed during the parameter tuning process. Inorder to explain this phenomenon, I develop a precise criterion to differentiate between possibleeigenvalue variation patterns. Moreover, I propose a method to directly compute the APLparameters Jg and Df , and in turn achieve desired APL dominant-mode behaviour duringnormal operation. Unlike previous VSG parameter tuning methods, the proposed methoddirectly computes parameter values that achieve desired transient and steady-state behaviour,and in so doing, greatly simplifies the tuning process.78Chapter 4Analyzing Feasible Pole-placement Region ofSynchronverter to Faciliate Its TuningFollowing the proposed synchronverter tuning method for normal operation in Chapter 3, thischapter further analytically provides the feasible pole-placement region of the synchronverterAPL. Within this region, we are able to place the APL dominant poles freely and achieve desiredsystem dynamics during normal operation. Further knowing the feasible pole-placement regionis of great importance for the synchronverter parameter tuning with the method in Chapter 3.This is because otherwise, we might require a lot of extra trial-and-error efforts to set differentdesirable APL dominant poles and repeat the parameter tuning process until the chosen APLpoles actually represent the dominant ones. Thus, my proposed feasible pole-placement regionprovides important guidance for the synchronverter parameter tuning process.The remainder of this chapter is organized as follows. In Section 4.1, I provide an overviewfor the direct computation method and highlight the necessity of knowing the feasible pole-placement region when tuning the synchronverter parameters. Then, Section 4.2 analyticallyderives the feasible pole-placement region, which provides guidance for the synchronverter APLtuning process and completely eliminates the trial-and-error process of choosing achievableAPL dominant poles. Next, Section 4.3 validates the proposed feasible synchronverter pole-placement region via numerical studies. Finally, in Section 4.4, I provide concluding remarksand compelling directions for the future work.79Figure 4.1: Synchronverter augmented with damping correction loop proposed in Chapter 2.My previous work in Chapter 3 proposes a method to directly compute control parameters andplace the APL dominant poles at desired locations. However, trial-and-error work is neededwhen specifying the desired dominant poles, since it may be impossible to achieve pole placementat some locations.4.1 Motivation of Studying Feasible Pole-placement RegionThe tuning method proposed in Chapter 3 directly computes the controller parameters ofthe synchronverter design proposed in Chapter 2 (Fig. 4.1) and places APL dominant polesat desired locations in the s-plane. This direct computation method in Chapter 3 avoidsthe trial-and-error tuning process in existing tuning methods, and thus significantly facilitatesthe synchronverter parameter tuning. However, I find that it is impossible to place the APLdominant poles at some locations in the s-plane, and trial-and-error work may be needed whenspecifying the desired dominant poles and computing control parameters with the methodin Chapter 3. In this section, I briefly overview the proposed direct computation methodin Chapter 3 and motivate the necessity of studying the feasible pole-placement region of thesynchronverter APL.80Figure 4.2: Desired APL pole locations in the s-plane used to directly compute parameters Jgand Df .4.1.1 Overview of Direct Computation MethodFor the synchronverter design in Fig. 4.1, the direct computation method in Chapter 3 computesparameters Jg and Df directly according to (4.1) and (4.2) as follows ((3.40) and (3.41) inChapter 3)Jg =√32ψ◦fU∞ cos θ◦g∞ − τfDpXtω?n2ω?n2Xt(1− 2τfω?n cosϕ?), (4.1)Df =2ψ◦f cosϕ?ω?n+τfψ◦f1− 2τfω?n cosϕ?−√23XtDpU∞ cos θ◦g∞(1 +τf2ω?n21− 2τfω?n cosϕ?), (4.2)where ϕ? = arccos ζ? ∈ (0, pi/2) rad, ω?n and ζ?, respectively, denote the specified APL naturalfrequency and damping ratio. Let s1, s2, and s3 denote the three roots of the third-order APLcharacteristic equation as follows ((3.11) in Chapter 3)s3 + bs2 +Ks+ d = 0. (4.3)whereb =Jg + τfDpτfJg, (4.4)K =1τfJg(Dp +Df√32U∞ cos θ◦g∞Xt), (4.5)d =√32ψ◦fU∞ cos θ◦g∞τfJgXt. (4.6)81With Jg and Df computed from (4.1) and (4.2), as depicted in Fig. 4.2, I place two of APLpoles (roots of (4.3)), s2 and s3, at the desired APL dominant pole locations λ?2 and λ?3, i.e.,s2 = λ?2 = ω?nej(pi−ϕ?) = −ω?n cosϕ? + jω?n sinϕ?, (4.7)s3 = λ?3 = ω?nej(pi+ϕ?) = −ω?n cosϕ? − jω?n sinϕ?. (4.8)In this way, I hope to achieve the following time domain response when varying P ?t∆Pt =ω?n2s2 + 2ζ?ω?ns+ ω?n2 ∆P?t =: G?1(s)∆P?t . (4.9)The expressions for Jg and Df in (4.1) and (4.2) are solved from the following Vieta’s formulasfor (4.3) ((3.36)–(3.38) in Chapter 3)−b = s1 + s2 + s3 = −α1 − 2ω?n cosϕ?, (4.10)K = s1s2 + s2s3 + s1s3 = 2α1ω?n cosϕ? + ω?n2, (4.11)−d = s1s2s3 = −α1ω?n2, (4.12)where s1 = −α1 (α1 > 0) denotes the unspecified root of (4.3). Note that before computing Jgand Df with (4.1) and (4.2), I first need to specify the desired APL natural frequency ω?n andthe damping ratio ζ?. However, via the Example 4 below, I find that for certain ω?n and ζ?,(4.1) and (4.2) place two APL poles s2 and s3 at the desired APL dominant pole locations λ?2and λ?3, but instead of s2 and s3, s1 is indeed the APL dominant pole and determines the APLdynamic response. In this case, we cannot achieve desired APL dynamics in (4.9). Thus, ω?nand ζ? may need to be repeatedly specified until s2 and s3 represent the APL dominant modeand we achieve desired APL dynamics.Example 4 (Impacts of ω?n and ζ? on Pole-placement Results and APL dynamics). In thisexample, I study the impacts of ω?n and ζ? on the pole-placement results and the APL dynamics.To achieve this, we first use (4.1) and (4.2) to compute Jg and Df based on specified ω?n and ζ?for the system in Fig. 4.1. Then, we substitute the computed Jg and Df into the small-signalAPL model in (3.10) to compute the APL poles and its step response. Note that the parameters82Table 4.1: Parameters of Synchronverter-connected System in Fig. 4.1Parameters Values Parameters Values Parameters ValuesRs 0.741 Ω Ls 20 mH S1, S2 1, 0Re 0.0 Ω Le 38.5 mH DC-link voltage 13 kVτf 0.01 s Kg 27980 Var· rad/V rated voltage 6.6 kVrmsDp 120 N·m· s/rad Dq 0 Var/V rated frequency 60 HzU∞ 6.6 kVrms ωN , ω?g 376.99 rad/s rated capacity 1 MVAbesides Jg and Df are reported in Table 4.1, in which I assume that Dp = 120 N ·m · s/radaccording to local grid code. I have two cases with ω?n set to be, respectively, 55 (Case I)and 100 rad/s (Case II). Note that in both cases, we choose ζ? = 0.707, which is the optimaldamping value. The APL pole-placement results and actual APL step responses (together withthe desired APL step response computed with (4.9)) are, respectively, plotted in Fig. 4.3(a)and (b). First, as shown in Fig. 4.3(a), I can place s2 and s3 at their desired locations λ?2 and λ?3in Cases I and II. When ω?n = 55 rad/s (Case I), as marked with× in Fig. 4.3(a), s2 and s3 are theAPL dominant poles and determine the APL dynamics, since s1 = −85.3 < −38.9 = Re(s2).Consequently, as shown in Fig. 4.3(b), the actual APL step response computed with G1(s)(trace (a1)) matches well with the desired one computed with G?1(s) (trace (b1)). However,when ω?n = 100 rad/s (Case II), as marked with 2 in Fig. 4.3(a), we find that s1 = −28.7 >−70.7 = Re(s2) and s1 instead becomes the APL dominant pole. Due to the influence of s1, asshown in Fig. 4.3(b), the actual APL step response (trace (a2)) is remarkably slower than thedesired one (trace (b2)). In this case, we do not achieve the desired APL dynamics. 4.1.2 Problem StatementBased on Example 4, we find that for certain ω?n and ζ?, after computing parameters Jg and Dfwith (4.1) and (4.2), we ensure that s2 and s3 are two APL poles, but they may not be theAPL dominant poles as we desire. In other words, we cannot place the APL dominant poleseverywhere in the s-plane. Note that this constraint results from the APL model structuredescribed by (3.10) instead of the tuning method. Thus, when tuning synchronverter parameterswith the direct computation method in Chapter 3, after specifying ω?n and ζ? and computing Jgand Df with (4.1) and (4.2), we need to further compute APL poles and verify that s1 < Re(s2).83Figure 4.3: Impacts of ω?n and ζ? on the pole-placement results and the APL dynamics. Wefind that for some specified ω?n and ζ?, though we place s2 and s3 at the desired APL dominantpole locations (4.7) and (4.8), s1 may instead be the APL dominant pole and we may notachieve desired APL dynamics in (4.9). (a) APL pole locations in the s-plane when ω?n takes,respectively, 55 and 100 rad/s (ζ?=0.707). (b) Actual and desired APL step responses when ω?ntakes, respectively, 55 and 100 rad/s (ζ?=0.707).Otherwise, if s1 ≥ Re(s2), we need to rechoose ω?n or ζ?, and repeat the above tuning processuntil s1 < Re(s2) is satisfied. In other words, trial-and-error effort may be needed whenspecifying ω?n and ζ?, and this complicates the tuning method in Chapter 3. In order to avoidthis, further study is required on the range of ω?n and ζ? which ensures that s2 and s3 areAPL dominant poles such that we achieve desired APL dynamics. In the next section, I firstanalytically study the range of ω?n for fixed ζ? under different conditions. Based on this, weprovide the feasible pole-placement region, in which the APL dominant poles can be placedfreely. Finally, I improve the direct computation method accordingly and completely avoid thetrial-and-error effort when specifying ω?n and ζ?.4.2 Analysis of Feasible Synchronverter Pole-placementRegionThis section presents the pole-placement region analytically, which is the main contributation ofthis chapter. The analyical pole-placement region allows us to know the range of specifying ω?nand ζ? before computing Jg and Df . In this way, we avoid the possible burdensome process ofrepeatedly computing the APL poles and verifying that s2 and s3 represent the APL dominantmode, i.e.,s1 < Re(s2) = Re(s3) = −ω?n cosϕ?. (4.13)84Here, I obtain the feasible pole-placement region that satisfies (4.13) via two steps. First, Ianalyze the possible range of ω?n for specific ζ? (or ϕ?). The key to the analysis is that dependingon the value of Dp, we have three different types of range for ω?n. Then, by varying ζ? from 0to 1 (or equivalently, varying ϕ? from pi/2 to 0 rad), I get the feasible pole-placement region inthe s-plane, which also consists of three cases depending the value of Dp. Finally, based on thepole-placement region analysis, I improve the direct computation method by eliminating thetrial-and-error process when specifying ω?n and ζ?.4.2.1 Range of ω?n With Fixed ζ? ∈ (0, 1]Here, I let ζ? remain constant, and study the range of ω?n which ensures that s2 and s3 sat-isfy (4.13) and represent the APL dominant poles. Note that since ζ? = cosϕ? ∈ (0, 1], wehave ϕ? ∈ [0, pi/2) rad. To begin our analysis on the range of ω?n that satisfies (4.13), weexpress s1 as a function of ω?n and ϕ?. First, according to (4.12), s1 can be expressed bys1 = −α1 = −d/ω?n2. (4.14)By substituting (4.1) into (4.6), and then substituting the resultant into (4.14), we haves1(ω?n, ϕ?) = −M2(2τf cosϕ? · ω?n − 1)τf (ω?n +M) (ω?n −M), (4.15)whereM =√√32ψ◦fU∞ cos θ◦g∞DpτfXt. (4.16)We note that M is inversely proportional to√Dp. Also, bear in mind that the frequency-droopcoefficient Dp in (4.16) varies depending on the different grid code requirements at differentplaces, thus s1(ω?n, ϕ?) varies and the feasible range of ω?n solved from (4.13) will be different.In order to differentiate them, I further define a variable µ dependent on Dp as followsµ =12τfM=√DpN∝√Dp, (4.17)85whereN =2√6τfψ◦fU∞ cos θ◦g∞Xt. (4.18)I find that depending on the value of µ compared with 0, the range of ω?n which ensures (4.13)are categorized into two different cases, as below.Case I: µ = 0 (Dp = 0)In this case, the synchronverter is not required by the grid code to provide the primary frequencyregulation, and thus I set Dp = 0. According to (4.16) and (4.17), we know that M → ∞and µ = 0. Also, the expression for s1 in (4.15) is simplified ass1 = 2 cosϕ? · ω?n −1τf. (4.19)Then, substitute (4.19) into (4.13) and bear in mind that ω?n > 0, we have the range of ω?n,which ensures that s2 and s3 are APL dominant poles, as followsω?n ∈(0,13τf cosϕ?). (4.20)Case II: µ > 0 (Dp > 0)In this case, the synchronverter is required by the local grid code to provide frequency regulationservice, and we have Dp > 0 and µ > 0. By substituting (4.7) and (4.15) into (4.13), we getF (ω?n)− 1(ω?n +M)(ω?n −M)> 0, (4.21)whereF (ω?n) := −τf cosϕ?M2· ω?n3 + 3τf cosϕ? · ω?n. (4.22)Bear in mind that ω?n > 0 and M > 0, (4.21) is equivalent to0 < ω?n < M,F (ω?n) < 1,(4.23)86Figure 4.4: Range of ω?n which ensures that s2 and s3 are the APL dominant poles (i.e., s1 <Re(s2)). (a) when 0 < µ < cosϕ? (0 < Dp < N cos2 ϕ?). (b) when µ ≥ cosϕ? (Dp ≥ N cos2 ϕ?).or ω?n > M,F (ω?n) > 1,(4.24)In order to solve the range of ω?n from (4.23) and (4.24), I plot the image of function F (ω?n) inFig. 4.4. First, we note that when ω?n = M , F (ω?n) takes its maximum value as followsF (ω?n)max = F (M) = 2τfM cosϕ? =cosϕ?µ. (4.25)Based on (4.25), the ratio of cosϕ? to µ determines the relationship between F (ω?n) and 1and influences the feasible range of ω?n solved from (4.23) and (4.24). If 0 < µ < cosϕ? (orequivalently, 0 < Dp < N cos2ϕ?), as shown in Figs. 4.4(a), we have F (ω?n)max > 1. Let ω1, ω2and ω3 denote the three roots of F (ω?n) = 1 (ω1 < 0 < ω2 < ω3), then based on Fig. 4.4(a), thesolution of (4.23) and (4.24) isω?n ∈ (0, ω2) ∪ (M,ω3) . (4.26)87If µ > cosϕ? (or equivalently, Dp ≥ N cos2 ϕ?), as shown in Fig. 4.4(b), we have F (ω?n) ≤ 1.Thus, (4.24) does not hold, and the solution of (4.23) isω?n ∈ (0,M) . (4.27)To sum up, for fixed ζ? = cosϕ?, the range of ω?n ensuring that s2 and s3 represent the APLdominant mode is ω?n ∈(0,13τfζ?)µ = 0ω?n ∈ (0, ω2) ∪ (M,ω3) 0 < µ < ζ?ω?n ∈ (0,M) µ ≥ ζ?.(4.28)Here, I name these conclusions on µ summarized in (4.28) as µ-criterion.Example 5 (Explaining Pole-placement Results in Example (4) with Proposed µ-Criterion).In this example, I use the proposed µ-criterion to explain the impacts of ω?n and ζ? on the pole-placement results observed in Example 4. Recall that Dp = 120 N ·m · /rad and ζ? = 0.707, wehave µ = 0.78 > ζ?. Also, M is computed to be 63.95. Thus, according to (4.28), (0, 63.95) rad/sis the range of ω?n that ensures s2 and s3 to be the APL dominant mode. In Case I, I set ω?nto be 55 ∈ (0, 63.95) rad/s, and as shown in Fig. 4.3(a), (4.13) is satisfied and s2 and s3 arethe APL dominant poles as desired. However, in Case II, we have ω?n = 100 /∈ (0, 63.95) rad/s,and thus as depicted in Fig. 4.3(a), (4.13) does not hold and the pole s1 instead dominates theAPL dynamics. Remark 11 (Range of ω?n to Ensure that s1 < nRe(s2), n = 2, 3, 4, · · · ). When specifying ω?nbefore computing Jg and Df , if we want to further ensure that s1 < nRe(s2), n = 2, 3, 4, · · · ,the range of ω?n is as followsω?n ∈(0,1(n+2)τfζ?)µ = 0ω?n ∈ (0, ω2) ∪ (M,ω3) 0 < µ <√(n+2)327nζ?ω?n ∈ (0,M) µ ≥√(n+2)327nζ?,(4.29)88Figure 4.5: Different patterns of feasible pole-placement region for APL dominant poles s2and s3 in the s-plane when (a) µ = 0 (Dp = 0), (b) 0 < µ < 1 (0 < Dp < N), and (c) µ ≥ 1(Dp > N).where ω2 and ω3 (ω2 < ω3) are the two positive roots of− nτfζ?M2· ω?n3 + (n+ 2)τfζ? · ω?n = 1. (4.30)89The derivation process of (4.29) is similar to that of (4.28), and thus I omit it here for simplicity.4.2.2 Feasible Pole-placement RegionBased on the feasible range of ω?n for fixed ζ? = cosϕ?, I vary ϕ? from 0 to pi/2, and explore thefeasible pole-placement region for the APL dominant poles s2 and s3 in the s-plane. Similarto the range of ω?n summarized in (4.28), the feasible pole-placement region for s2 and s3 hasthree kinds of patterns depending on the value of µ, as shown in Fig. 4.5.Pattern I with µ = 0 (Dp = 0)In this case, since (4.20) holds for all ϕ? ∈ [0, pi/2), the real part of s2 and s3 satisfyRe(s2) = Re(s3) = −ω?n cosϕ? ∈(− 13τf, 0). (4.31)Thus, as marked by the green part in Fig. 4.5(a), the region in which we can place s2 and s3as APL dominant poles isΩf ={si ∈ C∣∣∣∣− 13τf < Re(si) < 0}. (4.32)Pattern II with 0 < µ < 1 (0 < Dp < N)In this case, depending on the value of ϕ?, the feasible pole-placement region Ωf for the APLdominant poles is divided into two parts Ωf1 and Ωf2, i.e., Ωf = Ωf1 ∪ Ωf2.If 0 ≤ ϕ? < arccosµ, we have 0 < µ < cosϕ? = ζ. This corresponds to the secondcase summarized in (4.28), and thus as marked by the yellow part in Fig. 4.5(b), the feasiblepole-placement region for s2 and s3 isΩf1 = si = ω?nej(pi±ϕ?)∣∣∣∣∣∣∣0 ≤ ϕ? < arccosµ, 0 < ω?n < ω2(ϕ?) orM < ω?n < ω3(ϕ?) , (4.33)where ω2(ϕ?) and ω3(ϕ?) (0 < ω2(ϕ?) < ω3(ϕ?)), the two positive roots of F (ω?n) = 1, arefunctions of ϕ?, since they vary with ϕ?. Also, according to (4.16), we note that M remains90constant when ϕ? varies.If arccosµ ≤ ϕ? < pi/2, we have µ ≥ cosϕ? = ζ?. This corresponds to the third casesummarized in (4.28), and thus as marked by the fan-shaped green part in Fig. 4.5(b), thefeasible pole-placement region for s2 and s3 isΩf2 ={si = ω?nej(pi±ϕ?)∣∣∣∣ arccosµ ≤ ϕ? < pi/2, 0 < ω?n < M } . (4.34)Pattern III with µ ≥ 1 (Dp ≥ N)In this case, for ϕ? ∈ [0, pi/2), we have µ > 1 ≥ cosϕ? = µ. This corresponds to third casesummarized in (4.28), and consequently, as marked by the semicircle green part in Fig. 4.5(c),the feasible pole-placement region for the APL dominant poles isΩf ={si = ω?nej(pi±ϕ?)∣∣∣∣ 0 ≤ ϕ? < pi/2, 0 < ω?n < M } . (4.35)4.2.3 Updated Parameter Tuning ProceduresBased on the analysis of the feasible pole-placement region above, I improve the tuning methodfor the APL parameters Dp, τf , Jg and Df in Section 3.3 as follows. First, we compute Dpwith (2.2) according to local grid code. Next, we determine τf based on the LPF noise rejectionrequirements. With them, we solve the system equilibrium point x◦ and compute µ with (4.17).Then, we choose the desired damping ratio ζ? for the APL dominant mode. After that, wedetermine the desired natural frequency ω?n in the range (4.28) (or (4.29) if s1 < nRe(s2),n = 2, 3, 4, · · · is required) with due consideration for the desired APL response speed. Finally,we compute Jg and Df according to (4.1) and (4.2). Note that with the updated tuningmethod, we do not need to repetitively specify ω?n, and compute Jg, Df , and the APL polesuntil s2 and s3 represent the APL dominant mode. In the next section, I verify my derivedpole-placement region via simulation studies.91Figure 4.6: Verification of µ-criterion in (4.28) that specifies range of ω?n for certain ζ? (ζ? =cosϕ? = 0.707) when (a) µ = 0 (Dp = 0), (b) 0 < µ = 0.618 < ζ? (0 < Dp = 75 < N cos2 ϕ?),and (c) µ = 0.782 ≥ ζ? (Dp = 120 ≥ N cos2 ϕ?). Note that in Figs. 4.6(a)–(c), I use pinkshadow to mark the range of ω?n computed from (4.28), and by choosing ω?n within the markedrange, we ensure that s2 and s3 represent the APL dominant poles and achieve successful poleplacement when tuning synchronverter parameters.4.3 Case StudiesIn this section, I first verify the proposed µ-criterion that specifies the range of ω?n for cer-tain ζ? via numerical studies. It is found that by choosing ω?n within the range predicted bythe µ-criterion (after determining ζ?), we automatically satisfy (4.13) and ensure that s2 and s3represent the APL dominant mode. After that, I validate the feasible pole-placement regionderived based on the proposed µ-criterion. According to the numerical results, we ensure thespecified poles s2 and s3 to be the APL dominant mode only when choosing their desired loca-92tions within the developed feasible pole-placement region. We note that the system under studyin this section is the synchronverter-connected system in Fig. 4.1 and the system parametersexcept Dp are same as those reported in Table 4.1.4.3.1 Verification of µ-Criterion that Specifies Range of ω?nIn order to verify the proposed µ-criterion as summarized in (4.28), I choose the desired APLdamping ratio ζ? to be 0.707 and analyze three cases (cases I–III) with Dp set to be 0, 75,and 120 N ·m · s/rad, respectively. In this way, we have µ = 0, 0 < µ < ζ?, and µ > ζ?,respectively, in cases I–III, which correspond to the three cases in (4.28). With these parametersettings, I first vary ω?n from 0 to 160 rad/s, compute Jg and Df based on (4.1) and (4.2),substitute the resultant Jg and Df into the APL characteristic equation (4.3), and solve theAPL poles s1, s2, and s3 from (4.3). Next, bear in mind that the µ-criterion predicts the rangeof ω?n that ensures (4.13), I plot the relationship between Re(s1) and Re(s2) with respect to ω?nin Figs. 4.6(a)–(c) for cases I–III. At the same time, I compute the range of ω?n satisfying (4.13)based on the µ-criterion in (4.28) and highlight it with pink shadow in Figs. 4.6(a)–(c) for thepurpose of comparison. Then, by comparing the relationship between Re(s1) and Re(s2) andthe range of ω?n marked in pink in Fig. 4.6, we find that (4.13) holds only when ω?n lies in thepink region. In other words, only if specifying ω?n within the range predicted by the µ-criterion,we can ensure s2 and s3 to be the APL dominant poles and achieve desired APL dynamicperformance. We note that for other ζ? ∈ (0, 1], we could repeat the procedures above andobtain same conclusion. With these analyses, I verify the accuracy of the µ-criterion proposedin Section 4.2.1.4.3.2 Verification of Feasible Pole-placement RegionIn this study, I further verify that by placing s2 and s3 within the feasible pole-placement re-gion, we automatically satisfy (4.13) and ensure that s2 and s3 are the APL dominant poles.Since as shown in Fig. 4.5, the feasible pole-placement region consists of three cases (cases I–III)depending on the relationship between µ, 0, and 1, I verify these three cases by setting Dp tobe 0, 90, and 200 N ·m · s/rad, respectively. In this way, we have µ = 0, 0 < µ = 0.677 < 1,and µ = 1.001 > 1 in cases I, II, and III, respectively. Next, I compute the feasible pole-93Figure 4.7: Verification of feasible pole-placement region developed based on the µ-criterionwhen (a) µ = 0 (Dp = 0), (b) 0 < µ = 0.677 < 1 (0 < Dp = 90 < N), and (c) µ = 1.001 > 1(Dp = 200 > N). In this figure, I highlight the feasible pole-placement region with green oryellow colours. By placing the APL poles s2 and s3 within the feasible pole-placement region,we satisfy (4.13) and ensure that s2 and s3 represent the APL dominant poles.placement regions in cases I–III with (4.32)–(4.35) and highlight them with green or yellowcolours in Figs. 4.7(a)–(c). Then, we validate the pole-placement regions in (4.32)–(4.35)by placing the APL poles s2 and s3 within and outside these regions and comparing Re(s1)and Re(s2). In both cases I and III, the poles s2 and s3 marked with × within the green region94satisfy (4.13), while it is not true for those marked with 2 outside the green region. Similarly,in case II, we have (4.13) for the APL poles marked with × and 4 within the feasible pole-placement region marked in yellow and green colours, while not for those marked with 2 and #outside the highlighted region. Based on these observations, we are able to satisfy (4.13) only ifplacing s2 and s3 within the feasible pole-placement region computed from (4.32)–(4.35). Thus,I validate the derived feasible pole-placement region in Section 4.2.2.4.4 SummaryIn this chapter, I propose a µ-criterion that analytically computes the range of the APL naturalfrequency ω?n for chosen APL damping ζ?. Within the range of ω?n predicted by the µ-criterion,we are able to freely place the APL dominant poles and achieve desired APL dynamic perfor-mance during normal operation. Thus, by incorporating the µ-criterion into the tuning processof the synchronverter, we avoid the trial-and-error process of repeatedly specifying ω?n for theAPL dominant mode, computing the APL parameters, and checking whether s2 and s3 repre-sent the APL dominant poles. Based on the µ-criterion, I also analytically derive the feasiblepole-placement region that ensures the specified APL poles to be the dominant ones. Thisregion visualizes the area in the s-plane that we can place the APL dominant poles freely andprovides important guidance for the synchronverter parameter tuning for normal operation.95Chapter 5Achieving Fast Self-synchronization Speed withEasily Tuneable ParametersIn this chapter, I propose a fast self-synchronizing synchronverter controller design, which syn-chronizes the synchronverter inner voltage to the grid-side voltage without needing to measureits phase angle, prior to physical connection to the grid. The proposed design centres on theaddition of a virtual resistance branch (along with a suitable coordinate transformation), whichprovides the controller with virtual active- and reactive-power output feedback signals dur-ing self synchronization, even though the actual outputs are zero before grid connection. Thevirtual resistance branch enables fast self synchronization by avoiding inductance dynamicsin prevailing methods that use a virtual impedance branch. At the same time, the parame-ter tuning process is simplified as fewer parameters require tuning. With respect to analysis,I exploit separation-of-time-scales arguments and develop appropriate reduced-order models,which are well-suited for studying phase-angle and voltage-magnitude self-synchronization dy-namics independently. My work provides analytical justification for the effects of pertinentcontroller parameters and system initial conditions on self-synchronization dynamics observedempirically in time-domain simulations. As such, it offers practical guidance on favourableparameter-value settings to achieve fast self synchronization, and it yields accurate estimatesfor self-synchronization times with well-tuned parameters.The remainder of this chapter is organized as follows. First, section 5.1 proposes the self-synchronizing synchronverter and motivates the need to study its self-synchronization dynam-96Figure 5.1: Proposed self-synchronizing synchronverter, which has few parameters that requiretuning. Highlighted in red colour are aspects of particular relevance to the proposed design.(a) Grid interface. (b) Power computation block. (c) Active-power loop. (d) Reactive-powerloop.ics. Next, in Section 5.2, I develop reduced-order models to analyze these dynamics in detailand further recommend parameter values to achieve fast synchronization. Then, Sections 5.3and 5.4 validate my self-synchronization dynamic analyses via time-domain simulations andexperiments, respectively. Finally, concluding remarks are offered in Section 5.5.975.1 Self-synchronizing Synchronverter DesignIn this section, I first propose a fast self-synchronizing synchronverter design with easily tuneableparameters. Thereafter, I motivate the need to analyze its self-synchronization dynamics.5.1.1 Proposed Self-synchronizing Synchronverter DesignA voltage source converter (VSC), which is controlled via a synchronverter, is connected tothe point of common coupling (PCC) with voltage ut via an L-type filter Rs + jXs and abreaker, as shown in Fig. 5.1(a). The external grid, which is connected to the PCC, is modelledas a voltage source u∞ behind impedance Re + jXe. The proposed controller comprises thepower computation block, the active-power loop (APL), and the reactive-power loop (RPL),as shown in Figs. 5.1(b)–5.1(d), respectively. Below, we first focus on the power computationblock, which includes the key design point, i.e., the virtual resistance, for self synchronization.Then, for completeness, I briefly describe the APL and RPL dynamical models. Note thatthe synchronverter inner voltage eg is obtained by combining the rotating speed ωg and rotorangle θg from the APL, as well as the excitation flux ψf from the RPL, and its correspondingvoltage line-to-line RMS value is Eg =√3/2ωgψf .Power Computation BlockIn the power computation block, as shown in Fig. 5.1(b), I adopt a virtual resistance Rv insteadof the typical virtual impedance L˜vs + R˜v. Unlike L˜vs + R˜v, which has two parameters thatrequire tuning, Rv has only one. During self synchronization, the breaker in Fig. 5.1(a) is open,and the actual synchronverter active- and reactive-power outputs, P t and Qt, are both zero.The goal of self synchronization is to ensure that the inner voltage eg closely tracks ut, themeasured PCC voltage, which is equal to the grid voltage u∞ when the breaker is open. Toachieve this, the power computation block provides the APL and RPL with feedback signals Ptand Qt that result from their virtual analogues Pv and Qv. By setting Switch 1 in Fig. 5.1(b)to position 2, we obtain virtual current iv flowing through the virtual resistance Rv accordingtoiv =eg − utRv, (5.1)98where iv = [iva, ivb, ivc]T and the measured PCC voltage ut = [uta, utb, utc]T. With iv and utin place, I define virtual active and reactive power as [80],Pv = utaiva + utbivb + utcivc, (5.2)Qv =(uta − utb)ivc + (utb − utc)iva + (utc − uta)ivb√3. (5.3)Suppose that u∞ is a balanced three-phase voltage, as is ut because ut = u∞ when the breakeris open. Let U∞ and θ∞ denote, respectively, the line-to-line RMS value and the phase angleof u∞; and define θg∞ := θg − θ∞ as the phase-angle difference between eg and u∞, then Pvand Qv are given byPv =EgU∞Rvcos θg∞ − U2∞Rv, Qv = −EgU∞Rvsin θg∞, (5.4)respectively. Note that since I adopt a virtual resistance only, Pv and Qv are, respectively,closely related to Eg and θg∞. However, the APL and RPL are designed for predominantlyinductive grid conditions. In other words, the APL input Pt and RPL input Qt are, respectively,regulated by the rotor angle θg and the inner voltage magnitude Eg (or the excitation flux ψf ).Thus, during self synchronization, I cannot directly use Pv and Qv as the APL and RPL inputs.In fact, linearization of a self-synchronization design that uses only the virtual resistance revealsa pair of eigenvalues in the right half-plane, so such a design is unstable. In view of this, weleverage the following coordinate transformation:P vQv =0 −11 0PvQv , (5.5)and instead use the post-transformation variables P v and Qv as the APL and RPL inputs,respectively, so thatPt = P v =EgU∞Rvsin θg∞, (5.6)Qt = Qv =EgU∞Rvcos θg∞ − U2∞Rv. (5.7)99Figure 5.2: Equivalent representation of proposed synchronverter design in Fig. 5.1 duringself synchronization (Switch 1 in Fig. 5.1(b) is in position 2). (a) Equivalent grid interfacecorresponding to Figs. 5.1(a) and 5.1(b), in which Rv acts as virtual reactance jRv due to thealgebraic coordinate transformation in (5.5). (b) Active- and reactive-power feedback signals.(c)(d) APL and RPL.In this way, the APL and RPL regulate, respectively, Pt and Qt using θg and Eg (or ψf )during self synchronization. As shown in Fig. 5.2, via the coordinate transformation in (5.5),the virtual resistance Rv acts equivalently as a reactance for the purpose of computing virtualpower feedback signals. We also note that the proposed virtual-resistance design (together withthe coordinate transformation in (5.5)) cannot be replaced by a virtual inductance L˜vs only (i.e.,by setting R˜v to be zero in a virtual impedance branch L˜vs + R˜v). Via small-signal analysis,I find that the inductor dynamics introduced by L˜vs present a pair of unstable eigenvaluesthat lead to unsuccessful self synchronization. These undesirable dynamics are bypassed inthe proposed design via the algebraic coordinate transformation in (5.5). I refer readers toAppendix C for detailed analysis of the necessity of adopting the virtual resistance Rv togetherwith the coordinated transformation in 5.5.Active- and Reactive-power LoopsDuring self synchronization, the APL (Fig. 5.1(c)) and RPL (Fig. 5.1(d)), respectively, syn-chronize the phase angle and the voltage magnitude of eg to those of u∞ by setting Pt and Qtto zero. The APL regulates θg∞ to zero (phase-angle self synchronization), and the RPL reg-ulates Eg to be U∞ (voltage-magnitude self synchronization). Let Si represent the state ofSwitch i, i = 2, 3, 4, i.e., Si = 1 if Switch i is closed and Si = 0 if Switch i is open. Then, the100APL and RPL dynamics are described byJgdωgdt=P ?tωN− Tef − S2Dp(ωg − ω?g)−Dfddt(Tefψff), (5.8)Kgdψfdt= S3(Q?t −Qtf ) + S4√23Dq(U?t − Utf ). (5.9)In (5.8), Jg denotes the inertia constant, ω?g the reference value of ωg, P?t the active-powerreference, and ωN the rated rotating speed. The term S2Dp(ωg − ω?g) is the switchable power-frequency droop control; and the term Dfddt(Tefψff)represents the damping correction loop,which adjusts the APL damping freely. Furthermore, by integrating ωg over time, we get therotor angle, i.e., θg(t) =∫ t0 ωg(τ)dτ . In (5.9), Kg is a tuneable parameter, which determines theRPL response speed; Q?t and U?t are, respectively, the reference value of Qt and the line-to-lineRMS value Ut of ut; and the term S4√23Dq(U?t −Utf ) represents the switchable voltage-droopcontrol. In (5.8) and (5.9), Tef , ψff , Qtf , and Utf are filtered signals obtained fromτfdTefdt= −Tef + Te, τf dψffdt= −ψff + ψf , (5.10)τfdQtfdt= −Qtf +Qt, τf dUtfdt= −Utf + Ut, (5.11)where τf is the time constant of low-pass filters (LPFs), and Te = Pt/ωN is the electromagnetictorque. Since ψff is in the denominator in (5.8), I also include a limiter on ψff , ensuringthat ψff > 0. During self synchronization, I set S2 = 0 and P?t = 0, so that Pt = P vregulates to zero. Then, according to (5.6), we get θg∞ = 0 and thus achieve phase-angle selfsynchronization. Also, we close Switch 3 (S3 = 1), open Switch 4 (S4 = 0), and set Q?t = 0. Inthis way, we regulate Qt = Qv to be zero, as desired.Remark 12. By examining (5.6)and (5.7), we note that phase-angle self synchronization oughtto be achieved earlier than that of the voltage magnitude. Only after the APL regulates θg∞to be zero (or 2kpi, k ∈ Z) can we get Eg = U∞ when Qt = 0. Otherwise, according to (5.7),the RPL would regulate Eg to be U∞/ cos θg∞, which would not be desired. Once eg synchronizes with u∞, we close the breaker and set Switch 1 to position 1. Atthis point, the synchronverter is connected to the grid and operates normally. During normal101operation, the feedback active and reactive powers are the actual converter outputs P t and Qt,as shown in Fig. 5.1(b).Full-order Dynamical ModelSuitable algebraic manipulation of (5.6)–(5.11), along with appropriate switch settings, resultin the following sixth-order nonlinear model describing system dynamics during self synchro-nization:dθg∞dt= ωg∞, (5.12)dωg∞dt= −Df√32(ω∞ + ωg∞)ψfU∞JgτfωNRvψffsin θg∞ +(Dfψfτfψ2ff− 1)TefJg, (5.13)dTefdt=√32(ω∞ + ωg∞)ψfU∞τfωNRvsin θg∞ − 1τfTef , (5.14)dψfdt= − 1KgQtf , (5.15)dQtfdt=√32(ω∞ + ωg∞)ψfU∞τfRvcos θg∞ − U2∞τfRv− Qtfτf, (5.16)dψffdt=1τf(ψf − ψff ) , (5.17)where ω∞ is the angular-speed of u∞, and ωg∞ := ωg − ω∞ is the angular speed differencebetween eg and u∞. In the model described by (5.12)–(5.17), (5.13) is obtained by substitut-ing (5.6) and (5.10) into (5.8), (5.14) is obtained by substituting (5.6) into the first expressionin (5.10), (5.15) is obtained by setting S3 = 1 and S4 = 0 in (5.9), and (5.16) is obtained bysubstituting (5.7) into the first expression in (5.11). To simplify notation, let x denote the statevector, i.e., x = [θg∞, ωg∞, Tef , ψf , Qtf , ψff ]T.By setting (5.12)–(5.17) to zero and solving them (recall that ψff > 0), we find that thesystem has a family of equilibrium points x◦, as follows:x◦ =[2kpi, 0, 0,√23U∞ω∞, 0,√23U∞ω∞]T=: [θ◦g∞, ω◦g∞, T◦ef , ψ◦f , Q◦tf , ψ◦ff ]T, (5.18)where k ∈ Z. Thus, self synchronization is achieved when x converges to x◦. With regard to102initial conditions, since the grid-voltage phase angle θ∞ is unknown to the controller, the initialphase-angle difference θg∞(0) ∈ (−pi, pi) rad. Remaining state variables in x can be initializedwithin the controller, e.g., I set ωg(0) = ωN ≈ ω∞, so that ωg∞(0) ≈ 0. Under these initialconditions, θg∞(t) indeed converges to zero upon successful self synchronization, as desired.5.1.2 Dynamic Response of Self SynchronizationThe controller design described above is able to synchronize eg to u∞ before physical gridconnection. Based on (5.12)–(5.17), we know that tuning parameters Df and Kg, respectively,affects phase-angle and voltage-magnitude self-synchronization dynamics. Next, via a numeri-cal example, I show that while increasing Df accelerates phase-angle self synchronization, thereexists an upper bound to the θg∞(t)-convergence speed. I also show that while decreasing Kgaccelerates voltage-magnitude self synchronization, sufficiently small Kg values result in unde-sirable transient overshoots.Example 6 (Impacts of Df and Kg on Self Synchronization). In this example, I simulate thesynchronverter-connected system in Fig. 5.1 and observe the impacts of Df and Kg on self-synchronization dynamics in three scenarios (Cases I–III). Other relevant parameter values arefixed as follows: Rs = 0.741 Ω, Ls = 20 mH, Re = 0 Ω, Le = 38.5 mH, S2 = 0, S3 = 1, S4 = 0,τf = 0.01 s, Rv = 5 Ω, Jg = 11.2 kg·m2, ωN = ω?g = ω∞ = 376.99 rad/s, U∞ = 6.60 kV,udc = 13 kV, rated ac side voltage UN = 6.60 kV, and rated synchronverter capacity SN =1 MVA. All simulations are initialized at ωg∞(0) = 0 rad/s, Tef (0) = 0 N ·m, Qtf (0) = 0 Var,and ψf (0) = ψff (0) = 0.01 Wb.In Cases I and II, we observe effects of varying Df on self-synchronization dynamics bysettingDf = 4.38, 127, 239 V · s2/rad, as shown in Fig. 5.3 by traces marked as (i), (ii), and (iii),respectively. Furthermore, in order to discern the impacts of θg∞(0) on self synchronization,I set θg∞(0) to be 3.14 and −3.14 rad, respectively, in Cases I (shown in Figs. 5.3(a)–5.3(b))and II (shown in Figs. 5.3(c)–5.3(d)). Here, we make two key observations. First, in bothCases I and II, increasing Df from 4.38 to 127 V · s/rad results in θg∞(t) converging to zeromore quickly. This is because larger Df corresponds to greater APL damping, which helps toaccelerate the θg∞(t)-convergence rate. However, further increasingDf to 239 V · s/rad does notlead to even faster phase-angle self synchronization. In fact, I find that with Df > 127 V · s/rad,103Figure 5.3: Impacts of Df on phase-angle self-synchronization dynamics. We find that: (i)increasing Df accelerates the APL response speed and enables θg∞(t) to converge to θ◦g∞ morequickly, but there is an upper bound to phase-angle self-synchronization speed, (ii) reducedphase-angle self-synchronization speed results in slower voltage-magnitude self synchroniza-tion. (a)(b) Self-synchronization dynamics with θg∞(0) = 3.14 rad. (c)(d) Self-synchronizationdynamics with θg∞(0) = −3.14 rad.104Figure 5.4: Impacts of Kg on voltage-magnitude self-synchronization dynamics with θg∞(0) =3.14 ∈ (0, pi) rad. We find that decreasing Kg accelerates the RPL response speed so that ψf (t)converges to 1 p.u. more quickly, but sufficiently small Kg results in transient overshootsin ψf (t).if θg∞(0) = 3.14 rad, θg∞(t) converges to zero with speed near −ω∞; and if θg∞(0) = −3.14 rad,θg∞(t) first jumps to a positive value, and then also converges to zero with speed near −ω∞.Moreover, by comparing traces (i), (ii), and (iii) in Figs. 5.3(b) and 5.3(d), we observe that ψf (t)converges to 1 p.u. more slowly with much lower rate of θg∞(t) convergence. This is consistentwith Remark 12, as the RPL cannot track Eg(t) to U∞ (or ψf (t) to 1 p.u.) until the APL hasregulated θg∞(t) to zero.In Case III, we observe the impacts of Kg on self synchronization by setting Kg = 5.00 ×104, 1.00×104, 0.300×104 Var · rad/V. Additionally, I set Df = 32.1 V · s2/rad and θg∞(0) =3.14 rad. Here, we make another important observation. As shown in Fig. 5.4, decreasing Kgfrom 5.00×104 to 1.00×104 Var · rad/V causes ψf (t) to converge to 1.00 p.u. more quickly, andthus accelerates voltage-magnitude self synchronization. However, choosing even smaller Kg =0.300×104 Var · rad/V does not further improve voltage-magnitude self-synchronization speed;instead, it causes undesirable transient overshoots in ψf (t). Furthermore, setting θg∞(0) to anyother value gives rise to nearly identical simulation results as those shown in Fig. 5.4. As highlighted via Example 6, for the self-synchronizing synchronverter controller describedin Section 5.1.1, we can accelerate phase-angle and voltage-magnitude self synchronization byincreasing Df and decreasing Kg, respectively, but only up to a certain limit. Next, I offeranalytical justification for the empirical observations made in Example 6.1055.2 Analysis of Self-synchronization DynamicsThis section provides analytical insight into self-synchronization dynamics by studying the sys-tem in (5.12)–(5.17). Key to my analysis is the observation that, in practical settings and withwell-tuned parameters Df and Kg, phase-angle self synchronization is much faster than that ofthe voltage magnitude (see, e.g., Remark 12 and Fig. 5.3). This phenomenon uncovers a natu-ral separation of time scales, which I leverage to construct two different reduced-order modelsthat can be used to study phase-angle and voltage-magnitude self-synchronization dynamicsindependently. Then, based on my analyses, I recommend practical parameter settings thatachieve fast self synchronization under various initial conditions.5.2.1 Phase-angle Self-synchronization DynamicsIn order to approximate APL dynamics with a reduced-order model, we note that phase-angleself synchronization is achieved more quickly than voltage-magnitude self synchronization. Crit-ical to the development of this reduced-order model are the following assumptions.Assumption 1. I assume that Df  τfψ◦f for a well-tuned self-synchronizing synchronverter.This assumption rests upon the fact that, if Df > τfψ◦f , the equilibrium points in (5.18) areexponentially stable for the linear system obtained by linearizing (5.12)–(5.17) around x◦. Wededuce this by applying the Routh-Hurwitz criterion (see, e.g., [82]) on the linearized systemcharacteristic equation, which is given by(λ3 +1τfλ2 +√32Dfω∞U∞JgRvτfωNλ+U∞2JgRvτfωN)·(λ+1τf)·(λ2 +1τfλ+√32ω∞U∞KgRvτf)= 0.(5.19)Since x◦ is an exponentially stable equilibrium point for the linearized system under the con-dition that Df > τfψ◦f , this also guarantees that x◦ is exponentially stable for the nonlinearsystem in (5.12)–(5.17) near x◦ (see, e.g., Theorem 4.13 in [87]). Moreover, numerical resultsobtained in Example 6 indicate that larger Df values would speed up phase-angle self synchro-nization, hence Df  τfψ◦f for a well-tuned synchronverter is a reasonable assumption. Assumption 2. Since state variables ψf and ψff are associated with slower voltage-magnitude106Figure 5.5: Phase portrait of the reduced-order APL model consisting of (5.12) and (5.20).Trajectories marked as (i) and (ii) approach but do not cross the boundary delineated by ωg∞ =−ω∞ (traces marked as (iii)), so dθg∞dt > −ω∞ during phase-angle self synchronization. (a) Df =4.38. (b) Df = 20.0.self-synchronization dynamics, I assume thatψf (t)ψff (t)= c is a constant during time scales that arerelevant to phase-angle self synchronization. With the above assumptions in place, I approximate APL dynamics in (5.12)–(5.14) by areduced second-order nonlinear model (a derivation is provided in Appendix D), which consistsof (5.12) anddωg∞dt= − cτf(α(ω∞ + ωg∞) sin θg∞ + ωg∞) , (5.20)where c and α are, respectively, given byc =ψfψff, α =√32· DfJg· U∞ωNRv. (5.21)Note that tuneable parameters Df and Jg both appear in the expression for α. For ease ofparameter tuning, we set Jg to be the value tuned for normal operation and only varyDf . In thisreduced-order model, state variables may take initial values θg∞(0) ∈ (−pi, pi) rad and ωg∞(0) =0. I construct phase portraits of dynamic trajectories arising from all possible initial conditionsin Fig. 5.5.Via visual inspection of Fig. 5.5, we note that with θg∞(0) ∈ (0, pi) rad, as Df increases107from 4.38 (Fig. 5.5(a)) to 20.0 V · s/rad (Fig. 5.5(b)), the corresponding phase trajectories (see,e.g., trace (i) corresponding to θg∞(0) = 3.14 rad at the point A) get closer to, but remainabove, the boundary delineated by trace (iii) corresponding to ωg∞ = −ω∞. This offers ana-lytical justification for the first observation in Example 6, where we note that increasing Dfamplifies the rate of convergence of θg∞(t) to zero, up to an upper limit. Particularly, duringphase-angle self synchronization,dθg∞(t)dt = ωg∞(t) > −ω∞, for all t > 0. This is evident bychecking that for all points along trace (iii) in Fig. 5.5, i.e., ωg∞ = −ω∞, (5.20) can be simplifiedasdωg∞dt∣∣∣ωg∞=−ω∞=cτf· ω∞ > 0. (5.22)Treating c > 0 as a constant based on Assumption 2 and Appendix D, for all points alongtrace (iii), ωg∞ grows larger, i.e., trajectories do not cross boundary (iii) from above. Further-more, since Df does not appear in (5.22), we find that as observed in Example 6, regardless ofhow large a value Df takes, the rate of θg∞(t) convergence approaches but does not exceed −ω∞.With initial conditions θg∞(0) ∈ (−pi, 0) rad, the phase portraits in Fig. 5.5 first reveallarge positive ωg∞ values before trajectories eventually converge to the origin. As an example,consider trace (ii) in Fig. 5.5(a), which corresponds to the phase trajectory arising from initialcondition θg∞(0) = −3.14 rad, i.e., point B. The trajectory first climbs to point C correspondingto large positive ωg∞ value, then reaches point D with positive phase-angle value, before finallyconverging to the origin in a similar fashion as the trajectory marked by trace (i). By comparingtraces marked as (ii) in Figs. 5.5(a) and 5.5(b), we note that increasing Df causes larger initialexcursions in ωg∞(t). This explains the phenomenon observed in Example 6 (specifically inFig. 5.3(c)), where θg∞(t) jumps to a positive value before converging to zero with the choiceof large Df . Via numerical fitting, we can express the phase-angle self-synchronization time TAwith sufficiently large Df asTA =|θg∞(0)|ω∞+ , (5.23)where  represents an approximation error, which decreases as Df increases.1085.2.2 Voltage-magnitude Self-synchronization DynamicsTo analyze voltage-magnitude self-synchronization dynamics, we assume that the faster phase-angle self-synchronization dynamics have reached steady state. Accordingly, we set ωg∞ = 0and θg∞ = 0 in (5.16) to getdQtfdt=√32ω∞ψfU∞τfRv− U2∞τfRv− Qtfτf. (5.24)The system consisting of (5.15) and (5.24) represents an approximate reduced-order RPL model,which is decoupled from the faster APL dynamics. This RPL model is a linear system that canbe analyzed via its transfer function. To this end, we take the Laplace transformation of (5.15)and (5.24), and solve the resultant for ψf (s) asψf (s) =ω2ns2 + 2ζωns+ ω2n· ψ?f =: Gψ(s) · ψ?f , (5.25)where natural frequency ωn and damping ratio ζ are given byωn =√√32ω∞U∞KgτfRv, ζ =√√612KgRvω∞U∞τf, (5.26)respectively, and ψ?f =√2/3U∞/ω∞ = 1 p.u. According to (5.26), decreasing Kg reduces theRPL damping ratio ζ. Indeed, as shown in Fig. 5.4, decreasing Kg from 5.00 × 104 (trace (i))to 1.00×104 Var · rad/V (trace (ii)) causes voltage-magnitude synchronization to take less time.However, further decreasingKg to 0.300×104 Var · rad/V (trace (iii)) results in an underdampedsystem, which would lead to transient overshoots in the ψf (t) trajectory. We also note thatthe transfer function Gψ(s) is independent of initial condition θg∞(0). This explains the finalobservation in Example 6 that θg∞(0) has little to no influence on the dynamic response of ψf (t).5.2.3 Parameter Values to Achieve Fast Self SynchronizationTo begin, I tune the virtual resistance Rv in the power computation block to a value withsimilar magnitude as the total reactance Xt := Xs +Xe. Particularly, I choose Rv asRv = 0.15 · U2NSN, (5.27)109where UN and SN , respectively, denote the rated voltage and capacity of the synchronverter.Next, recall that as observed in Example 6, significantly slower phase-angle self synchronizationcauses delays in voltage-magnitude self synchronization. In order to achieve fast phase-angleself synchronization, I recommend the following value of Df :Df = η · JgωNUNSN, (5.28)where Jg is tuned for normal operation (i.e., after grid connection) using the method proposedin Chapters 3 and 4 and η is a tuneable coefficient. The choice of η ≥ 0.4 ensures that Dfis sufficiently large to achieve fast phase-angle self synchronization, so as to satisfy Assump-tion 1 and justify the separation-of-time-scales arguments that led to the development of thereduced-order APL and RPL models in Sections 5.2.1 and 5.2.2. This aspect is detailed inRemark 13. At the other extreme with η = 6,  ≈ 0.002 s in (5.23), which is reasonably small.Although larger Df leads to faster APL synchronization, the practical choice of Df is limitedby the processor sampling time Ts. As shown in Remark 14 later in this section, setting Df tobe too large leads to instability when the synchronverter is implemented in discrete time, andthe maximum allowable Df depends on Ts. In typical implementations with Ts = 50 µs, werecommend setting η = 0.6 to strike a balance between ensuring system stability and achievingreasonably fast APL synchronization. If the sampling time is smaller, we may increase η tofurther accelerate the phase-angle self synchronization. Next, based on the analysis in Sec-tion 5.2.2, I set the desired damping ratio in (5.26) to be ζ = 1/√2, so that short settling timeis achieved while avoiding large transient overshoots. Then, I compute Kg asKg =√6τfω∞U∞Rv. (5.29)With the choices outlined in (5.27) and (5.29), along with setting τf = 0.01 s to guaranteethe LPFs’ noise rejection ability [17], the expected RPL settling time is [82]TR =4ζωn= 8τf = 0.08 s, (5.30)where the second equality above results by substituting (5.26). In fact, with the parameter set-110tings in (5.27)–(5.29), I can prove the self-synchronization capability of the proposed controllervia stability analysis of the reduced-order APL and RPL models. Interested readers may referto Appendix E for details.Remark 13 (Verifying that (5.28) satisfies Df  τfψ◦f ). According to (3.40) and also used inthe present work, we setJg =√32ψ◦fU∞ cos θ◦g∞ω?n2Xt(1−2τfω?nζ?), (5.31)which is valid with Dp = 0 as switch 2 in Fig. 5.1 is open. Also, by solving SN from (5.27), wehaveSN = 0.15 · U2NRv. (5.32)Substituting (5.31) and (5.32) into (5.28), we getDf = η ·√32ψ◦fU∞ cos θ◦g∞ω?n2Xt(1− 2τfω?nζ?)· ωNRv0.15UN≈ 10√6 η ωNψ◦f3ω?n2(1− 2τfω?nζ?), (5.33)where the approximation above results by assuming that Rv ≈ Xt, U∞ ≈ UN , and cos θ◦g∞ ≈ 1.Further suppose that the desired APL damping ratio ζ? = 0.707 and natural frequency ω?n ∈(0, 50). Then the choice of η ≥ 0.4 yieldsDf > 1.38ψ◦f  τfψ◦f , (5.34)as desired. Remark 14 (Effect of Processor Sampling Time Ts on Df ). In practical implementation,relevant signals are sampled by the controller at fixed time period. I use the reduced-order APLmodel in (5.12) and (5.20) to show that the choice of Df depends on the processor samplingtime Ts. Let θg∞[n] = θg∞(nTs) and ωg∞[n] = ωg∞(nTs), n = 1, 2, . . . Then, the system modelin (5.12) and (5.20) can be discretized asθg∞[n] = θg∞[n− 1] + Ts ωg∞[n− 1], (5.35)111Figure 5.6: Root loci patterns of the linearized discrete-time APL model in the complex plane.ωg∞[n] =ωg∞[n− 1]− Tscτf(α(ω∞ + ωg∞[n− 1]) · sin θg∞[n− 1] + ωg∞[n− 1]) , (5.36)and successful self synchronization is achieved when its state vector (θg∞[n], ωg∞[n]) convergesto the equilibrium x◦d1 = (2kpi, 0), k ∈ Z. Next, linearize the discrete APL model (5.35)and (5.36) around x◦d1 to getθg∞[n]ωg∞[n] = 1 Ts−αcTsωg∞τf 1−cTsτfθg∞[n− 1]ωg∞[n− 1]=: Adθg∞[n− 1]ωg∞[n− 1] , (5.37)with the characteristic equation |λdI −Ad| = 0, i.e.,λd2 −(2− cTsτf)λd +(1− cTsτf+αcTs2ω∞τf)= 0, (5.38)where λd denotes the eigenvalues of the system matrix Ad. Then, we study the impact of Dfon λd via the root locus analysis. To do this, I express the characteristic equation in (5.38) asfollows:1 +K1(λd − p1) (λd − p2) = 0, (5.39)112where K = αcTs2ω∞τf, p1 = 1 − cTsτf and p2 = 1. We note that K is proportional to α, and inturn Df as well, so variations in K and Df produce the same trends in the root loci of (5.39).By increasing K from 0 to +∞, and also bearing in mind that cTs  τf in practice, we obtainthe root loci as shown in Fig. 5.6. The eigenvalues of Ad are within the unit circle only ifDf <√23JgωNRvTsω∞U∞. (5.40)In other words, though larger Df accelerates the phase-angle self synchronization as the analysisin Section 5.2.1 shows, making Df too large causes system instability. Moreover, accordingto (5.40), the upper limit of Df is inversely proportional to the sampling time Ts. Thus, inpractical implementations where Ts is larger, the maximum value that Df can take before thediscrete-time system becomes unstable is smaller. 5.3 Simulation VerificationVia computer simulations in PSCAD/EMTDC, I verify the analyses and the recommendedparameter settings from Section 5.2. I also validate that the reduced second-order nonlinearAPL model accurately reflects phase-angle self-synchronization dynamics. Since the ideas pre-sented in Section 5.2 can be readily validated for the system used in Example 6 by examiningFigs. 5.3 and 5.4, here, I opt for a different set of system parameters, as follows: Rs = 1.62 Ω,Ls = 43 mH, Re = 1.51 Ω, Le = 40 mH, ωN = ω?g = ω∞ = 376.99 rad/s, UN = U∞ = 13.8 kV,udc = 25 kV, and SN = 2 MVA.5.3.1 Verification of Self-synchronization AnalysisUsing the parameter tuning method in Chapters 3 and 4 customized for normal operation (i.e.,after grid connection), I first choose Jg = 34.0 kg ·m2. Then, according to (5.27), (5.28),and (5.29), I compute synchronverter parameters during self synchronization (i.e., before gridconnection) and obtain Rv = 14.3 Ω, Df = 531 V · s2/rad, and Kg = 8.92 × 103 Var · rad/V.In my simulations, the solution time step (which is analogous to controller sampling timein practice) is very small (1 µs), so setting η = 6 does not destabilize the system. Withthe above parameter values, I simulate the synchronverter-connected system in Fig. 5.1 in113Figure 5.7: Self-synchronization simulation results using the proposed controller design. Theseverify my analyses of self-synchronization dynamics, which leverage two suitable reduced-ordermodels to study phase-angle and voltage-magnitude dynamics independently, as detailed inSection 5.2.PSCAD/EMTDC with initial phase-angle difference θg∞(0) = −3.14, 0, 3.14 rad. In all cases,the synchronverter begins self synchronization at t = 0 s and reaches steady state before t =0.15 s. At this point, I close the breaker, set Switch 1 from position 2 to 1, and fix Df =1142.17 V · s2/rad to begin normal operation.Key simulation results are plotted in Fig. 5.7. As shown in Fig. 5.7(a) by traces (i) and (iii)respectively, θg∞(t) converges to nearly zero at TA ≈ 0.01 s for both θg∞(0) = −3.14, 3.14 rad.This agrees well with the phase-angle self-synchronization time predicted in (5.23), and italso validates the effectiveness of the choice of Df recommended in (5.28). Moreover, asshown in Fig. 5.7(b), the actual ψf (t) trajectories arising from all phase-angle initial condi-tions (traces (i)–(iii)) nearly overlap with the step response of Gψ(s) in (5.25) (trace (iv)). Thesettling time for all ψf (t) trajectories shown as traces (i)–(iii) in Fig. 5.7(b) is TR ≈ 0.08 s,as predicted in (5.30). Moreover, varying the initial condition θg∞(0) does not significantlyaffect ψf (t), as expected from the analysis performed in Section 5.2.2. Thus, simulation resultsshown in Fig. 5.7(b) verify the reduced-order RPL model and subsequent analysis for voltage-magnitude self-synchronization dynamics, as well as the parameter settings given by (5.27)and (5.29). By comparing Figs. 5.7(a) and 5.7(b), we note that there indeed exists a separa-tion of time scales between phase-angle and voltage-magnitude dynamics. Particularly, θg∞(t)converges to zero very quickly, followed by ψf (t) to 1 p.u. by t = 0.15 s. This ensures synchro-nization of ega(t) to u∞a, as shown in Fig. 5.7(c). Furthermore, no significant start-up currentsare observed after the synchronverter is physically connected to the grid at t = 0.15 s, as shownin Fig. 5.7(d). Note that we can indeed connect the breaker safely at as early as t = 0.035 s,since as depicted in Fig. 5.7(c), the difference between ega and u∞a has almost decreased tozero by that time. Thus, the self-synchronization speed of my design is at least two times fasterthan that of conventional ones with the virtual impedance branch and PI controller, which stilltake at least 0.08 to 0.1 s even after tuning the controller to the best of my capability (viapainstaking trial-and-error process) [88].Remark 15 (Acknowledging processor sampling time). Suppose that the processor that im-plements the proposed controller samples at Ts = 50 µs, as is the case for the experimen-tal setup in Section 5.4. To emulate this in my simulation, I set the solution time step tobe 50 µ s in PSCAD/EMTDC. Here, the synchronverter cannot achieve self synchronizationif I adopt Df = 531 V · s2/rad as before, since this value is computed with the assump-tion that Ts is much smaller. Instead, I set η = 0.6, recompute Df according to (5.28) to115Figure 5.8: Self-synchronization simulation results of the proposed controller design. Assumingthat the processor sampling time is 50 µs, Df is computed with η = 0.6.Figure 5.9: Grid interface of the LCL-filter-based synchronverter.get Df = 53.1 V · s2/rad. Again, I simulate the synchronverter-connected system in Fig. 5.1with initial phase-angle difference θg∞(0) = −3.14, 0, 3.14 rad. Key results are shown in Fig. 5.8,where the phase-angle difference θg∞(t) and the excitation flux ψf (t), respectively, converge to 0and 1 p.u. before t = 0.15 s, and we can safely close the breaker after that. Based on a visualinspection of Fig. 5.8(a), θg∞(t) converges to nearly zero at approximately 0.03 s, which is ex-pectedly greater than that observed in Fig. 5.7 obtained using larger value of Df . On the otherhand, note that the RPL settling time is still 0.08 s, as shown in Fig. 5.8(b). Thus, the use ofa practical sampling time does not affect the total time needed to achieve self synchronization,which is limited by the RPL. 116Figure 5.10: Self-synchronization simulation results of the proposed controller design with anLCL filter.Remark 16 (Self-synchronization dynamics with LCL filter). Since an LCL filter is commonlyused in the synchronverter, I further validate the effectiveness of my proposed design on theLCL-filter-based synchronization design. As shown in Fig. 5.9, the LCL filter consists of aconverter-side inductance L1 = 25 mH with parasitic resistance R1 = 1.40 Ω, a grid-sideinductance L2 = 6.7 mH with parasitic resistance R2 = 0.38 Ω, and a filter capacitor Cf =1.4 µF with damping resistance Rf = 7.70 Ω. Other system parameters remain unchanged,and the synchronverter parameters during self synchronization are recomputed as follows: Jg =42.0 kg ·m2, Rv = 14.3 Ω, Df = 661 V · s2/rad, and Kg = 8.92 × 103 Var · rad/V. Weconsider initial condition θg∞(0) = −3.14 rad, and plot the a-phase grid voltage uta, LCL-filteroutput voltage eCa, and synchronverter inner voltage ega in Fig. 5.10. Via visual inspection,we find that both eCa (trace (i)) and ega (trace (ii)) converge to uta (trace (iii)) before t =0.15 s. Adopting the LCL filter does not impede successful self synchronization because thesynchronverter controller tracks eg to ut and, for well tuned LCL filter, eC ≈ eg (for thefundamental-frequency component) before connection. Moreover, by comparing ega of the L-filter-based synchronverter (trace (i) in Fig. 5.7) and that of the LCL-filter-based synchronverter(trace (ii) in Fig. 5.10), we find that the two traces are nearly identical. In fact, the L-and the LCL-filter-based synchronverters achieve self synchronization with nearly identicaldynamics. This is because during self synchronization, the two synchronverters can be modelledby the same full-order dynamical system, i.e., (5.12)–(5.17). As such, my analysis for selfsynchronization dynamics via model-order reduction and the resultant parameters to achievefast self synchronization are valid for both L- and LCL-filter-based synchronverter designs. 117Figure 5.11: Verification of the reduced second-order APL model (Model A) via comparisonswith the full-order self-synchronizing synchronverter model in (5.12)–(5.17) (Model B) andModel B with (5.13) replaced by (D.2) (Model C). (a) θg∞(t) dynamics. (b) Phase portraits(ωg∞-θg∞ plots).5.3.2 Verification of Reduced Second-order APL ModelHere, I verify the suitability of the reduced second-order APL model described by (5.12)and (5.20) (Model A) developed in Section 5.2.1 as well as pertinent assumptions that leadto it. I do so by comparing the dynamics of Model A with those of the full-order synchron-verter model (5.12)–(5.17) (Model B) and the full-order Model B except with (5.13) replacedby (D.2) (Model C). Note that the constant c = 3.4 is found via trial and error.Simulations are conducted using Models A, B, and C with initial phase-angle difference θg∞(0)= 3.14 rad. The resulting time-domain trajectories of θg∞(t) and phase portraits (ωg∞-θg∞plots) are depicted in Fig. 5.11. As shown in Fig. 5.11(a), the dynamics resulting from Mod-els A (trace (i)), B (trace (ii)), and C (trace (iii)) are nearly identical. This is also observedin their respective phase portraits, as shown in Fig. 5.11(b). These numerical results verifythat with sufficiently large Df , replacing (5.13) with (D.2) and making Assumption 2 do notcause large modeling errors in the resultant APL dynamics, as assumed in Appendix D. Thus,the reduced second-order APL model indeed accurately captures the actual phase-angle self-118Figure 5.12: Schematic diagram of self-synchronizing synchronverter experimental setup.Table 5.1: Experimental Hardware and Controller Parameters in Fig. 5.12Parameters Values Parameters Values Parameters ValuesR3 0.25 Ω L3 8 mH rated capacity 3 kVARf3 0.5 Ω Cf3 5.6 µF rated frequency 50 HzRe 0.0 Ω Le 0.0 Ω DC-link voltage 650 Vgrid voltage U∞ 380 V LPF time constant τf 0.01 s rated voltage 380 Vswitch S1 2 switches S2, S3, S4 0, 1, 0 ωN , ω?g , ω∞ 314.15radssampling frequency 20 kHz switching frequency 10 kHzsynchronization dynamics.5.4 Experimental VerificationI implement the proposed self-synchronization synchronverter design experimentally via thesetup shown in Fig. 5.12. The synchronverter is instantiated in a three-phase two-level voltagesource inverter with an LC filter. The proposed control algorithm is implemented in thedSPACE DS1103 processor board, and the grid voltage is emulated by the Chroma 61830grid simulator. Relevant signals are measured using the Tektronic MDO 3034 oscilloscope. Thedetailed experimental hardware and controller parameters are reported in Table.5.1.System parameters are as follows: R3 = 0.25 Ω, L3 = 8.0 mH, Cf3 = 5.6 µF, Rf3 = 0.5 Ω,Re = 0, Le = 0, ωN = ω?g = ω∞ = 314.16 rad/s, UN = U∞ = 380 V, udc = 650 V,119Figure 5.13: Experimental results of the self-synchronization dynamics using the proposedcontroller design when (a)–(d) θg∞(0) = −3.14 rad, (e)–(h) θg∞(0) = 0, and (i)–(l) θg∞(0) =3.14 rad/s.and SN = 3.0 kW. Note that due to safety considerations, I adopt a low-voltage synchron-verter experimentally, but this is sufficient to validate the effectiveness of the proposed self-synchronization design.Based on the method in Chapters 3 and 4 and (5.27)–(5.29), I choose tuneable param-eters relevant to self synchronization as follows: Jg = 0.672 kg ·m2, Rv = 7.22 Ω, Df =16.0 V · s2/rad, and Kg = 405 Var · rad/V. Similar to the simulation conducted in Remark 15,the actual sampling time Ts = 50 µs, so Df is computed using (5.28) with η = 0.6. With theexperimental setup in conjunction with the parameters above, I consider three cases in whichthe initial phase-angle difference θg∞(0) is −3.14, 0, and 3.14 rad. Experimental results areshown in Fig. 5.13, with the self synchronization process beginning at t = 0.04 s, at which pointthe phase-angle differences θg∞(t) in Figs. 5.13(b)(f)(j) start to be measured. I find that in120all three cases, the proposed controller achieves self synchronization and all trajectories reachsteady state within 0.15 s, similar to the simulation results in Section 5.3.1. Moreover, as shownin Figs. 5.13(c)(g)(k), ψf (t) trajectories in all cases reach 1 p.u. with settling time 0.08 s. Also,ψf (t) trajectories with different initial phase-angle differences are similar to each other. Theseobservations verify my analysis in Sections 5.2.2 and 5.2.3. Furthermore, upon closer inspectionof Figs. 5.13(a)(b) and (i)(j), we find that θg∞(t) converges to nearly zero approximately 0.03 safter self synchronization begins for both θg∞(0) = −3.14, 3.14 rad. This matches well with thesimulations and discussion in Remark 14. We note that our observations here do not contradictthose made in Example 6, since the speed of θg∞(t) converging to zero is only slightly delayedand this small delay has little effect on the RPL response speed. In summary, the experimentalresults echo simulations in Section 5.3, and they verify the proposed self-synchronization con-troller, my analysis of self-synchronization dynamics, and the recommended parameter settingsin (5.27)–(5.29).5.5 SummaryThis chapter proposes a fast self-synchronizing synchronverter design that uses a virtual resis-tance branch and the damping correction loop. The virtual power feedback signals are computedvia a coordinate transformation to ensure expected operation of APL and RPL during self syn-chronization. The proposed design not only has fast self-synchronization speed, but also has fewparameters that need to be tuned. After proposing this controller design, I further analyze itsself-synchronization dynamics. My analyses leverage suitable reduced-order models, which aredeveloped based on separation-of-time-scales arguments, to study the faster phase-angle andslower voltage-magnitude self-synchronization dynamics independently. The system-theoreticperspectives provide analytical justification for the effects of controller parameters and initialconditions on self-synchronization dynamics, yield accurate estimates for self-synchronizationtimes, and offer guidance on parameter-value settings.121Chapter 6Reducing Transient Active- and Reactive-powerCoupling in SynchronverterAs studied in Chapter 2, in order to vary the synchronverter output active power response speedfreely during normal operation, we can augment its controller with the damping correction loopor the transient droop function. By combining these, this chapter presents a synchronverterdesign that reduces the coupling between its active- and reactive-power outputs during normaloperation while allowing the response speed to be tuned freely. The proposed combinationreduces the impact of the RPL input on the APL output, which is more severe than thatof the APL on the RPL. Also, I provide analytical justification for the proposed design viatransfer-function analysis. Closed-form expressions for parameter values are derived to facilitatecontroller tuning.The remainder of this chapter is organized as follows. In Section 6.1, I describe the proposedsynchronverter controller design. Then, in Section 6.2, I analyze the transfer function of theproposed design, study the output-power coupling, and derive parameter settings to achievedesired dynamic behaviour. Next, in Section 6.3, I validate the effectiveness of the proposedcontroller via extensive simulations. Finally, I summarize this chapter in Section 6.4.6.1 Proposed Controller DesignAs shown in Fig. 6.1(c), the synchronverter controller, which is embedded within the VSC, isconnected to the grid via filter Rs + jXs and transmission line Re + jXe, with the assumptionthat Xt = Xs + Xe  Rs + Re. In order to adjust the synchronverter response speed freely122Figure 6.1: Proposed synchronverter design that combines the damping correction loop and thetransient droop function. Specifically, by setting Dm = 0, this figure represents the synchron-verter with the damping correction loop only, and by setting Df = 0, this figure represents thesynchronverter with the transient droop function only). This controller design is able to reducethe coupling between active- and reactive-power loops regardless of the tuned synchronverterresponse speed.and simultaneously reduce the coupling between the APL and the RPL, I propose to add boththe damping correction loop and the transient droop function into the APL, as depicted inFig. 6.1(a). With these augmented, the dynamics for the synchronverter rotating speed ωg canbe expressed asJgdωgdt= Tm − Tef −Dp(ωg − ω?g)− (T1 + T2), (6.1)where Jg is a tuneable inertia parameter, Tm = P?t /ωN is the input torque (with P?t as thereference value of active-power output Pt and ωN as the rated value of ωg), Tef is the filteredelectrical torque Te, and ω?g is the reference value of ωg. In (6.1), the term −Dp(ωg−ω?g) achievesfrequency-droop control, with Dp being the droop constant and determined by Dp = ∆Tm/∆ωg,where ∆ωg = ωg − ω?g denotes the angular speed deviation, and ∆Tm represents the amount ofinput torque change required by local grid code [17].In order to present the core ideas behind T1 and T2, I neglect the low-pass filters (LPFs),123marked as LPF1 and LPF2 in Fig. 6.1(a). (On the other hand, later in Section 6.2.3, I willfully consider these LPFs for the purpose of parameter tuning [89].) Then, the outputs of thedamping correction loop and the transient droop function are expressed as [17, 42]T1 ≈ Df ddt(iTg sin θ˜g), T2 ≈ DmdPtdt, (6.2)respectively, where Df [V · s2/rad] and Dm [s2/rad] are tuneable parameters, ig = [iga, igb, igc]Tis the output current, and sin θ˜g = [sin θg, sin (θg − 2pi3 ), sin (θg + 2pi3 )]T (with θg denoting therotor angle). In (6.2), the term T1 represents the realization II of the damping correction loop,and features that it avoids the potential shortcoming of saturating the controller when ψff = 0.Let θg∞ denote the phase-angle difference between the inner voltage eg and the grid voltage u∞.Note that eg = ωgψf sin θ˜g and ωg ≈ ωN . Then, neglecting circuit (see Fig. 6.1(c)) dynamics inthe time scales that we consider, the synchronverter active-power output Pt an can be expressedasPt ≈ iTg eg ≈√32ωNψfU∞Xtsin θg∞, (6.3)and the term iTg sin θ˜g in (6.2) can be expressed asiTg sin θ˜g = iTgegωgψf≈ PtωNψf=√32U∞Xtsin θg∞, (6.4)where U∞ is the line-to-line RMS value of u∞. Then, by substituting (6.3) and (6.4) into (6.2),and further substitutingdθg∞dt = ωg − ω∞ into the resultant, we getT1 = Df√32U∞ cos θg∞Xt(ωg − ω∞) , (6.5)T2 = Dm∂Pt∂θg∞(ωg − ω∞) +Dm ∂Pt∂ψfdψfdt. (6.6)Based on expressions for T1 and T2 in (6.5) and (6.6), respectively, I plot the small-signal APLblock diagram in Fig. 6.2 and make two key observations regarding the proposed design inFig. 6.1. First, both T1 and T2 are zero at steady state, so neither affects the steady-statefrequency-droop characteristics. Also, similar in form to Dp(ωg − ω?g), both T1 and T2 provide124Figure 6.2: Block diagram for small-signal model of the APL (omitting LPF1 and LPF2). Theblock marked in red is associated with the damping correction loop, and those in blue arerelated to the transient droop function.tuneable damping torque components that adjust the APL response speed by varying APLdamping.6.2 Transfer-function AnalysisIn this section, I develop the transfer-function model for the APL in the proposed synchronverterdesign. Then by analyzing the resulting model, I show that the proposed synchronverter design,which combines the damping correction loop and the transient droop function, reduces output-power coupling.6.2.1 Transfer-function Model of the APLTo show that the inclusion of both the damping correction loop and the transient droop functionwith outputs T1 and T2, respectively, reduces coupling between the APL and RPL, I constructthe small-signal model for the synchronverter APL in Fig. 6.1(a) by linearizing (6.1), (6.3),(6.5), and (6.6) around the equilibrium point (denoted by the superscript ◦) and taking theLaplace transformation of the resultant linear system. In this model, I consider small variationsin the APL input variables P ?t , ω∞, and ψf , denoted by ∆P ?t , ∆ω∞, and ∆ψg, respectively (ω?gremains unchanged as it is a reference value). Further let ∆θg∞ and ∆Pt denote the variationsin θg∞ and Pt caused by the variations in the APL inputs. Then, as shown in Fig. 6.2, I getthe following APL transfer-function model:∆Pt = G1(s)∆P?t +G2(s)∆ω∞ +G3(s)∆ψf , (6.7)125whereG1(s) =ω2ns2 + 2ζωns+ ω2n, (6.8)G2(s) =−M(s+ α)s2 + 2ζωns+ ω2n, (6.9)G3(s) =N(s2 + βs)s2 + 2ζωns+ ω2n. (6.10)with coefficients M and N expressed as, respectively,M =√32ωNψ◦fU∞ cos θ◦g∞Xt, (6.11)N =√32ωNU∞ sin θ◦g∞Xt. (6.12)In (6.8)–(6.10), the parameter α = Dp/Jg; and further, damping ratio ζ, natural frequency ωn,and parameter β are given byζ =A√Jg(Dp +(Df +DmωNψ◦f)√32U∞ cos θ◦g∞Xt), (6.13)ωn =B√Jg, (6.14)β =1Jg(Dp +Df√32U∞ cos θ◦g∞Xt), (6.15)respectively, where coefficients A and B satisfy, respectively,A =√ √6Xt12ψ◦fU∞ cos θ◦g∞, (6.16)B =√√6ψ◦fU∞ cos θ◦g∞2Xt. (6.17)6.2.2 Analysis of Output-power CouplingI focus my analysis on transfer function G3(s) in (6.8)–(6.10), as it represents the effect of theRPL output ψf on the APL output Pt, thus revealing the active- and reactive-power coupling.This coupling can be reduced by varying β in G3(s), which is linearly dependent on the tuneable126parameter Df and inversely proportional to Jg according to (6.13)–(6.15). To see the influenceof β, decompose G3(s) asG3(s) = G31(s) +G32(s), (6.18)whereG31(s) =Ns2s2 + 2ζωns+ ω2n, (6.19)G32(s) =Nβss2 + 2ζωns+ ω2n. (6.20)In the above, setting β = 0 eliminates the effect of G32(s) in G3(s). I recommend setting β < 0,since in so doing, G32(s) would further partially offset the effect of G31(s) in the resultantG3(s). In this way, we reduce the impact of the RPL output ψf on the APL output Ptdynamics. Additionally, the desired APL dynamic response can be achieved by setting ζ andωn to suitable values. As revealed in (6.13)–(6.15), the combination of the damping correctionloop and the transient droop function provides three tuneable parameters Jg, Df , and Dm,which give enough control degrees of freedom to set β, ζ, and ωn to their desired values. Thus,the proposed synchronverter achieves both output-power decoupling and desired APL dynamicresponse speed. Such an outcome cannot be achieved with either the damping correction loopor the transient droop function alone, as in both of those cases, the APL has only two tuneableparameters Jg and Df (or Dm). In fact, after tuning these two parameters for desired APLdynamic response, β is always positive, which causes greater APL and RPL coupling [17]. InRemark 17 below, I use the model in (6.7) to quantitatively compare the output-power couplingcaused by the damping correction loop to that by the transient droop function in synchronverter.Remark 17 (Comparison of Coupling in synchronverter Augmented with Damping CorrectionLoop vs. Transient Droop Function). Let ω?n and ζ?, respectively, denote the desired APLnatural frequency and damping ratio. I find that by setting ω?n >Bζ?ADpso that the APLresponds quickly, the damping correction loop leads to less coupling; if 0 < ω?n <Bζ?ADp, thetransient droop function results in less coupling; and if ω?n =Bζ?ADp, the two designs have sameoutput-power coupling. To see this, I first obtain the APL models of the two designs by setting,respectively, Dm and Df to be zero in (6.7). Specifically, for the synchronverter augmented127with only the damping correction loop, set Dm = 0 in (6.13)–(6.15), and we getζ =A√Jg(Dp +Df√32U∞ cos θ◦g∞Xt)=: ζDCL, (6.21)ω =B√Jg=: ωn,DCL, (6.22)β =1Jg(Dp +Df√32U∞ cos θ◦g∞Xt)=: βDCL. (6.23)On the other hand, for the synchronverter augmented with only the transient droop function,set Df = 0 in (6.13)–(6.15) to getζ=A√Jg(Dp+DmωNψ◦f√32U∞ cos θ◦g∞Xt)=: ζTDF, (6.24)ω=B√Jg=: ωn,TDF, (6.25)β=DpJg=: βTDF. (6.26)Next, to ensure a fair comparison, via suitable choices for the values of Jg and Df (or Dm), I setthe damping ratios and natural frequencies of these two designs to the same reference values ζ?and ω?n, i.e.,ζDCL = ζTDF = ζ?, (6.27)ωn,DCL = ωn,TDF = ω?n. (6.28)Then, substituting Jg and Df solved from (6.21) and (6.22) into (6.23), and Jg and Dm solvedfrom (6.24) and (6.25) into (6.26), and taking the difference between the resultant expressionsfor βDCL and βTDF, we getβDCL − βTDF = −Dpω?nB2(ω?n −Bζ?ADp). (6.29)Based on (6.29), if ω?n >Bζ?ADp, then βDCL − βTDF < 0 and βTDF > βDCL > 0; if 0 < ω?n < Bζ?ADp,then βDCL − βTDF > 0 and βDCL > βTDF > 0; and if ω?n = Bζ?ADp, then βDCL − βTDF = 0and βTDF = βDCL > 0. Thus, with a larger value for ω?n, which ensures faster response128speed, the synchronverter augmented with the damping correction loop has lower output-powercoupling than that with the transient droop function. On the other hand, with a smaller valuefor ω?n, which achieves slower response speed, the synchronverter augmented with the dampingcorrection loop has larger output-power coupling. Furthermore, if ω?n =Bζ?ADp, the two methodsresult in identical APL and RPL coupling. Via transfer-function analysis, we conclude that by combining the damping correction loopand the transient droop function, the proposed synchronverter design has reduced output-powercoupling regardless of whether the APL response speed is tuned to be faster or slower. Next, Ioutline the parameter tuning procedure to achieve desired transient behaviour with respect tooutput-power coupling and response speed.6.2.3 APL Parameter TuningAlthough the model developed in (6.7) is sufficiently accurate to reveal the effects of syn-chronverter active- and reactive-power coupling, it cannot be used directly to tune controllerparameters [89]. This is because the filters LPF1 and LPF2 in Fig. 6.1(a) are neglected in (6.7)for ease of analysis. Thus, here, for purposes of parameter tuning, we fully include the effectsof LPF1 and LPF2 to ensure accurate parameter values are chosen. To this end, denote by β?,ζ?, and ω?n, the reference values for β, ζ, and ωn to achieve desired output-power decouplingand APL response speed. Then, we obtain the following closed-form expressions for the APLparameters:Jg =√32ψ◦fU∞ cos θ◦g∞ − τfDpXtω?n2ω?n2Xt(1− 2τfω?nζ?), (6.30)Df =√23Xt (β?Jg −Dp)U∞ cos θ◦g∞, (6.31)Dm =2ζ?ωNω?n+√23JgXt(ω?n2τf − β?)ωNψ◦fU∞ cos θ◦g∞. (6.32)Interested readers may refer to Remark 18 as follows for detailed derivation of (6.30)–(6.32).Remark 18 (Derivation of APL Parameter Settings in (6.30)–(6.32)). To obtain the closed-form expressions for APL parameters Jg, Df , and Dm, I include LPF1 and LPF2 in the syn-chronverter APL model and construct the corresponding small-signal model based on Fig. 6.1.129Figure 6.3: Block diagram for small-signal model of the APL (including LPF1 and LPF2 markedin purple colour). Blocks marked in red are associated with the damping correction loop, andthe one in blue is related to the transient droop function.The block diagram of the resulting small-signal APL model is shown in Fig. 6.3. The corre-sponding transfer-function model is as follows:∆Pt =N1(s)∆P?t +N2(s)∆ω∞ +N3(s)∆ψfs3 + bs2 +Ks+ d, (6.33)whereb =Jg +DpτfJgτf, (6.34)d =√32ψ◦fU∞ cos θ◦g∞JgτfXt, (6.35)K =1τfJg(Dp +(Df +DmωNψ◦f)√32U∞cos θ◦g∞Xt). (6.36)I refrain from providing expressions for N1(s), N2(s), and N3(s) for brevity. By tuning Jg, Df ,and Dm, we wish to endow (6.33) with two dominant poles λ?2,3 = −ω?nζ? ± jω?n√1− (ζ?)2(which are roots of the characteristic equation of (6.33)), where ω?n and ζ?, respectively, denotethe desired natural frequency and the damping ratio for the APL dominant mode, correspondingto desired APL dynamic behaviour. I also set β to its desired value β?. Further let λ?1 = −α1 <0, denote the remaining unspecified real-valued root of the characteristic equation of (6.33).130Table 6.1: Parameter values used to verify proposed synchronverter design tuned to respondquickly (case I) and slowly (case II) in Section 6.3.1.Method Jg (kg·m2) Df (V·s/rad) Dm (s/rad)Case IA 10 −6.0 6.7×10−4B 10 −2.57 N/AC 10 N/A −4.7×10−4Case IIA 803 −7.0 2.3×10−3B 803 5.1 N/AC 803 N/A 9.6×10−4According to Vieta’s theorem [84],−b = s1 + s2 + s3 = −α1 − 2ω?nζ?, (6.37)K = s1s2 + s2s3 + s1s3 = 2α1ω?nζ? + ω?n2, (6.38)−d = s1s2s3 = −α1ω?n2. (6.39)Then, by solving Jg, Df , and Dm from (6.37)–(6.39), we arrive at the closed-form expres-sions (6.30)–(6.32) for control parameters Jg, Df , and Dm.6.3 Simulation ValidationIn this section, via numerical studies, I verify that the proposed synchronverter design indeedreduces its output-power coupling regardless of whether the APL is tuned to respond quicklyor slowly. I also verify that the response speed of the proposed synchronverter can be tunedfreely without affecting its steady-state frequency-droop characteristics. The simulated systemas shown in Fig. 6.1 is modelled in PSCAD/EMTDC in conjunction with parameters as follows:Rs+jXs = 0.74+j7.5 Ω, Re+jXe = 1.5+j15 Ω, Dp = 1407 N ·m · s/rad, ωN = ω?g = 377 rad/s,and U∞ = 6.6 kVrms.6.3.1 Active- and Reactive-power CouplingIn this case study, I show that the proposed synchronverter design has lower output-power cou-pling than synchronverter augmented with either the damping correction loop or the transientdroop function alone. I also validate several analytical insights highlighted in Section 6.2.2.131Reduced CouplingWe consider two cases in which the APL is tuned to respond (i) quickly (Case I: ω?n=15 rad/s,ζ? = 0.8, and β? = −67), and (ii) slowly (Case II: ω?n = 2.5 rad/s, ζ? = 0.8, and β? = −67).The corresponding parameter values are reported in Table 6.1. Note that parameters of theproposed design (method A) is computed according to (6.30)–(6.32), and satisfies ω?n, ζ?, and β?requirements, while using damping correction loop (method B) or transient droop function(method C) achieves only ω?n and ζ? due to their limited control degrees of freedom. In Case I,as shown in Fig. 6.4(a), method A results in the least transient overshoot in Pt when Q?tincreases from 0.0 to 0.4 MVar at t = 4.0 s compared with methods B and C. This is alsoobserved in Fig. 6.4(b) for Case II, where the APL is tuned to respond slowly. Thus, indeed,the proposed synchronverter design effectively reduces the coupling between the APL and theRPL, both when the APL is tuned to respond quickly and slowly. Moreover, following thestep change in active-power reference P ?t from 0.0 to 0.5 MW at t = 1.0 s, method A hasidentical dynamic response with methods B and C, effectively demonstrating that the responsespeed of the proposed design is fully adjustable. Moreover, these results validate the analyticalexpressions (6.30)–(6.32) that determine values of APL control parameters Jg, Df , and Dmbased on the desired transient behaviour.Impact of β on CouplingAs stated in Section 6.2.2, having β < 0 achieves better performance in reducing active- andreactive-power coupling. I verify this by comparing two cases, one with β = −67 < 0 and theother with β = 0. The controller is tuned to respond quickly, i.e, ω?n = 15 rad/s and ζ? = 0.8.We note, however, that similar observations can be made when the synchronverter is tuned torespond slowly, and I refrain from further discussions thereof. As shown in Fig. 6.5, followingthe increase in Qt from 0 to 0.4 MVar, the active power Pt in the case with β < 0 (trace (a1))indeed has a smaller transient overshoot than that with β = 0 (trace A1).Damping Correction Loop vs. Transient Droop FunctionI verify the analysis presented in Remark 17 on the comparison between synchronvertersaugmented with either the damping correction loop or the transient droop function. Here,132Figure 6.4: Comparison of dynamic response of proposed synchronverter design (method A)with synchronverter augmented with only the damping correction loop (method B) and onewith only the transient droop function (method C). Indeed, method A has the least couplingwith (a) fast and (b) slow APL response speed.Figure 6.5: Impact of β on active- and reactive-power coupling. By tuning parameters suchthat β < 0, active- and reactive-power coupling is reduced when compared with setting β = 0.set ω?n =Bζ?ADp= 5.6 rad/s and consider a step change in the reactive-power reference value Q?tfrom 0 to 0.4 MVar at t = 4.0 s. As shown in Fig. 6.6, synchronverters augmented with eitherthe damping correction loop or the transient droop function have nearly identical active- andreactive-power coupling. Thus, Bζ?ADpis indeed the critical value for ω?n, and their relative valuesdetermine whether the damping correction loop or the transient droop function results in lowercoupling.133Figure 6.6: Active- and reactive-power coupling in synchronverters augmented with eitherdamping correction loop (method B) or transient droop function (method C) are nearly identi-cal with ω?n =Bζ?ADp. Indeed, the relative values of ω?n andBζ?ADpdetermine whether the dampingcorrection loop or the transient droop function results in lower coupling.Figure 6.7: Steady-state frequency-droop characteristics are maintained under both (i) fast and(ii) slow APL response speeds.6.3.2 Steady-state Frequency-droop CharacteristicsIn this case study, I validate that adjusting the APL response speed of the proposed syn-chronverter does not affect its steady-state frequency-droop characteristics. Suppose the gridfrequency f∞ drops from 60 to 59.9 Hz at t = 4.0 s. As depicted in Fig. 6.7, whether thesynchronverter is tuned to respond quickly (trace (i)) or slowly (trace (ii)), the active power Ptconverges to the same value at t = 7.0 s following the frequency step change at t = 4.0 s. Thesteady-state deviation is dictated by Dp, which is set to the same value 1407 N ·m · s/rad forboth scenarios of fast and slow response speed.6.4 SummaryIn this chapter, I propose to reduce the synchronverter active- and reactive-power couplingduring normal operation by augmenting it with both the damping correction loop and the134transient droop function. Unlike synchronverters equipped with either of these two designsalone, combining them provides more control degrees of freedom and thus results in betterAPL and RPL coupling reduction performance. Also, the proposed design is able to adjust theAPL response speed without affecting the steady-state frequency-droop characteristics.135Chapter 7Improving Active-power Transfer Capacityof Synchronverter in Weak GridBy providing virtual inertia, the synchronverter improves system stability. However, underweak-grid conditions, system stability may also be adversely affected by the synchronverterexceeding its active-power transfer capacity, leading to a phenomenon known as voltage in-stability. Conventionally, voltage stability is often compromised by heavy loading conditions,and the impacts of grid voltage variations on the voltage stability has been studied in [78]. Inmy setting, however, loss of stability stems from greater synchronverter active-power outputthan that can be delivered to the rest of the system. In this chapter, I highlight this problemvia simulations conducted for a microgrid test system. Then, I offer analytical justificationfor the mechanism and root cause of the observed voltage instability. I further propose twocountermeasures to improve the active-power transfer capacity: (i) activate the voltage-droopcontroller, which requires the synchronverter to provide more reactive-power output, and (ii) useadditional reactive-power compensation devices, so as to enable full use of the synchronvertercapacity. It is worth noting that though voltage droop control and reactive-power compensa-tion have been extensively studied before, they have not been used to improve synchronverteractive-power transfer capacity under weak-grid conditions.The remainder of this Chapter is organized as follows. First, in Section 7.1, I describea synchronverter-connected microgrid test system and demonstrate the necessity of studyingsynchronverter active-power transfer capacity. Then, analysis is provided in Section 7.2, based136Figure 7.1: Synchronverter augmented with damping correction loop. (a) Active-power loop.(b) Reactive-power loop.on which I develop methods to improve the active-power transfer capacity. Next, in Section 7.3,I verify the effectiveness of proposed solutions via numerical simulations. Finally, Section 7.4offers concluding remarks and directions for future work.7.1 PreliminariesIn this section, I first provide an overview of the synchronverter design. Thereafter, I introducea microgrid test system, in which a synchronverter operates under weak-grid conditions since itis connected to the external grid via a long transmission line. Finally, using this system as anexample, I show the necessity of studying the synchronverter active-power transfer capacity.7.1.1 SynchronverterHere, I use synchronverter with realization II of the damping correction loop in Chapter 2 asan example. As shown in Fig. 7.1, the active- and reactive-power loops of the synchronverterregulate its active-power output Pt and reactive-power output Qt by varying its rotor angle θgand excitation flux ψf , respectively. Spefically, as shown in Fig. 7.1(b), the dynamics of the137synchronverter reactive-power loop (RPL) in Fig. 7.1(b) is described byKgdψfdt= S1(Q?t −Qtf ) + S2√23Dq(U?t − Utf ). (7.1)where the coefficient Kg determines the RPL response speed, Q?t and U?t , respectively, denotethe reference value of the synchronverter reactive-power output Qt and output voltage mag-nitude Ut, and Qtf and Utf , respectively, represent the filtered signals of Qt and Ut. We notethat depending on the statuses of Switches 1 and 2, denoted by S1 and S2, respectively, thesynchronverter can achieve voltage-droop controls (Si = 1 if Switch i is ON, and Si = 0 ifSwitch i is OFF, i = 1, 2). In (7.1), the voltage droop coefficient Dq is tuned based on the re-quired amount of variation in Qt for certain change in Ut. In the remainder of paper, I omit thedynamics of the low-pass filters (LPFs) in Fig. 7.1 and assume that Qtf ≈ Qt and Utf ≈ Ut. Wenote that neglecting their dynamics facilitates my analysis without loss of accuracy, since volt-age stability, which limits the synchronverter active-power transfer capacity, acts on a slowertime-scale and is usually analyzed via static models (e.g., PV and QV curves, continuationpower flow, modal analysis, etc.) that are independent of these dynamics [77].As for the synchronverter active-power loop (APL) in Fig. 7.1(a), it emulates SG rotordynamics described byJgdωgdt=P ?tωN− Tef −Df ddt(Tefψff)−Dp(ωg − ω?g), (7.2)dθgdt= ωg, (7.3)where Jg denotes the inertia constant, P?t the active-power reference value, ωN the rated angularfrequency, and Tef the filtered signal of the synchronverter electromagnetic torque Te. In (7.2),the terms Dfddt(Tefψff)and Dp(ωg−ω?g), respectively, represent the damping correction loop,which adjusts the APL damping ratio, and the frequency-droop controller, which achievesprimary frequency control.As shown in Fig. 7.1, with the RPL output ψf as well as the rotating speed ωg and therotor angle θg from the APL, we obtain the synchronverter inner voltage eg with the line-to-lineRMS value Eg =√3/2ωgψf .138Figure 7.2: Microgrid test system used to demonstrate the active-power transfer capacity ofthe synchronverter under weak-grid conditions.Table 7.1: Parameters of Synchronverter in Fig. 7.2Parameters Values Parameters Values Parameters ValuesDf 2.13 V · s2/rad S1, S2 1, 0 DC-link voltage 13 kVJg 28.7 kg·m2 Kg 5.60×104 Var· rad/V rated voltage 6.6 kVrmsDp 0 N·m· s/rad Dq 7.42× 103 Var/V rated frequency 60 HzU∞ 6.6 kVrms ωN , ω?g 376.99 rad/s rated capacity 1.5 MVA7.1.2 Test SystemMy aim is to study the active-power transfer capacity of the synchronverter under weak-grid conditions. In order to do so, I consider the microgrid test system shown in Fig. 7.2,where a synchronverter and two SGs (SG1 and SG2) provide power, behind three transform-ers, to three constant-impedance loads (Load1, Load2, and Load3). Furthermore, the syn-chronverter is connected to the rest of the system via a long transmission line (Line47) withimpedance Z47 = 13.2+j132 Ω, which corresponds to a low short circuit ratio SCR ≈ 0.8. Thissetup is representative of the scenario in which synchronverter-interfaced devices (e.g., RESsand battery storage) must be located farther away from load centres due to geographical andsafety considerations. Note that to stabilize grid frequency and regulate grid voltage, both SGs139Figure 7.3: Without voltage-droop controller: system dynamics caused by increasing the syn-chronverter active-power reference P ?t from 0 to 0.5 (case I) and 0.7 MW (case II), respectively.(a)(b) Case I. (c)(d) Case II.are equipped with a modified Woodward governor (see Fig. 3.9) and a standard excitation con-troller (see Fig. 3.10). I report the synchronverter parameters in Table 7.1. As for values usedfor SG and network parameters except the line impedance Z47, I refer readers to Appendix B.140Figure 7.4: Equivalent circuit of the system in Fig. 7.2 as seen from the synchronverter.7.1.3 Problem StatementWith the the synchronverter controller and system described above, I motivate the necessity ofstudying the synchronverter active-power transfer capacity under weak-grid conditions via anexample.Example 7 (Motivation). In this example, I use the system in Fig. 7.2 to study the active-power transfer capacity of the synchronverter by increasing its active-power reference value P ?tfrom 0 to 0.5 (case I) and 0.7 MW (case II) at t = 5.0 s. In both cases, I set Q?t to bezero and deactivate frequency- and voltage-droop controls, which results in unity power factorand enables Pt (Qt) to track P?t (Q?t ) without steady-steady error. In case I, as shown inFigs. 7.3(a), the synchronverter active-power output Pt (trace (i)) grows from 0 to 0.5 MWat t = 5.0 s following the change in P ?t . Also, as depicted in Fig. 7.3(b), the synchronverterinner voltage Eg (trace (i)) decreases from 6.5 to 6.1 kV and remains stable. On the other hand,as shown in Fig. 7.3(c), following the change in P ?t from 0 to 0.7 MW, the synchronverter outputsinitially converge to the desired setpoints but then begins to oscillate wildly about 20 s after thepower-reference change. As shown in Fig. 7.3(d), the synchronverter inner voltage Eg graduallydecreases beginning from t = 5.0 s and finally collapses at t = 24.0 s. After this moment, asshown in Fig. 7.3(c), Pt and Qt becomes oscillatory and they are unable to converge to desiredP ?t and Q?t . Based on the observations made in Example 7, I am motivated to seek analytical justificationfor the oscillatory behaviour in Fig. 7.3(c) and further propose countermeasures to prevent iteven for large active-power transfer.1417.2 Synchronverter Active-power Transfer CapacityIn this section, I provide analytical justification for voltage instability due to violation of syn-chronverter active-power transfer capacity. Then based on the analysis, I propose countermea-sures to improve the active-power transfer capacity of the synchronverter and avoid voltageinstability in the system.7.2.1 Analysis of Synchronverter Active-power Transfer CapacityMy analysis of the synchronverter active-power transfer capacity begins with the derivation of anequivalent single-synchronverter infinite-bus system shown in Fig. 7.4, in which the infinite busvoltage is U∞ = 6.8 kV, the LCL filter impedance is ZLCL ≈ jXs = j2.72 Ω, and the equivalentimpedance is Ze ≈ jXe = j33.62 Ω (I retain only the reactance Xe  Xs, since the networkin Fig. 7.2 is assumed to be predominantly inductive). A full exposition of the derivation canbe found in Chapter 3. Let Xt denote the total system reactance, i.e., Xt = Xs + Xe. Thenthe synchronverter active-power output Pt and reactive-power output Qt can be expressed as,respectively,Pt =EgU∞ sin θg∞Xt, Qt ≈E2gXt− EgU∞ cos θg∞Xt, (7.4)where U∞ denotes the grid voltage, and θg∞ denotes the phase-angle difference between Egand U∞. By eliminating θg∞ from (7.4), we get the following expression:E4g − (2XtQt + U2∞)E2g +X2t (P 2t +Q2t ) = 0, (7.5)which is quadratic in E2g , and its discriminant is∆ = U4∞ + 4XtQtU2∞ − 4X2t P 2t . (7.6)To ensure that one solution of (7.5) corresponds to a valid system operating point, the discrim-inant ∆ ≥ 0, i.e.,Qt ≥ XtU2∞P 2t −U2∞4Xt. (7.7)142This is because by satisfying ∆ ≥ 0, we have2XtQt + U2∞ ≥ 2Xt(XtU2∞P 2t −U2∞4Xt)+ U2∞ =2X2t P2tU2∞+U2∞2> 0, (7.8)and the solution of (7.5) satisfiesE2g =(2XtQt + U2∞)±√∆2≥ (2XtQt + U2∞)−√∆2=((2XtQt + U2∞)−√∆)((2XtQt + U2∞) +√∆)2((2XtQt + U2∞) +√∆)=2X2t (P2t +Q2t )(2XtQt + U2∞) +√∆> 0. (7.9)Also, we bear in mind that due to the constraint of the synchronverter capacity SN , the followingcondition should be satisfied:P 2t +Q2t ≤ S2N . (7.10)Let P t(Qt) denote the synchronverter active-power transfer capacity, which is the maxi-mum value that Pt can take for a given value of reactive-power output Qt within the space offeasible solutions delineated by (7.7) and (7.10). I plot this space in Fig. 7.5(a)–(c) for threedifferent ranges of SN in green hatched pattern, and I further mark P t(Qt) in thick traces.Then, we focus on P t(Qt) by flipping the Pt- and Qt-axes in Figs. (7.5)(a)–(c) to arrive atFigs. 7.5(d)–(f). Simultaneously, I reinterpret the conditions on SN as ranges of values that thegrid impedance Xt can take. Next, we derive closed-form expressions for P t(Qt) in the threecases corresponding to Figs. 7.5(d)–(f). We will find the following functions useful:f1(Qt) :=U2∞Xt√XtU2∞Qt +14≥ 0, (7.11)f2(Qt) :=√S2N −Q2t ≥ 0. (7.12)143Figure 7.5: Synchronverter active-power transfer capacity P t(Qt) for rated capacity SN (gridimpedance Xt) in the ranges of (a)(d) 0 < SN ≤ U2∞4Xt(0 < Xt ≤ U2∞4SN), (b)(e) U2∞4Xt< SN ≤ U2∞2Xt( U2∞4SN< Xt ≤ U2∞2SN), and (c)(f) SN >U2∞2Xt(Xt >U2∞2SN).Case I: 0 < Xt ≤ U2∞4SNAs shown in Fig. 7.5(d), we haveP t(Qt) = f2(Qt), if − SN ≤ Qt ≤ SN . (7.13)As marked by L2 in Figs. 7.5(a) and (d), P t obtains its maximum value Pmaxt = SN whenQt = 0.Case II: U2∞4SN< Xt ≤ U2∞2SNCorresponding to Fig. 7.5(e), we haveP t(Qt) =f1(Qt), if − U2∞4Xt≤ Qt ≤ SN − U2∞2Xt,f2(Qt), if SN − U2∞2Xt< Qt ≤ SN .(7.14)144Similar to above, P t obtains its maximum value Pmaxt = SN when Qt = 0. This is markedby M3 in Figs. 7.5(b) and (e).Case III: Xt >U2∞2SNAs shown in Fig. 7.5(f), in this case, P t(Qt) can also be expressed as (7.14). However, asmarked by N2 in Figs. 7.5(c) and (f), P t reaches its maximum valuePmaxt =U2∞Xt√XtSNU2∞− 14, (7.15)whenQt = Qmaxt = SN −U2∞2Xt> 0. (7.16)Example 8 (Explanation for Observations in Example 7). In this example, based on the anal-ysis above, I provide analytical justification for the observations made in Example 7. Since thesynchronverter in Fig. 7.2 operates under the weak-grid conditions with large Xt = 36.34 Ω >U2∞2SN= 15.41 Ω and its reactive-power output Qt is regulated to zero, according to (7.14),the synchronverter active-power transfer capacity is P t(0) = 0.63 MVA. Hence, in Exam-ple 7, case (I) with Pt = 0.5 MW ≤ P t(0) remains stable. On the other hand, case (II)with Pt = 0.7 MW > P t(0) does not belong to the space of feasible solutions. Below, I furtheranalyze the mechanism of the voltage instability in detail. 7.2.2 Connection to Voltage StabilityIf the synchronverter active-power output Pt exceeds P t(Qt), the so-called voltage instabilitywill occur. Here, in order to reveal this, I assume that the synchronverter capacity SN issufficiently large so that it is not exceeded as Pt increases, i.e., the synchronverter active-powertransfer capacity is located on the parabola between N1 and N2 in Figs. 7.5(c) and (f). Furtherdefine α such that Qt = αPt, which represents a straight line crossing the origin with slope αin the Pt-Qt plane, as shown in Figs. 7.5(c) and (f). Then, I solve (7.5) for Eg asEg =√(U2∞2+ αPtXt)±√∆α2, (7.17)145Figure 7.6: Synchronverter PV curves parameterized by α with Qt = αPt.where∆α = U4∞ + 4XtαPtU2∞ − 4X2t P 2t (7.18)is the discriminant of (7.5) with Qt = αPt. The expression in (7.17) describes the so-calledPV curve (i.e., plot of Eg versus Pt) parameterized by α. Note that the introduction of αenables us to study the influence of Qt on the synchronverter active-power transfer capacity.The synchronverter PV curve has a “nose point” that represents the point of maximum possibleactive-power transfer capacity and the corresponding voltage. For a given α, the nose point islocated atP t(α) =U2∞2Xt(α+√α2 + 1), (7.19)Eg(α) =√U2∞2+ αPtXt. (7.20)where P t(α) is the solution to ∆α = 0, and Eg(α) is obtained by setting ∆α = 0 in (7.17).According to (7.19), P t is inversely proportional to Xt, so the synchronverter active-powertransfer capacity is limited with large Xt, i.e., under weak-grid conditions.Example 9 (Voltage Instability in Example 7). To further explain the observed voltage in-stability in Example 7, in Fig. 7.6, I plot the synchronverter PV curves for α = −0.5, 0, 0.4and delineate their nose points as A, B, and C, respectively. These correspond to operating146points marked with the same labels in Figs. 7.5(c) and (f). Particularly, if Qt = 0 (i.e., α = 0),corresponding to the scenario in Example 7, trace (ii) in Fig. 7.6 reveals that the nose point Brepresents active-power transfer capacity P t = 0.64 MW. This matches the observations madein Example 7, where the regulation of Pt to P?t = 0.7 MW causes large oscillations in the out-put power. The increase in active-power output necessitates larger phase angle difference θg∞between the synchronverter and bus 2 in Fig. 7.4. However, Qt also grows with larger θg∞(see (7.4) and trace (ii) in Figs. 7.3(a) and (c)), which is at odds with the second goal of reg-ulating Qt to Q?t = 0. Thus, the RPL resorts to decreasing the value of Eg in order to reducethe reactive-power output of the synchronverter (see Fig. 7.3(d)). This, in turn, reduces theactive-power transfer capacity according to (7.4). If P ?t < P t as in case I, the synchronverterfinally converges to a point above the nose point on the PV curve. However, if P ?t > P t asin case II, the system cannot converge to a viable power-flow solution, which leads to voltageinstability. 7.2.3 Improving Synchronverter Active-power Transfer CapacityIndeed, by comparing points A, B, and C in Fig. 7.5(f) or traces (i)–(iii) in Fig. 7.6, I find thatwith greater reactive-power output Qt, or equivalently α, the active-power transfer capacity P tof the synchronverter increases. We also see this by taking the derivative of P t(α) in (7.19)with respect to α and checkingdP tdα=U2∞2Xt(1 +α√α2 + 1)> 0, ∀α ∈ R. (7.21)Based on this, I propose to improve the synchronverter active-power transfer capacity by (i)activating voltage-droop control, and (ii) using reactive-power compensation.Voltage-droop ControlAccording to (7.1), the voltage-droop controller allows the synchronverter to increase Qt pro-portionally as Ut decreases (note that Ut ≈ Eg). This is evident by setting the derivative in (7.1)to zero to getQt = Q?t + S2√23Dq(U?t − Ut). (7.22)147With this in mind, I propose to enable the synchronverter voltage-droop controller in order toimprove its active-power transfer capacity under weak-grid conditions. This is equivalent toincreasing α in Fig. 7.6, thus improving the synchronverter active-power transfer capacity. Wenote that the transfer capacity is also influenced by the values of parameters Q?t , Dq, and U?t .Reactive-power CompensationInstead of the synchronverter itself, we can use reactive-power compensation devices, e.g.,switched capacitors and static VAR compensators, to provide reactive power and improve thesynchronverter active-power transfer capacity. For example, for our system in Fig. 7.2, we caninstall a switched capacitor Cw at bus 1 and provide more reactive power to maintain Eg athigher value. This method is also equivalent to increasing α in Fig. 7.6, and the additionalreactive-power compensation devices allow us to make full use of the synchronverter capacity.Remark 19 (Trade-off of Increasing Synchronverter Reactive Power). Above, we find thatgreater synchronverter reactive-power output increases its active-power transfer capacity. How-ever, if the synchronverter reactive-power output is too high (e.g., see trace (iii) in Fig. 7.6),its output voltage remains close to the rated grid voltage value when its active-power outputis about to reach the synchronverter active-power transfer capacity. In this case, the voltageinstability is more difficult to predict, as it is not preceded by a significant voltage drop [77]. 7.3 Case StudiesIn this section, via numerical simulations, I validate the effectiveness of the proposed methodsto improve the synchronverter active-power transfer capacity. First, I show that activatingvoltage-droop control allows us to achieve so by increasing the synchronverter reactive-poweroutput following increase in the active-power output. After that, I demonstrate that by adoptinga switched capacitor, we can also support the synchronverter output voltage and increase itsactive-power transfer capacity. The simulated system is shown in Fig. 7.2, with parametervalues reported in Table 7.1 and Appendix B (except that Z47 = 13.2+j132 Ω), unless otherwisespecified.148Figure 7.7: With voltage-droop controller: system dynamics resulting from increasing the syn-chronverter active-power reference P ?t from 0 to 0.7 MW.7.3.1 Activating Voltage-droop ControlIn order to validate the efficacy of activating the voltage-droop controller, I close Switch 2 inFig. 7.1, implement the reference signals from case II in Example 7, and plot resulting dynamicsin Fig. 7.7. First, as shown by trace (i) in Fig. 7.7(a), I find that with the voltage-droop controlin place, the synchronverter is able to inject Pt = 0.7 MW into the grid, and system does notsuffer from the oscillations that were observed in Example 7. This is because after sensing thevoltage drop caused by the increase in Pt at t = 5.0 s, the voltage-droop controller enables thesynchronverter to increase its reactive-power output from 0.09 to 0.20 MVar. This helps tomaintain the synchronverter output voltage Eg (see trace (i) in Fig. 7.7(b)), operate the systemabove the nose point, and thus avoid the voltage collapse in Example 7.7.3.2 Using Reactive-power Compensation DevicesAfter increasing Pt from 0 to 0.7 MW at t = 5.0 s, I connect a fixed capacitor Cw = 10 µFto bus 1 at t = 10.0 s. We note that the voltage-droop control is deactivated in this case.Simulation results are plotted in Fig. 7.8. The synchronverter output voltage Eg (trace (i)in Fig. 7.8(b)) drops following the increase of Pt (trace (i) in Fig. 7.8(a)) from 0 to 0.7 MW149Figure 7.8: With reactive-power compensation devices: system dynamics resulting from in-creasing the synchronverter active-power reference P ?t from 0 to 0.7 MW.at t = 5.0 s. Similar to case II in Example 7, this puts the system in danger of voltage instability.However, after connecting Cw at t = 10.0 s, Eg is restored to a higher value, and as a result,the system avoids the voltage collapse observed in Example 7.7.4 SummaryIn this chapter, I motivate the necessity of studying synchronverter active-power transfer ca-pacity under weak-grid conditions via a numerical example. I find that voltage stability iscompromised if the maximum transfer capacity of the line connecting the synchronverter to therest of the grid is exceeded. Then, I provide analytical justification for the synchronverter trans-fer capacity and determine the root cause of the observed voltage instability. Finally, basedon my analysis, I propose two countermeasures to improve the synchronverter active-powertransfer capacity and provide simulation verification.150Chapter 8Research Contributions and Future DirectionsIn this chapter, I first summarize the research results and contributions presented in Chapters 2–7 of the dissertation. After that, I offer several future research directions based on the presentedwork.8.1 Contribution SummaryProviding virtual inertia and improving system stability, synchronverter technology is one com-pelling technical route to achieve the high-penetration renewable energy integration in ourpower system. Though conventional synchronverter designs have numerous advantages overother converter controller designs, I find that they still lack in control degrees of freedom,require trial-and-error tuning process, synchronize with the grid slowly, and suffer from output-power coupling during normal operation or start-up process. Also, their active-power transferlimits have not been studied, especially under weak-grid conditions. Aiming at addressing thesechallenges, this dissertation has contributions as follows:8.1.1 A Synchronverter Design With Freely Adjustable Response SpeedIn Chapter 2, I firstly propose a synchronverter design with freely adjustable response speedduring normal operation. First, I identify a shortcoming in conventional synchronverter design,which has neglected for more than five years, via a numerical example. That is, the responsespeed of conventional synchronverter design cannot be freely adjusted without affecting itssteady-state frequency-droop characteristics during normal operation. This is also true formost of other VSG designs. The root cause of this shortcoming is that numerous conventional151designs do not contain a separate damping term and thus lack in control degrees of freedom.Also, the droop control −Dp(ωg−ωN ), which should be tuned based on the local grid code andaffects steady-state characteristics, is misinterpreted as the damping term in most of previousstudies. Next, in order to address this problem, I propose to augment the synchronverterdesign with a damping correction loop, which adjusts the synchronverter APL damping ratiofreely without affecting the steady-state frequency-droop characteristics. Four realizations ofthe damping correction loop are presented in my dissertation, and in fact, they are reminiscentof the damper winding in conventional SGs. I show this core idea by conducting a transfer-function analysis for the synchronverter APL. Then, I validate the proposed design via detailedsmall-signal analysis and extensive simulation studies. With freely adjustable response speed,the proposed synchronverter design can be widely used in SVGs, HVDC links, and wind powerintegration. Since the proposed controller modifies the synchronverter damping ratio, anotherrelevant application that can be tackled is that of power system stabilizers, where one attemptsto damp system oscillations in order to help stabilize the grid.8.1.2 A Tuning Method to Directly Computing Synchronverter ParametersIn Chapters 3 and 4, I propose a tuning method to directly compute parameters of the synchron-verter proposed in Chapter 2 for its normal operation. Compared with conventional iterativetuning methods based on small-signal analysis, the proposed method completely avoids re-peated computation of system eigenvalues, bypasses onerous trial-and-error tuning, and thussignificantly simplifies the tuning process. The proposed tuning method consists of two partsof work, as following.Pole-placement MethodIn Chapter 3, I propose a method to place the APL poles at desired locations in the s-plane,and in so doing, achieve desired dynamic response. Firstly, I identify a shortcoming in conven-tional iterative parameter tuning method for the synchronverter augmented with the dampingcorrection loop. Particularly, I reveal that during the parameter tuning process, changing thesame parameter influences system eigenvalues differently under various operating conditions.This is indeed one shortcoming of the typically used small-signal analysis, but has always been152neglected in most of previous studies based on this analysis. Analytical justification for thisphenomenon is provided using root locus analysis for the characteristic equation of a reduced-order model. Through this analysis, I develop a precise criterion to predict different eigenvaluevariation patterns in my case. Thereafter, I bypass this uncertainty in the parameter tuningprocess and propose a direct computation method, which obtains the synchronverter APL pa-rameters according to prescribed pole locations. Since the proposed method relies on the exactsolution of a set of three algebraic equations, it ensures parameter tuning accuracy while beingcomputationally tractable. Finally, I validate the accuracy of the proposed criterion and thedirect computation method via numerical case studies. It is worth noting that, I include theLPFs in the reduced-order APL model, which improves the model accuracy and allows us tocompute APL parameters in analytical closed form given desired dynamic response.Pole-placement RegionIn order to complete the proposed tuning method for normal operation, Chapter 4 proposes amethod to compute the feasible pole-placement region of the synchronverter APL in the s-plane,i.e., the region in which I can place the synchronverter APL poles freely and ensure that theyrepresent the APL dominant mode. Since the APL dynamics are mainly determined by itsdominant poles, the feasible pole-placement region represents the set of the achievable systemdynamics via the parameter tuning. First, I develop a criterion to directly compute the range ofthe achievable synchronverter APL natural frequency under given APL damping ratio. Withinthe range predicted by this criterion, I am able to choose the desired APL natural frequencyfreely and achieve desired dynamics via the pole-placement method in Chapter 3. In this way,we completely avoid the trial-and-error process when tuning the synchronverter. Also, basedon this criterion, I further analytically derive the feasible pole-place region and visualize it inthe s-plane. These provides important guidance for the synchronverter parameter tuning.8.1.3 A Fast Self-synchronizing Synchronverter DesignIn Chapter 5, I propose a fast self-synchronizing synchronverter, which synchronizes the syn-chronverter inner voltage to the grid-side voltage without complicated phase-angle measurementunits, prior to physical connection to the grid. The proposed design enables the “plug-and-play”153operation of the synchronverter-based RESs. Unlike conventional designs with the PI controllerand the virtual impedance, the proposed design adopts a damping correction loop and an al-gebraic virtual resistance branch (along with a suitable coordinate transformation) with lessparameters that requires tuning. In this way, the proposed design eliminates at least two pa-rameters that require trial-and-error tuning in conventional designs, and thus is more easilytuneable. On the other hand, the self-synchronization speed of my proposed design is at leasttwo times faster than that of conventional self-synchronization designs.In order to facilitate its parameter tuning, I analyze the dynamics of the proposed con-troller in details. First, via time-domain simulations, I empirically observe the impacts of con-troller parameters as well as system initial conditions on self-synchronization dynamics. ThenI develop suitable reduced-order models to analyze the faster phase-angle self-synchronizationdynamics and slower voltage-magnitude ones separately. These perspectives offer analytical jus-tification for the effects of controller parameters and initial conditions on self-synchronizationdynamics. They also provide accurate estimates for phase-angle and voltage-magnitude self-synchronization times. Moreover, building on the aforementioned analyses, I recommend prac-tical parameter settings to achieve fast self synchronization. Finally, I validate the analysesperformed based on the reduced-order models and verify parameter-value settings via numeri-cal case studies and experiments.8.1.4 A Synchronverter Design With Reduced Output-power CouplingIn Chapter 6, I propose to combine the damping correction loop and the transient droopfunction in order to reduce the coupling between the APL and RPL during normal operationregardless of the tuned synchronverter response speed. Unlike synchronverters equipped witheither of these two designs alone, combining them provides more control degrees of freedom andthus results in better APL and RPL coupling reduction performance during normal operation.Also, the proposed design adjusts the APL response speed without affecting the steady-statefrequency-droop characteristics. Next, I provide analytical justification for the proposed designvia transfer-function analysis. Finally, I derive the closed-form expressions for parameter valuesto facilitate controller tuning. The proposed synchronverter design may be widely adopted inapplications related to renewable energy integration, HVDC transmission systems, and flexible154AC transmission systems.8.1.5 Improvement of Synchronverter Active-power Transfer Capacity inWeak GridsIn Chapter 7, I firstly study the active-power transfer capacity of the synchronverter-basedrenewable energies, especially those connected to a weak grid. First, I motivate the necessityof studying the synchronverter active-power transfer limit via a numerical example. I find thatif the actual active power delivered from the RESs to the load centre exceeds the transfercapacity, the system would not converge to a viable power-flow solution, and this leads tovoltage instability. Conventionally, voltage stability is often compromised by heavy loadingconditions. In our setting, however, loss of stability stems from greater synchronverter active-power output than that can be delivered to the rest of the system. I reveal an importantcause of voltage instability in synchronverter-integrated power systems. Particularly, I provideanalysis for the mechanism of voltage instability under weak-grid conditions. I further proposetwo countermeasures to improve synchronverter active-power transfer capacity: (i) activatethe voltage-droop controller, which requires the synchronverter to provide more reactive-poweroutput, and (ii) use additional reactive-power compensation devices, so as to enable full use ofthe synchronverter capacity. It is worth noting that though voltage droop control and reactive-power compensation have been extensively studied before, they have not been used to improvesynchronverter active-power transfer capacity under weak-grid conditions.8.2 Future DirectionsBased on the research results and contributions in Chapters 2–7, this dissertation suggests fourfuture directions worth exploring, as follows.8.2.1 Unified Synchronverter Design for Various Grid ConditionsIn order to improve the synchronverter performance, numerous modifications from various as-pects have been proposed for the synchronverter design (also, VSG), e.g., increasing its controldegrees of freedom [17], improving its transient stability [30], limiting its fault currents [51],and adding adjustable virtual impedance [15, 90]. However, most of previous researches justfocuses on one certain aspect, and we need a unified synchronverter design which combines at155least all of the following desired functions listed as below.(1) Contributing virtual inertia to the power system;(2) Regulating the dc-side voltage, ac-side voltage, or the ac-side power (fully considering theeffects of the dc-side capacitor);(3) Eliminating complicated PLLs in the controller;(4) Having freely adjustable dynamic response speed;(5) Providing frequency- and voltage-droop control;(6) Adopting freely adjustable virtual impedance;(7) Operating well under unbalanced grid conditions;(8) Operating well under weak grid conditions, even with large ratio of the resistance to thereactance;(9) Limiting its current outputs when faults occur in the system;(10) Decoupling active- and reactive-power output control completely.Also, based on my tuning method in Chapters 3 and 4 which directly computes the synchron-verter parameters and achieves pole placement, we can further include a grid impedance mea-surement unit and propose an adaptive synchronverter controller. The proposed controller auto-matically updates its control parameters to achieve desired dynamics when the grid impedancevaries due to load variations, grid typology changes, etc.8.2.2 Application of Proposed Synchronverter Designs in Power SystemConsidering the numerous advantages of proposed synchronverter designs over conventionalconverter controllers, we can further explore their application in various power-electronics-basedcomponents in our power system, including(1) Doubly fed induction generator (DFIG) wind turbine system;(2) Permanent magnet synchronous generator (PMSG) wind turbine system;156(3) Solar photovoltaic (PV) system;(4) High-voltage direction-current transmission system;(5) Flexible AC transmission system (FACTS);(6) Electric vehicle charging station;(7) Energy storage system.On the other hand, the other compelling research direction is the application of my proposeddesign in microgrid and shipboard power system.8.2.3 Parameter Tuning Method of High-order SystemIn Chapter 3, I propose a tuning method by analyzing the third-order APL model via Vieta’sformulas that connect the desired system poles and the parameters that require tuning [84].The derivation of this method is indeed equivalent to substituting the desired poles into thecharacteristic equation(λ?2)3 + b(λ?2)2 +Kλ?2 + d = 0, (8.1)(λ?3)3 + b(λ?3)2 +Kλ?3 + d = 0, (8.2)where coefficients b, K, and d are functions of parameters Df and Jg, and then solving theparameters Df and Jg from this set of equations. Also, we can transplant this idea to simplifythe tuning process of other third-order system, e.g., the droop controller in [16, 42]. Basedon this idea, one interesting further research direction is to generalize this parameter tuningmethod such that we can directly compute the parameters of high-order system and explorethe feasible pole-placement region. In this way, we are able to avoid the trial-and-error effortsarising from small-signal analysis and significantly simplify the tuning method of the high-ordersystem.8.2.4 Studying Dynamics of Power-Electronics-Based Power SystemThe increasing penetration of power-electronics-interfaced RESs not only challenges the opera-tion of our power system, but also significantly alters its dynamic behaviours. The dissertation157mainly focuses on the converter controller design, however, systematic system-level studies onthe power system dynamics are also required. Particularly, we should explore the difference ofthe dynamics between modern power-electronics-based and conventional SG-dominated powersystems. These studies provide foundations for the stability analysis, protection setting, andoperation planning of our modern power system. We note that some analysis methods and toolsin conventional power system studies might also need to be updated. For example, my work inChapter 3 finds that the second-order swing equation model might not evaluate the synchron-verter APL dynamics with high accuracy, and we should further consider a third-order APLmodel which considers the LPF dynamics. Based on this, I infer that the Lyapunov functionconstructed for multi-machine system and used for transient stability analysis might not bedirectly applied to our modern power system including power-electronics-interfaced renewableenergy units, even those adopting synchronverter or other VSG controller designs. 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Power electron., 27(11):4734–4749, Nov. 2012.ISSN 1941-0107. doi:10.1109/TPEL.2012.2199334.169Appendix ALinearized State-space Model of Synchron-verter Proposed in Chapter 2Linearized State-space Modeld∆xdt= A∆x+B∆u;∆y = C∆x+D∆u, (A.1)in which∆x = [∆ωg,∆θg∞,∆ψf ,∆ψff ,∆Tef ,∆Qtf ,∆Utf ]T,∆u = [∆P ?g ,∆Q?g,∆U?t ,∆ω?g ,∆ω∞]T,∆y = [∆Pt,∆Qt]T,A is a 7 × 7 matrix, B is a 7 × 5 matrix, C is a 2 × 7 matrix, and D is a 2 × 5 matrix. Werespectively denote by A[i, j], B[i, j], C[i, j], and D[i, j] as the entry in the i-th row and j-thcolumn of the matrix A, B, C, and D.Matrix AEntries of matrix A are (the 1st row of A will be included last) expressed as follows:170• Row 1 of A:A[1, 1] =∂F1 1∂ωg∣∣∣x◦= −√32DfU∞ sin θ◦g∞JgτfωNXt− DpJg,A[1, 2] =∂F1 1∂θg∞∣∣∣x◦= −√32Dfω◦gU∞ cos θ◦g∞JgτfωNXt,A[1, 3] =∂F1 1∂ψf∣∣∣x◦=Df (XtωNT◦ef −√32ω◦gU∞ψ◦ff sin θ◦g∞)Jg(ψ◦ff )2τfωNXt,A[1, 4] =∂F1 1∂ψff∣∣∣x◦= − DfT◦efJg(ψ◦ff )2τf,A[1, 5] =∂F1 1∂Tef∣∣∣x◦=Df − ψ◦ff τfJgψ◦ff τf. (A.2)and all other entries in Row 1 of A are zero.• Row 2 of A:A[2, 1] =∂F1 2∂ωg∣∣∣x◦= 1,and all other entries in Row 2 of A are zero.• Row 3 of A:A[3, 6] =∂F1 3∂Qtf∣∣∣x◦= −S1K,A[3, 7] =∂F1 3∂Qtf∣∣∣x◦= −√23S2DqK, (A.3)and all other entries in Row 3 of A are zero.• Row 4 of A:A[4, 3] =∂F1 4∂ψf∣∣∣x◦=1τf,A[4, 4] =∂F1 4∂ψff∣∣∣x◦= − 1τf, (A.4)and all other entries in Row 4 of A are zero.171• Row 5 of A:A[5, 1] =∂F1 5∂ωg∣∣∣x◦=√32ψ◦fU∞ sin θ◦g∞τfωNXt,A[5, 2] =∂F1 5∂θg∞∣∣∣x◦=√32ω◦gψ◦fU∞ cos θ◦g∞τfωNXt,A[5, 3] =∂F1 5∂ψf∣∣∣x◦=√32ω◦gU∞ sin θ◦g∞τfωNXt,A[5, 5] =∂F1 5∂Tef∣∣∣x◦= − 1τf, (A.5)and all other entries in Row 5 of A are zero.• Row 6 of A:A[6, 1] =∂F1 6∂ωg∣∣∣x◦=1τf( ∂fq∂ωg)∣∣∣x◦=1τf(3XeX2tω◦g(ψ◦f )2 +√32Xs −XeX2tψ◦fU∞ cos θ◦g∞),A[6, 2] =∂F1 6∂θg∞∣∣∣x◦=1τf( ∂fq∂θg∞)∣∣∣x◦=1τf((−1) ·√32Xs −XeX2tω◦gψ◦fU∞ sin θ◦g∞),A[6, 3] =∂F1 6∂ψf∣∣∣x◦=1τf( ∂fq∂ψf)∣∣∣x◦=1τf(3XeX2t(ω◦g)2ψ◦f +√32Xs −XeX2tω◦gU∞ cos θ◦g∞),A[6, 6] =∂F1 6∂Qtf∣∣∣x◦= − 1τf, (A.6)and all other entries in Row 6 of A are zero.• Row 7 of A:A[7, 1]=∂F1 7∂ωg∣∣∣x◦=1τf(∂fu∂ωg)∣∣∣x◦=3X2eX2tω◦g(ψ◦f )2+√6XeXsX2tψ◦fU∞ cos θ◦g∞2τf√X2eX2t(√32ω◦gψ◦f )2+X2sX2tU2∞+2XeXsX2t(√32ω◦gψ◦f )U∞ cos θ◦g∞,A[7, 2]=∂F1 7∂θg∞∣∣∣x◦=1τf( ∂fu∂θg∞)∣∣∣x◦=−√6XeXsX2tω◦gψ◦fU∞ sin θ◦g∞2τf√X2eX2t(√32ω◦gψ◦f )2+X2sX2tU2∞+2XeXsX2t(√32ω◦gψ◦f )U∞ cos θ◦g∞,A[7, 3]=∂F1 7∂ψf∣∣∣x◦=1τf( ∂fu∂ψf)∣∣∣x◦172=3X2eX2t(ω◦g)2ψ◦f+√6XeXsX2tω◦gU∞ cos θ◦g∞2τf√X2eX2t(√32ω◦gψ◦f )2+X2sX2tU2∞+2XeXsX2t(√32ω◦gψ◦f )U∞ cos θ◦g∞,A[7, 7]=∂F1 7∂Qtf∣∣∣x◦= − 1τf, (A.7)and all other entries in Row 7 of A are zero.Matrix BEntries of matrix B are expressed as follows:• Row 1 of B:B[1, 1] =∂F1 1∂P ?g∣∣∣x◦=1JgωN,B[1, 4] =∂F1 1∂ω?g∣∣∣x◦=DpJg, (A.8)and all other entries in Row 1 of B are zero.• Row 2 of B:B[2, 5] =∂F1 2∂ω∞∣∣∣x◦= −1, (A.9)and all other entries in Row 2 of B are zero.• Row 3 of B:B[3, 2] =∂F1 3∂Q?g∣∣∣x◦=S1K,B[3, 3] =∂F1 3∂U?t∣∣∣x◦=√23S2DqK, (A.10)and all other entries in Row 3 of B are zero.• Entries of Row 3–7 of B: all entries are zero.Matrix CEntries of matrix C are expressed as follows:173• Row 1 of C:C[1, 1] =∂Pt∂ωg∣∣∣x◦=√32ψ◦fU∞ sin θ◦g∞Xt,C[1, 2] =∂Pt∂θg∞∣∣∣x◦=√32ω◦gψ◦fU∞ cos θ◦g∞Xt,C[1, 3] =∂Pt∂ψf∣∣∣x◦=√32ω◦gU∞ sin θ◦g∞Xt, (A.11)and all other entries in Row 1 of C are zero.• Row 2 of C:C[2, 1] =∂Qt∂ωg∣∣∣x◦= 3XeX2tω◦g(ψ◦f )2 +√32Xs −XeX2tψ◦fU∞ cos θ◦g∞,C[2, 2] =∂Qt∂θg∞∣∣∣x◦= −√32Xs −XeX2tω◦gψ◦fU∞ sin θ◦g∞,C[2, 3] =∂Qt∂ψf∣∣∣x◦= 3XeX2t(ω◦g)2ψ◦f +√32Xs −XeX2tω◦gU∞ cos θ◦g∞, (A.12)and all other entries in Row 2 of C are zero.Matrix DAs the output variables Pt and Qt are independent of the input variables ∆P?g , ∆Q?g, ∆U?t ,∆ω?g , and ∆ω∞, the matrix D = 02×5.174Appendix BParameters of Components in the Test Sys-tem in Fig. 3.8The rated capacity, rated line-to-line RMS voltage and rated frequency of SG1 and SG2 are,respectively, 5 MVA, 6.6 kV and 60 Hz. Other parameters in the six-bus system in Fig. 3.8 arereported in Tables B.1–3.8.Table B.1: Parameters of synchronous generator SG1 and SG2 in Fig. 3.8.Reactances (p.u.) Machine constants (s)xd xq x′d x′q x′′d x′′q Hsg T′d0 T′q0 T′′d0 T′′q0SG1 1.56 1.06 0.296 0.296 0.177 0.177 8.00, 3.00 3.70 0.500 0.0500 0.0500SG2 1.56 1.06 0.296 0.296 0.177 0.177 8.00, 3.00 3.70 0.500 0.0500 0.0500Table B.2: Parameters of modified Woodward governors used in SG1 and SG2 in Fig. 3.8.Reference, gain and output limit (p.u.) Time constants (s)ω?sg P?sg Kw Rw Tmax Tmin T1 T2 T3 T4 T5 T6 TdSG1 1.00 1.00 40.0 0.0100 1.10 0.00 0.0100 0.0200 0.200 0.250 0.00900 0.0384 0.0240SG2 1.00 1.00 40.0 0.0100 1.10 0.00 0.0100 0.0200 0.200 0.250 0.00900 0.0384 0.0240Table B.3: Parameters of excitation system used in SG1 and SG2 in Fig. 3.8.U?sg (p.u.) KA (p.u.) TA (s) Emaxfd (p.u.) Eminfd (p.u.)SG1 1.03 140 0.0500 6.00 0.00SG2 1.03 140 0.0500 6.00 0.00175Table B.4: Parameters of the synchronverter in Fig. 3.8.Rs + jXs (Ω) S1 S2 τf (s) Jg (kg · m2) Df (V·s2rad ) Dq (VarV ) Kg (Var·radV ) ωN , ω?g ( rads ) udc (kV)0.741+j7.54 1 0 0.0100 21.3, 129 0.953, 2.26 0.00 27980 377 13Table B.5: Parameters of transformers T1, T2 and T3 in Fig. 3.8.Turns ratio (kV) Rated frequency (Hz) Rated capacity (MVA) Leakage reactance (p.u.)T1 6.60/13.8 60.0 ST1 = 1.50 Xp.u.T1 = 0.100T2 6.60/13.8 60.0 ST2 = 6.00 Xp.u.T2 = 0.100T3 6.60/13.8 60.0 ST3 = 6.00 Xp.u.T3 = 0.100Table B.6: Parameters of lines Line45, Line46 and Line56 in Fig. 3.8.Rated frequency (Hz) Rated voltage (kV) Impedance (Ω)Line45 60.0 13.8 Rh45 + jXh45 = 0.150 + j1.47 ΩLine46 60.0 13.8 Rh46 + jXh46 = 0.100 + j1.980 ΩLine56 60.0 13.8 Rh56 + jXh56 = 0.100 + j0.980 ΩTable B.7: Parameters of constant-impedance loads Load1, Load2 and Load3 in Fig. 3.8.Rated frequency (Hz) Rated voltage (kV) Rated capacity (MVA) Impedance (Ω)Load1 60.0 13.8 3.00 + j0.60 61.0 + j12.2 ΩLoad2 60.0 13.8 3.00 + j0.60 61.0 + j12.2 ΩLoad3 60.0 13.8 4.00 + j0.50 46.9 + j5.86 Ω176Appendix CNecessity of Adopting Virtual-Resistance inSelf-synchronizing SynchronverterIn this appendix, I elaborate on the advantages of my design (i.e., virtual resistance combinedwith coordinate transformation) in Chapter 5 over either virtual resistance or inductance only.In order to show the necessity of defining virtual active and reactive power via coordinatetransformation, I consider three cases as follows for comparison:(i) Synchronverter with only virtual resistance R˜v (using the virtual impedance L˜vs+ R˜v butsetting L˜v = 0).(ii) Synchronverter with only virtual inductance L˜v (using the virtual impedance L˜vs + R˜vbut setting R˜v = 0).(iii) Proposed synchronverter with the virtual resistance Rv together with the coordinate trans-formation.We find that unlike the proposed design with coordinate transformation (case (iii)), theadoption of either virtual resistance only (case (i)) or virtual inductance only (case (ii)) causesinstability and leads to unsuccessful self synchronization.Below, I detail analyses and simulations for cases (i)–(iii) that led to the above conclusions.Parameter values are reported in Table C.1. Note that to ensure a fair comparison, I choosethe virtual resistance R˜v in case (i), the virtual inductance L˜v in case (ii), and the virtualresistance Rv in case (iii) such that they satisfy R˜v = ω∞L˜ = Rv, where ω∞ denotes the177grid frequency. Other system and control parameters are set to be same in all cases (i)–(iii).I provide small-signal analysis with R˜v in case (i) and L˜v in case (ii) being varied in largerange. In this way, I show that the unsuccessful synchronization in cases (i) and (ii) is indeeda universal problem resulting from their particular virtual impedance designs rather than anexceptional case caused by poorly chosen parameter values.Table C.1: Parameters of the synchronverter in cases (i)–(iii) during self synchronization.UN , U∞(kV)fN(Hz)SN(MVA)Rs(Ω)Ls(mH)Re(Ω)Le(mH)Case (i) 13.8 60 2 1.6 43 1.5 40Case (ii) 13.8 60 2 1.6 43 1.5 40Case (iii) 13.8 60 2 1.6 43 1.5 40S1 S2 S3 S4τf(s)Jg(kg·m2)R˜v(Ω)L˜v(mH)Rv(Ω)Df(V·s2rad)Kg(Var·radV)P ?t(W)Q?t(Var)ω?g( rads)Case (i) 2 0 1 0 0.01 31 14 481.3 8.9 × 103 0 0 377Case (ii) 2 0 1 0 0.01 31 38 481.3 8.9 × 103 0 0 377Case (iii) 2 0 1 0 0.01 31 14 481.3 8.9 × 103 0 0 377C.1 Synchronverter with only virtual resistance R˜v.The block diagram for the synchronverter in this case is depicted in Fig. C.1. I first demonstrate,via simulations, that this design does not achieve successful self synchronization. Then I provideanalytical justification for the simulation results.Simulation Results. We simulate the system in Fig. C.1 in PSCAD/EMTDC and theself-synchronization process starts from t = 0 s. In the simulation, the phase-angle differenceis initialized at θg∞(0) = −3.14 rad and the excitation flux to ψf (0) = 0.01 V · s. Time-domain trajectories of key state variables θg∞ and ψf are plotted in Fig. C.2. Recall that forsuccessful self synchronization, θg∞(t) and ψf (t) should converge to 0 rad (or 2kpi, k ∈ Z)and 1 p.u., respectively. However, via visual inspection of Fig. C.2, we observe that θg∞(t)oscillates between −pi and pi and ψf (t) does not converge to 1 p.u.. Hence, adopting only thevirtual resistance R˜v does not ensure successful self synchronization.Analytical Justification. I explain the unsuccessful self synchronization observed inFig. C.2 from two perspectives. First, with regard to the application setting, the synchron-verter controller is designed for predominantly inductive grid conditions, i.e., its APL and RPL178Figure C.1: Case (i)—synchronverter equipped with only virtual resistance R˜v ((a1)–(d1)) andits equivalent representation during self synchronization ((a2)–(d2)).regulate the active- and reactive-power outputs (Pt and Qt) by adjusting θg∞ and ψf (or Eg),respectively. However, adopting only the virtual resistance Rv results in purely resistive gridconditions, according to Fig. C.1, the APL and RPL inputs are expressed as, respectively,Pt =EgU∞Rvcos θg∞ − U∞2Rv, (C.1)Qt = −EgU∞Rvsin θg∞. (C.2)As such, Pt and Qt are closely related to Eg and θg∞, respectively. This incompatibilityresults in unsuccessful self synchronization. Next, based on small-signal analysis, I provideanalytical justification for the unsuccessful self synchronization observed in Fig. C.2. For thesynchronverter system in Fig. C.1, the state vector containsx1 = [ωg∞, θg∞, Tef , ψf , Qtf , ψff ]T . (C.3)179Figure C.2: Time-domain simulation of self synchronization with only virtual resistance R˜v.We note that both the active- and reactive-power reference values, i.e., P ?t and Q?t , are set to bezero (as noted in Table C.1), and Switches 2 and 4 are disconnected during self synchronization.According to Fig. C.1(c1), we get the following equations describing the APL dynamics:Jgddtωg∞ = −Tef −Df ddt(Tefψff), (C.4)ddtθg∞ = ωg∞, (C.5)τfddtTef = −Tef + PtωN. (C.6)Also, based on Fig. C.1(d1), RPL dynamics can be expressed asKgddtψf = −Qtf , (C.7)τfddtQtf = −Qtf +Qt, (C.8)τfddtψff = −ψff + ψf . (C.9)Substituting (C.1) and (C.2) into (C.6) and (C.8), and also considering that Eg =√32ωgψf =180√32 (ωg∞ + ω∞)ψf , we get thatddtTef = − 1τfTef +√32(ωg∞ + ω∞)ψfU∞ cos θg∞ωN R˜vτf− U∞2ωN R˜τf, (C.10)ddtQtf = − 1τfQtf −√32(ωg∞ + ω∞)ψfU∞ sin θg∞R˜vτf. (C.11)Then substitute (C.9) and (C.10) into (C.4) to getddtωg = − 1JgTef −√32Df (ωg∞ + ω∞)ψfU∞ cos θg∞Jgψff ωN R˜vτf+DfU∞2ωN R˜vτf+DfTef ψfJgψff2τf. (C.12)With the above in place, the dynamical model of the synchronverter with only the virtualresistance R˜v comprises (C.5), (C.7), (C.9) and (C.10)–(C.12). Setting these to zero and solving(recall that ψff > 0), we find that the system has a family of equilibrium points x◦1, as follows:x◦1 =[0, 2kpi, 0,√23U∞ω∞, 0,√23U∞ω∞]T= [ω◦g∞, θ◦g∞, T◦ef , ψ◦f , Q◦tf , ψ◦ff ]T, (C.13)where k ∈ Z. Then, we linearize the nonlinear model around x◦1 to get the following linearstate-space model:d∆x1dt= A1∆x1, (C.14)where matrix A1 is reported in (C.15). With the parameter values reported in Table C.1, wefind that the eigenvalues of system matrix A1 areλ1 1 = −100.00 + j0.0000, λ1 2 = −499.88 + j0.0000A1 =−√32DfU∞JgτfωN R˜v0 − 1Jg +√32Dfω∞JgU∞τf −32Dfω∞2JgωN R˜vτf0 01 0 0 0 0 0U∞2ωNω∞R˜vτf0 − 1τf√32ω∞U∞ωN R˜vτf0 00 0 0 0 − 1Kg 00 − U∞2R˜vτf0 0 − 1τf 00 0 0 1τf 0 − 1τf. (C.15)181Figure C.3: Eigenvalues of the synchronverter with only virtual resistance R˜v > 0 in case (i).When changing R˜v from 1 to 799 Ω, we find that one pair of eigenvalues, i.e., λ1 5 and λ1 6,are always on the right half-plane in the complex plane. This explains the unsuccessful self-synchronization simulation results in case (i), as observed in Fig. C.2.λ1 3 = −167.94 + j0.0000, λ1 4 = −0.062100 + j0.0000λ1 5 = 33.933 + j99.376, λ1 6 = 33.933− j99.376.Both λ1 5 and λ1 6 have positive real parts, which indicates that the linearized system (C.14)is unstable, and the family of equilibrium points x◦1 are not attractors. In fact, as shown inFig. C.3, with R˜v > 0, the state-space model (C.14) has a pair of eigenvalues with positive realparts. Since small-signal stability is a necessary condition for large-signal stability, the systemis not stable in the large. Thus, trajectories of the nonlinear dynamical system do not convergeto the equilibrium points x◦1.With the analysis above, we conclude that the synchronverter equipped with only the virtualresistance R˜v does not achieve self synchronization successfully.C.2 Synchronverter with only virtual inductance L˜v.The block diagram for the synchronverter in this case is depicted in Fig. C.4. I first demonstrate,via simulations, that this design does not achieve successful self synchronization. Then, viasmall-signal analysis, I provide analytical justification for the simulation results.Simulation Results. We simulate the system in Fig. C.4 in PSCAD/EMTDC and theself-synchronization process starts from t = 0 s. In the simulation, the phase-angle difference isinitialized at θg∞(0) = −3.14 rad and the excitation flux to ψf (0) = 0.01 V · s. Time-domain182Figure C.4: Case (ii)—synchronverter equipped with only virtual inductance L˜v ((a1)–(d1))and its equivalent representation during self synchronization ((a2)–(d2)).trajectories of key state variables θg∞ and ψf are plotted in Fig. C.5. Via visual inspection,we observe that θg∞(t) oscillates around zero but does not converge to it. Furthermore, ψf (t)overshoots and then oscillates within the range (1.0, 2.0) p.u. instead of converging to 1 p.u.,as desired. Hence, adopting only the virtual inductance L˜v does not ensure successful selfsynchronization.Analytical Justification. The unsuccessful self synchronization observed in Fig. C.5 canbe substantiated via small-signal analysis. For the synchronverter system in Fig. C.4, the statevector containsx2 = [ωg∞, θg∞, Tef , ψf , Qtf , ψff , Igd, Igq]T , (C.16)where Igd and Igq are additional states associated dynamics introduced by the virtual induc-tance L˜v in Fig. C.4(b1) (or Fig. C.4(a2)). Note that I assume that the d-axis is in alignwith the grid voltage phasor√23U∞∠0◦, and q-axis leads d-axis by 90◦ as shown in Fig. C.6.183Figure C.5: Time-domain simulation of self synchronization with only virtual inductance L˜v.Figure C.6: Phasor diagram of proposed synchronverter with only virtual inductance L˜v inFig. C.4.Moreover, there is no zero sequence components in the circuit.With x2 in place, we derive the full-order dynamical model for the synchronverter with onlyvirtual inductance L˜v in Fig. C.4. Denote, by Egd and Egq, the d-axis and q-axis componentsof the synchronverter inner voltage Eg, and denote, by U∞d and U∞q, the d-axis and q-axis184components of the grid voltage U∞. We can relate these byEgd − U∞d = L˜v ddtIgd − ω∞L˜vIgq, (C.17)Egq − U∞q = L˜v ddtIgq + ω∞L˜vIgd. (C.18)Further considering thatEgd = ωgψf cos θg∞ = (ωg∞ + ω∞)ψf cos θg∞, (C.19)Egq = ωgψf sin θg∞ = (ωg∞ + ω∞)ψf sin θg∞, (C.20)U∞d =√23U∞, (C.21)U∞q = 0, (C.22)(C.17) and (C.18) can be rewritten to obtain state equations describing dynamics of Igd and Igq,as follows:ddtIgd =1L˜v(ωg∞ + ω∞)ψf cos θg∞ −√23U∞L˜v+ ω∞Igq, (C.23)ddtIgq =1L˜v(ωg∞ + ω∞)ψf sin θg∞ − ω∞Igd. (C.24)Dynamics pertaining to the APL and RPL, which are described by (C.4)–(C.6) and (C.7)–(C.9),respectively, remain the same as in case (i). Unlike case (i), however, the APL and RPL inputsare expressed as, respectively,Pt = Pv =32U∞dIgd =√32U∞Igd, (C.25)Qt = Qt = −32U∞dIgq = −√32U∞Igq. (C.26)Now, substituting (C.25) and (C.26) into (C.6) and (C.8), respectively, we getddtTef = − 1τfTef +√32U∞IgdωNτf, (C.27)ddtQtf = − 1τfQtf −√32U∞Igqτf. (C.28)185Leveraging the quotient rule for ddt(Tefψff)in (C.4), and further substitute (C.27) and (C.9) intothe resultant (C.4), we getddtωg = − 1JgTef −√32DfU∞IgdJgωNτfψff+DfTef ψfJgψff2τf. (C.29)With the above in place, the dynamical model of the synchronverter with only the virtualinductance L˜v comprises (C.23), (C.24), (C.5), (C.7), (C.9), and (C.27)–(C.29), collected below:ddtωg∞ = − 1JgTef −√32DfU∞IgdJgωNτfψff+DfTef ψfJgψff2τf. (C.30)ddtθg∞ = ωg∞, (C.31)ddtTef = − 1τfTef +√32U∞IgdωNτf, (C.32)ddtψf = − 1KgQtf , (C.33)ddtQtf = − 1τfQtf −√32U∞Igqτf. (C.34)ddtψff = − 1τfψff +1τfψf . (C.35)ddtIgd =1L˜v(ωg∞ + ω∞)ψf cos θg∞ −√23U∞L˜v+ ω∞Igq, (C.36)ddtIgq =1L˜v(ωg∞ + ω∞)ψf sin θg∞ − ω∞Igd, (C.37)Setting (C.30)–(C.37) to zero and solving (recall that ψff > 0), we find that the system has afamily of equilibrium points x◦2, as follows:x◦2 =[0, 2kpi, 0,√23U∞ω∞, 0,√23U∞ω∞, 0, 0]T=: [ω◦g∞, θ◦g∞, T◦ef , ψ◦f , Q◦tf , ψ◦ff , I◦gd, I◦gq]T, (C.38)where k ∈ Z. Then, we linearize the nonlinear model (C.30)–(C.37) around x◦2 to obtain thefollowing linear state-space model:d∆x2dt= A2∆x2, (C.39)186where matrix A2 is expressed in (C.40). With the parameter values reported in Table C.1, wefind that the eigenvalues of system matrix A2 areλ2 1 = −100.00 + j0.0000, λ2 2 = −0.062100 + j0.0000λ2 3 = −49.969 + j1356.0, λ2 4 = −49.969− j1356.0,λ2 5 = −51.815 + j49.873, λ2 6 = −51.815− j49.873,λ2 7 = 1.8145 + j370.57, λ2 8 = 1.8145− j370.57. (C.41)Both λ2 7 and λ2 8 have positive real parts, which indicates that the linearized system in (C.39)is unstable, and the family of equilibrium points x◦2 are not attractors. In fact, as shown inFig. C.7, with L˜v > 0, the system (C.39) has a pair of eigenvalues with positive real parts.Since small-signal stability is a necessary conditions for large-signal stability, the system is notstable in the large. Thus, trajectories of the nonlinear dynamical system do not converge tothe equilibrium point x◦2.Unstable Mode in Linearized System. To show that the unstable eigenvalues, i.e.,λ2 7 and λ2 8, are indeed caused by dynamics associated with the virtual inductance L˜v, we setthe derivatives in (C.36) and (C.37) to zero (effectively assuming that the virtual-inductanceA2 =0 0 − 1Jg +√32Dfω∞JgU∞τf 0 0 0 −32Dfω∞JgωN τf01 0 0 0 0 0 0 00 0 − 1τf 0 0 0√32U∞ωN τf00 0 0 0 − 1Kg 0 0 00 0 0 0 − 1τf 0 0 −√32U∞τf0 0 0 1τf 0 − 1τf 0 0√23U∞L˜vω∞0 0 ω∞L˜v0 0 0 ω∞0√23U∞L˜v0 0 0 0 −ω∞ 0=:[A¯ B¯C¯ D¯]. (C.40)187Figure C.7: Eigenvalues of the synchronverter with only virtual inductance L˜v > 0 in case (ii).When changing L˜v from 1 to 100 mH, we find that one pair of eigenvalues, i.e., λ2 7 and λ2 8,are always on the right half-plane in the complex plane. This explains the unsuccessful self-synchronization simulation results in case (ii), as observed in Fig. C.5.dynamics are infinitely fast) to get0 =1L˜v(ωg∞ + ω∞)ψf cos θg∞ −√23U∞L˜v+ ω∞Igq, (C.42)0 =1L˜v(ωg∞ + ω∞)ψf sin θg∞ − ω∞Igd. (C.43)Combining the above with (C.30)–(C.35), we obtain the following model:d∆x¯2dt= A¯∆x¯2 + B¯∆y¯2, (C.44)0 = C¯∆x¯2 + D¯∆y¯2, (C.45)wherex¯2 = [ωg∞, θg∞, Tef , ψf , Qtf , ψff ]T , y¯2 = [Igd, Igq]T , (C.46)and matrices A¯, B¯, C¯, and D¯ are defined in (C.40). Then, by solving ∆y¯2 from (C.45) andsubstituting the resultant into (C.44), we get the state-space model neglecting the inductancedynamics, as follows:d∆x¯2dt=(A¯− B¯D¯−1C¯)∆x¯2 =: A¯2∆x¯2, (C.47)where A¯2 is shown in (C.48). With A¯2 in place and using the parameter settings for case (ii)188in Table C.1, we compute the eigenvalues for system in (C.47) and getλ¯2 1 = −100.00 + j0.0000, λ¯2 2 = −0.062100 + j0.0000λ¯2 3 = −49.969 + j1356.0, λ¯2 4 = −49.969− j1356.0,λ¯2 5 = −50.000 + j49.975, λ¯2 6 = −50.000− j49.975. (C.49)By comparing the eigenvalues in (C.41) and (C.49), we find that by omitting the virtual-inductance dynamics, the unstable eigenvalue pair is removed. In contrast, other (stable)eigenvalues remain nearly unchanged. Thus, the unstable eigenvalues of the linearized systemin (C.39) are associated with the virtual-inductance dynamics.With the analysis above, we conclude that the synchronverter equipped with only the virtualinductance L˜v does not achieve self synchronization successfully, and this results from thedynamics introduced by the virtual inductance.C.3 Proposed synchronverter with the virtual resistance Rvalong with coordinate transformation.The block diagram for the synchronverter in this case is depicted in Fig. C.8. As shown inFigs. C.9(a) and (b), unlike simulations results in cases (i) and (ii), the phase-angle differ-ence θg∞ converges to zero within about 0.01 s, and the excitation flux ψf converges to 1 p.u.within around 0.12 s. Below, I offer two perspectives that help to explain why the proposedsynchronverter design is advantageous over those in cases (i) and (ii).Application Setting. The use of the virtual resistance Rv together with the coordinatetransformation emulates inductive grid conditions, as shown in Fig. C.8(a2). This matches wellA¯2 =0 −√32DfU∞JgωN τf L˜v− 1Jg +√32Dfω∞JgU∞τf 0 0 01 0 0 0 0 00 U∞2ωN τfω∞L˜v− 1τf 0 0 00 0 0 0 − 1Kg 0U∞2τfω∞2L˜v0 0√32U∞τf L˜v− 1τf 00 0 0 1τf 0 − 1τf. (C.48)189Figure C.8: Case (iii)—synchronverter with virtual resistance R˜v together with a coordinatetransformation ((a1)–(d1)) and its equivalent representation during self synchronization ((a2)–(d2)).with the APL and RPL designs in Figs C.8(c1) and (d1), i.e., the active- and reactive-powerfeedback signals are, respectively, closely related to θg∞ and Eg. It also avoids the incompat-ibility between the controller and the computation block, which leads to the unsuccessful selfsynchronization in case (i).Avoiding Undesirable Dynamics. By using the resistance together with the coordinatetransformation instead of the virtual inductance, we avoid the inductance dynamics arisingfrom the integrator in Fig. C.8(b1), which causes unsuccessful self synchronization in case (ii).As shown in Fig. C.8(b1), my virtual resistance design involves only algebraic manipulations.In fact, by linearizing the nonlinear dynamical system for this case ((5.12)–(5.17)), we obtainthe following state-space model:d∆x3dt= A3∆x3, (C.51)where the state matrix A3 is given by (C.50), which we can recover by setting L˜v = Rv/ω∞190Figure C.9: Self-synchronization process with virtual resistance Rv together with the coordinatetransformation.in (C.48). This further verifies that the resistance Rv together with the coordinate acts as thevirtual inductance L˜v without the undesirable inductance dynamics.A3 =0 −√32DfU∞ω∞JgωN τfRv− 1Jg +√32Dfω∞JgU∞τf 0 0 01 0 0 0 0 00 U∞2ωN τfRv− 1τf 0 0 00 0 0 0 − 1Kg 0U∞2τfω∞Rv 0 0√32U∞ω∞τfRv− 1τf 00 0 0 1τf 0 − 1τf. (C.50)191Appendix DDerivation of Reduced Second-order APL Modelin Chapter 5During time scales that are relevant to the faster phase-angle self-synchronization dynamics, thefollowing assertions are valid: (i) 0 < ψf (t) < ψ◦f because ψf (t) increases from ψf (0) > 0 buthas not yet converged to ψ◦f (see traces marked as (ii) in Figs. 5.3(b) and 5.3(d)), and (ii) 0 <ψff (t) < ψf (t) as the filtered signal ψff (t) is delayed compared with ψf (t). Combining thestatements above, we have that 0 < ψff (t) < ψf (t) < ψ◦f . Also recall that Df  τfψ◦f fromAssumption 1. With these in mind, we get thatDfψfτfψ2ff=Dfτfψ◦f· ψ◦fψff· ψfψff 1. (D.1)Thus, we can approximate (5.13) asdωg∞dt= −Df√32(ω∞ + ωg∞)ψfU∞JgτfωNRvψffsin θg∞ +Dfψfτfψ2ff· TefJg. (D.2)Furthermore, by rearranging (5.14) and (5.17), we get that√32(ω∞ + ωg∞)ψfU∞τfωNRvsin θg∞ =dTefdt+Tefτf, (D.3)ψfτf=dψffdt+ψffτf. (D.4)192Then, by substituting (D.3) and (D.4), respectively, into the first and second terms on theright-hand side of (D.2), and further simplifying the resultant expression, we get thatdωg∞dt= −DfJg· ddt(Tefψff), (D.5)where we make use of the quotient rule for derivatives. Next, by assuming that ωg∞(0) ≈ 0and Tef (0) = 0, we integrate both sides of (D.5) to yieldTef = −Jg ωg∞ψffDf. (D.6)Finally, substituting (D.6) into (D.2) and bearing in mind Assumption 2, we obtain (5.20),as desired. The second-order model consisting of (5.12) and (5.20) approximates the APLdynamics during phase-angle self synchronization.193Appendix EProof of Self-synchronization Capability ofSynchronverter Proposed in Chapter 5Analytical proof of successful self synchronization is necessary, since repeated simulations andexperiments are valid only on a case-by-case basis and do not guarantee the self-synchronizationcapability of the synchronverter under all initial conditions. Here, via stability analysis, weshow that the proposed design in Chapter 5 successfully achieves self synchronization for allθg∞(0) ∈ (−pi, pi) rad (the initial value for other state variables are ωg∞(0) = 0, Tef (0) = 0,ψf (0) = ψff (0) = 0.01, Qtf (0) = 0, since they can be initialized in the controller). Successfulself synchronization is achieved when θg∞ and ψf , respectively, converge to θ◦g∞ = 2kpi, k ∈ Z(k = 0 in most cases), and ψ◦f =√23U∞ω∞ .Since studying the dynamics of the full-order system (5.12)–(5.17) is analytically intractable,I resort to two reduced-order models, i.e., the APL model in (5.12) and (5.20) and the RPLmodel in (5.15) and (5.24), which capture phase-angle synchronization dynamics in the APLand voltage-magnitude synchronization dynamics in the RPL. As shown in Section 5.2 andverified in Sections 5.3 and 5.4, these two reduced-order models are valid for a well-tunedsynchronverter.Convergence of θg∞(t) to zero. The set of possible initial conditions of the reduced second-order APL model in (5.12) and (5.20) is given byB = {(θg∞, ωg∞) | −pi < θg∞ < pi and ωg∞ = 0} . (E.1)194Figure E.1: Lyapunov function V (θg∞, ωg∞) of the second-order APL model in (5.12)and (5.20).For the system in (5.12) and (5.20), consider the Lyapunov function candidateV (θg∞, ωg∞) =τfcα(ωg∞ + ω∞ ln(ω∞ω∞ + ωg∞))+ (1− cos θg∞) . (E.2)As shown in Fig. E.1, letD ={(θg∞, ωg∞)∣∣∣∣ −pi < θg∞ < pi, ωg∞ > −ω∞, and V (ωg∞, θg∞) < 2} ,so that B ⊂ D; V is positive definite in set D, andV˙ (θg∞, ωg∞) =∂V∂θg∞· dθg∞dt+∂V∂ωg∞· dωg∞dt=− ω2g∞α(ω∞ + ωg∞)≤ 0, (E.3)for all (θg∞, ωg∞) ∈ D. Let S = {(θg∞, ωg∞) ∈ D | V˙ (θg∞, ωg∞) = 0}. Note that V˙ (θg∞, ωg∞) =0 only if ωg∞ = 0. Hence, S = {(θg∞, ωg∞) ∈ D | ωg∞ = 0}, which contains only the triv-ial trajectory θg∞(t) = 0 and ωg∞(t) = 0. To see this, consider ωg∞ = 0 and θg∞ 6= 0,195thendωg∞dt 6= 0, so the trajectory will not remain within S. Therefore, according to LaSalle’stheorem (see, e.g., [87]), the origin is asymptotically stable and we conclude that all trajectoriesstarting from initial points in set B converge to the origin. In other words, for a well-tuned self-synchronizing synchronverter, both θg∞(t) and ωg∞(t) converge to zero with initial phase-angledifference θg∞(0) ∈ (−pi, pi) rad, as desired.Convergence of ψf (t) to ψ◦f . The reduced-order RPL model in (5.15) and (5.24) is linear.By defining ψf = ψf − ψ◦f , where ψ◦f =√23U∞ω∞ , we get the following equivalent system:ddtψfQtf = 0 − 1Kg√32ω∞U∞τfRv− 1τfψfQtf . (E.4)Also, substituting (5.29) into (E.4), we find that the eigenvalues of the system in (E.4) areλ1 = − 12τf+ j12τf, λ2 = − 12τf− j 12τf, (E.5)which have negative real parts. Thus, ψf (t) and Qtf (t) both converge to zero. Equivalently, ψfconverges to ψ◦f =√23U∞ω∞ , as desired.196

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