Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Nowak, Severin
Doctor of Philosophy - PhD
University of British Columbia
Widespread integration of distributed energy resources (DERs), such as solar photovoltaic, wind, and storage systems helps to alleviate environmental concerns of conventional fossil fuel-based generation. Moreover, DERs can provide valuable benefits to the reliable and efficient operation of the integrated power grid when dispatched accurately and timely. Furthermore, emerging advanced sensors provide vast amounts of data, which enables new opportunities in the operation of power grids. These trends have motivated recent research into developing techniques to dispatch DERs for optimization and control of power distribution systems. Existing research involves repeated solutions of optimization problems constrained by the nonlinear network power balance, DER capacity limits, and other operational limits. Such methods may be computationally burdensome in practical implementations, especially with rapidly varying operating points. Moreover, many require an accurate and up-to-date network model that may not be available in real time, resulting in DER setpoints that lead to unexpected or undesired system behaviour. In this thesis, the overarching goal is to develop new techniques to optimize and control modern power distribution systems with high penetrations of DERs by leveraging real-time measurements provided by advanced sensors such as distribution-level phasor measurement units. Specifically, the research focuses on (i) centrally dispatching optimal DER active- and reactive-power outputs by recursively estimating sensitivity models relating bus voltages and power injections, (ii) leveraging distributed estimation and optimization methods to alleviate the scalability concerns of the centralized measurement-based DER dispatch method, (iii) capitalizing on system-wide optimization and local measurements to regulate system voltages in cases with highly intermittent renewable generation, and (iv) minimizing network power losses and cost of DER reactive-power outputs based on a measurement-based gradient-descent method. Overall, the proposed research advances the efficient and reliable operation of modern power distribution systems and facilitates the widespread integration of DERs by using measurement-based techniques. As a consequence, this work contributes to transitioning toward a more environmentally-friendly electric power supply by enabling higher penetrations of renewables.
Measurement-based Techniques for Optimization and Controlof Power Distribution Systems Using Distributed EnergyResourcesbySeverin NowakB.Sc., University of Applied Sciences and Arts Western Switzerland – Fribourg, 2008a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe college of graduate studies(Electrical Engineering)The University of British Columbia(Okanagan)February 2021© Severin Nowak, 2021The following individuals certify that they have read, and recommend to The College of Grad-uate Studies for acceptance, the dissertation entitled:Measurement-based Techniques for Optimization and Control of Power Dis-tribution Systems Using Distributed Energy Resourcessubmitted by Severin Nowak in partial fulfillment of the requirements for the degree of Doc-tor of Philosophy in Electrical Engineering.Examining Committee:Liwei Wang, School of Engineering, UBC OkanaganSupervisorYu Christine Chen, Electrical and Computer Engineering, UBC VancouverCo-SupervisorWilson Eberle, School of Engineering, UBC OkanaganSupervisory Committee MemberJonathan Holzman, School of Engineering, UBC OkanaganSupervisory Committee MemberZheng Liu, School of Engineering, UBC OkanaganUniversity ExaminerBala Venkatesh, Electrical and Computer Engineering, Ryerson UniversityUniversity ExamineriiAbstractWidespread integration of distributed energy resources (DERs), such as solar photovoltaic,wind, and storage systems helps to alleviate environmental concerns of conventional fossil fuel-based generation. Moreover, DERs can provide valuable benefits to the reliable and efficientoperation of the integrated power grid when dispatched accurately and timely. Furthermore,emerging advanced sensors provide vast amounts of data, which enables new opportunities inthe operation of power grids. These trends have motivated recent research into developing tech-niques to dispatch DERs for optimization and control of power distribution systems. Existingresearch involves repeated solutions of optimization problems constrained by the nonlinear net-work power balance, DER capacity limits, and other operational limits. Such methods maybe computationally burdensome in practical implementations, especially with rapidly varyingoperating points. Moreover, many require an accurate and up-to-date network model that maynot be available in real time, resulting in DER setpoints that lead to unexpected or undesiredsystem behaviour.In this thesis, the overarching goal is to develop new techniques to optimize and control mod-ern power distribution systems with high penetrations of DERs by leveraging real-time mea-surements provided by advanced sensors such as distribution-level phasor measurement units.Specifically, the research focuses on (i) centrally dispatching optimal DER active- and reactive-power outputs by recursively estimating sensitivity models relating bus voltages and powerinjections, (ii) leveraging distributed estimation and optimization methods to alleviate the scal-ability concerns of the centralized measurement-based DER dispatch method, (iii) capitalizingon system-wide optimization and local measurements to regulate system voltages in cases withhighly intermittent renewable generation, and (iv) minimizing network power losses and costof DER reactive-power outputs based on a measurement-based gradient-descent method.iiiOverall, the proposed research advances the efficient and reliable operation of modern powerdistribution systems and facilitates the widespread integration of DERs by using measurement-based techniques. As a consequence, this work contributes to transitioning toward a moreenvironmentally-friendly electric power supply by enabling higher penetrations of renewables.ivLay SummaryWidespread integration of distributed energy resources (DERs), such as solar photovoltaic,wind, and storage systems helps to alleviate environmental concerns of conventional fossilfuel-based generation. Such DERs can provide valuable benefits to the reliable and efficientoperation of the integrated power grid when dispatched accurately and timely. Furthermore,emerging advanced sensors provide vast amounts of data, which enables new opportunities inthe operation of power grids. These trends have motivated recent research into developingtechniques to dispatch DERs for optimization and control of power distribution systems. Suchmethods may be challenging to solve due to rapidly varying renewable generation and inaccuratenetwork topology information, which could compromise their performance.In this thesis, the overarching goal is to develop new techniques to optimize and control mod-ern power distribution systems by leveraging real-time measurements. Overall, the proposedresearch within this thesis advances the efficient and reliable operation of modern power distri-bution systems and facilitates the widespread integration of DERs by using measurement-basedtechniques.vPrefaceThis dissertation has been written and prepared by the author under supervision of Prof. Li-wei Wang and Prof. Yu Christine Chen. Most of the research results in this thesis have beenpublished in (or were submitted to) scientific journals and conference proceedings. In all ofthem, the author was responsible for developing the ideas, deriving the mathematical models,performing the theoretical analyses, and conducting simulation studies. The supervisors pro-vided detailed guidance, highly valuable feedback, and extremely helpful comments during allstages of these research works, from developing the ideas to writing the papers. All publicationsresulting from this Ph.D. program are listed below:Chapter 3• S. Nowak, Y. C. Chen, and L. Wang, “Measurement-based optimal DER dispatch witha recursively estimated sensitivity model,” IEEE Trans. Power Syst., vol. 35, no. 6, pp.4792–4802, Nov. 2020.• S. Nowak, L. Wang, and Y. C. Chen, “Measurement-based optimal power flow with linearpower-flow constraint for DER dispatch,” in Proc. North American Power Symposium,Oct. 2019, pp. 1–5.Chapter 4• S. Nowak, Y. C. Chen, and L. Wang, “Distributed measurement-based optimal DERdispatch with estimated sensitivity models,” IEEE Trans. Power Syst., submitted.viChapter 5• S. Nowak, L. Wang, and M. S. Metcalfe, “Two-level centralized and local voltage controlin distribution systems mitigating effects of highly intermittent renewable generation,”Int. J. Electr. Power Energy Syst., vol. 119, p. 105858, Jul. 2020.• S. Nowak, M. S. Metcalfe, W. Eberle, and L. Wang, “Comparison of voltage controlmethods in distribution systems using Q-V based PI and droop controls of solar inverters,”in Proc. IEEE Power Energy Soc. General Meeting, Jul. 2017, pp. 1-5.Chapter 6• S. Nowak, Y. C. Chen, and L. Wang, “A measurement-based gradient-descent methodto optimally dispatch DER reactive power,” in Proc. IEEE Photovoltaics Specialist Conf.,Jun. 2020, pp. 1-5. Best Paper Award.Other publications• S. Nowak, L. Wang, W. Eberle, A.M. Shotorbani, and M. S. Metcalfe, “Measurement-based network model reduction of distribution systems using two-port networks,” inProc. IEEE Canadian Conf. Elec. Comp. Eng., May 2019, pp. 1-4.• S. Nowak, N. Tehrani, M. S. Metcalfe, W. Eberle, and L. Wang, “Cloud-based DERMStest platform using real-time power system simulation,” in Proc. IEEE Power Energy Soc.General Meeting, Jul. 2018, pp. 1-5.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1 The Electric Power System . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 Transition to Sustainable Electricity Supply . . . . . . . . . . . . . . . . . 51.2.3 Traditional Distribution System Control . . . . . . . . . . . . . . . . . . . 51.2.4 Distributed Energy Resources in Power Systems . . . . . . . . . . . . . . 61.2.5 Challenges and Opportunities Integrating High Penetrations of DERs inDistribution Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8viii1.2.6 Monitoring of Distribution Systems . . . . . . . . . . . . . . . . . . . . . . 101.3 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3.1 Optimal DER Dispatch Methods . . . . . . . . . . . . . . . . . . . . . . . 111.3.2 Voltage Control Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3.3 Measurement-based Applications . . . . . . . . . . . . . . . . . . . . . . . 161.4 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.4.1 Objective I: Centralized Method with a Recursively Estimated SensitivityModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.4.2 Objective II: Distributed Method with Estimated Per-area SensitivityModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.4.3 Objective III: Mitigating Effects of Highly Intermittent Renewable Gen-eration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.4.4 Objective IV: Gradient-descent Method for Optimal Reactive-Power Dis-patch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2 Network- and Power-flow Model Representation . . . . . . . . . . . . . . . . . . . 232.3 Linearized Power-flow Model Representation . . . . . . . . . . . . . . . . . . . . . 252.4 Optimal DER Dispatch Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5 Linearized Optimal DER Dispatch Problem . . . . . . . . . . . . . . . . . . . . . 273 Centralized Method with a Recursively Estimated Sensitivity Model . . . 293.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2 Estimation of Sensitivity Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2.2 RWPLS-based Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3 DER Dispatch Problem with Estimated Sensitivity Model . . . . . . . . . . . . . 353.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3.2 ADMM-based Solution Approach . . . . . . . . . . . . . . . . . . . . . . . 38ix3.4 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.4.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.4.2 Adaptability to Topology and Operating-point Changes . . . . . . . . . . 413.4.3 Effect of RWPLS Forgetting Factor . . . . . . . . . . . . . . . . . . . . . . 443.4.4 Effect of DER Dynamics and Delays . . . . . . . . . . . . . . . . . . . . . 453.4.5 Consideration of Typical DER Constraints . . . . . . . . . . . . . . . . . 473.4.6 Execution Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 Distributed Method with Estimated Per-area Sensitivity Models . . . . . . 494.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.2 Per-area Sensitivity Model Estimation . . . . . . . . . . . . . . . . . . . . . . . . 514.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2.2 Solution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.3 Distributed Optimal DER Dispatch . . . . . . . . . . . . . . . . . . . . . . . . . . 544.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.3.2 ADMM-based Solution Approach . . . . . . . . . . . . . . . . . . . . . . . 574.4 Application to Multi-phase System . . . . . . . . . . . . . . . . . . . . . . . . . . 614.4.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.4.2 Execution Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.4.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 Two-level Voltage Control Method Mitigating Effects of Highly Intermit-tent Renewable Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.1.1 Branch Flow Model Formulation . . . . . . . . . . . . . . . . . . . . . . . 685.1.2 Linearized Branch Flow Model Formulation . . . . . . . . . . . . . . . . . 695.1.3 DER Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.2 Lower-Level Local Control Scheme and Stability Assessment . . . . . . . . . . . . 705.2.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70x5.2.2 State-of-the-Art Local Controllers . . . . . . . . . . . . . . . . . . . . . . 715.2.3 Proposed Local PI Controller . . . . . . . . . . . . . . . . . . . . . . . . . 725.2.4 Evaluation of Local Control Schemes . . . . . . . . . . . . . . . . . . . . . 775.3 Upper-Level Centralized Controller . . . . . . . . . . . . . . . . . . . . . . . . . . 775.4 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.4.1 Different Control Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.4.2 Configuration of Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . 835.4.3 Simulation Results with Balanced Loading Conditions . . . . . . . . . . . 855.4.4 Simulation Results with Unbalanced Loading Conditions . . . . . . . . . . 915.4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946 Gradient-descent Method for Optimal Reactive-Power Dispatch . . . . . . 956.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.2 Measurement-based DER Dispatch via Gradient Updates . . . . . . . . . . . . . 986.2.1 Estimation of Sensitivity Models . . . . . . . . . . . . . . . . . . . . . . . 986.2.2 Optimal Reactive-power Dispatch with Gradient Descent Method . . . . . 1006.3 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.3.1 Loss Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.3.2 Loss and DER Cost Minimization . . . . . . . . . . . . . . . . . . . . . . 1046.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057 Research Contributions and Future Directions . . . . . . . . . . . . . . . . . 1077.1 Contribution Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077.2 Concluding Remarks and Future Directions . . . . . . . . . . . . . . . . . . . . . 110Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112A Deriving Partial Least-squares Estimate in (3.7) of Chapter 3 . . . . . . . . 121B Reformulating (3.12) as Standard Quadratic Program in Chapter 3 . . . . 123C Derivation of (5.12) in Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . 124xiList of TablesTable 3.1 Comparison of relative errors in cost function, voltage magnitudes, and active-and reactive-power dispatch among (i) model-based DER dispatch with ac-curate system model, (ii) model-based DER dispatch with inaccurate systemmodel, and (iii) proposed measurement-based DER dispatch. . . . . . . . . . . 43Table 3.2 Computation times for (i) sensitivity model estimation (test), (ii) optimal DERdispatch via ADMM (tADMM), and (iii) optimal DER dispatch via MATLAB’sstandard solver quadprog (tIP) . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Table 4.1 Comparison of cost-function and relative-errors among (i) the centralizedmodel-based nonlinear DER dispatch, (ii) the centralized measurement-basedDER dispatch, and (iii) the proposed distributed measurement-based DERdispatch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Table 4.2 Comparison of execution times. . . . . . . . . . . . . . . . . . . . . . . . . . . 63Table 5.1 Overview of control methods for case studies . . . . . . . . . . . . . . . . . . . 83Table 5.2 Summary of simulation results for balanced operation . . . . . . . . . . . . . . 90Table 5.3 Summary of simulation results with OLTC transformer . . . . . . . . . . . . . 90Table 5.4 Summary of simulation results for unbalanced operation . . . . . . . . . . . . 93xiiList of FiguresFigure 1.1 Massive deployment of DERs, such as solar PV, wind, and storage systemsis crucial in the transition to a more sustainable energy future. Additionalbenefits for grid operations are enabled when DERs are suitably dispatchedvia online optimization routines. However, optimal DER dispatch may becompromised by inaccurate offline network models due to incorrect topologyinformation and fast-changing operating points, resulting in suboptimal orundesired system behaviours. The research in this thesis is motivated by theuse of high-precision measurements from D-PMUs to develop measurement-based optimal DER dispatch methods. . . . . . . . . . . . . . . . . . . . . . . 3Figure 1.2 Overview of research objectives. . . . . . . . . . . . . . . . . . . . . . . . . . 18Figure 1.3 Centralized measurement-based optimal DER dispatch method with (i) modelestimation and (ii) optimal DER dispatch. . . . . . . . . . . . . . . . . . . . . 19Figure 1.4 Distributed measurement-based optimal DER dispatch method with (i) modelestimation and (ii) optimal DER dispatch for each area. . . . . . . . . . . . . 20Figure 3.1 IEEE 33-bus test system with DERs at buses 6, 12, 18, 25, and 33; measure-ments are collected at bus 1 and all DER buses. Switches SW1 and SW2 arenormally open, and switches SW3 and SW4 are normally closed. . . . . . . . 35xiiiFigure 3.2 Time-domain simulation via proposed measurement-based optimal DER dis-patch. DERs are activated at time tstart = 5 s and the network topologyand operating point are changed at time t∆ = 10 s. Top pane: bus voltagemagnitudes Vi at measured buses; Middle pane: bus active-power injectionsPi at measured buses; Bottom pane: substation reactive-power injection Q1and DER reactive-power outputs Qgeni at controllable buses. . . . . . . . . . . 42Figure 3.3 Bus voltage magnitudes Vi at t = 16 s obtained via (i) model-based methodwith accurate system model, (ii) model-based method with inaccurate sys-tem model, and (iii) proposed measurement-based method. Voltage profileresulting from the proposed measurement-based method matches that frommodel-based benchmark with accurate system model, and it outperforms themodel-based approach when the system model is not updated. . . . . . . . . 43Figure 3.4 Effect of forgetting factor in RWPLS algorithm on dispatch accuracy. (a) Meancost-function RMSDs among model-based dispatch and measurement-baseddispatch with different forgetting factors σ for 100 repeated simulations.(b) Histogram of RMSD values for the repeated simulations, with best pa-rameter choice (σ = 0.6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Figure 3.5 Evaluation of cost function value over time for different DER time constants τwith the measurement-based DER dispatch method, compared to the model-based method with ideal DERs. . . . . . . . . . . . . . . . . . . . . . . . . . . 45Figure 3.6 Effect of delays between measurement acquisition and DER actuation. DERsare activated at time tstart = 5 s and the network topology and operatingpoint are changed at time t∆ = 10 s. Top pane: evaluation of cost functionvalue over time for different DER dispatch delays ξ with the measurement-based DER dispatch method, compared to the model-based method withoutdelay; Bottom pane: bus voltage magnitudes Vi at measured buses in caseDER actuation is delayed by ξ = 3 s. . . . . . . . . . . . . . . . . . . . . . . . 46Figure 3.7 DER active- and reactive-power outputs obtained via the measurement-based method when each DER is constrained by (a) independent active-and reactive-power limits, (b) its maximum apparent power. . . . . . . . . . 47xivFigure 4.1 IEEE 33-bus test system [1] with DERs connected at buses collected in D ={6, 12, 18, 22, 25, 33}. The network is partitioned into five areas collected inthe set A = {1, 2, . . . , 5}. Assume D-PMUs are installed at all buses. Thistest system is single phase, so Pi = {a}, ∀ i ∈ N . As an example of thenotation used throughout this chapter, consider area 1. Within area 1, busesequipped with D-PMUs are collected in E1 = {1, 2, 3, 4, 5, 6, 23, 24, 25} andbuses connected to DERs are collected in D1 = {6, 25}. The extended set ofbuses is E1+ = E1∪{7, 19, 26}. Adjacent areas are collected in A1 = {2, 3, 4}.Boundary buses are collected in B1 = (E1+ ∩ E2+) ∪ (E1+ ∩ E3+) ∪ (E1+ ∩E4+) = {2, 6, 7, 19, 26}. Finally, consider boundary bus 6, which belongs tothe extended set of buses in areas collected in A6 = {1, 3, 4}. . . . . . . . . . 54Figure 4.2 Time-domain simulation via proposed distributed measurement-based opti-mal DER dispatch when DERs are activated at time tstart = 5 s, and at timet∆ = 10 s, the system operating point is changed. Top pane: bus voltages Viat buses with DERs; Middle pane: bus active-power injections Pi at buseswith DERs; Bottom pane: substation reactive-power injection Q1 and DERreactive-power injections Qgeni at buses with DERs. . . . . . . . . . . . . . . . 60Figure 4.3 IEEE 123-bus multi-phase test system partitioned into 6 areas. . . . . . . . . 62Figure 4.4 Normalized daily solar PV output for different locations with intermittentsolar irradiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Figure 4.5 Time-domain simulation of the scenario without active DER control. Toppane: bus voltages Vi at monitored buses; Middle pane: bus active-powerinjections Pi at substation and buses with DERs; Bottom pane: bus reactive-power injections Qi at substation and buses with DERs. . . . . . . . . . . . . 65Figure 4.6 Time-domain simulation via proposed estimation and optimization frame-work. Top pane: bus voltages Vi at monitored buses; Middle pane: bus active-power injections Pi at substation and buses with DERs; Bottom pane: busreactive-power injections Qi at substation and buses with DERs. . . . . . . . 65xvFigure 5.1 (a) Single-phase distribution system network model representation with linesegments between buses i, j, l ∈ N ; (b) Operating zone of DERs (shaded area). 70Figure 5.2 Comparison of different local controllers: (a) VV droop control, (b) delayedVV droop control, (c) scaled droop-based local control, and (d) proposed PIcontrol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Figure 5.3 Overview of proposed two-level control system with the centralized controllerexecuted at intervals ∆tτ and the local PI controllers at bus i and bus jexecuted at intervals ∆tk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Figure 5.4 IEEE 123-bus test system with 10 three-phase DERs (enclosed with solidlines) used for the case studies. Additional 4 single-phase DERs are addedfor the unbalanced case studies (enclosed with dashed lines). . . . . . . . . . 82Figure 5.5 Normalized load and solar generation profiles for the 24-hour simulation period. 83Figure 5.6 Locations of eigenvalues of A˜ within unity circle. . . . . . . . . . . . . . . . . 84Figure 5.7 (a) Bus voltages Vi without reactive-power control (Method 0); (b) Active-power generations P geni for Methods 0-5; (c) Bus voltages Vi and (d) Reactive-power generations Qgeni for Method 1 and their minimum and maximumreactive-power limits (dashed lines) Qgeni,min and Qgeni,max; (e) Bus voltages Viand (f) Reactive-power generations Qgeni for Method 2 and their minimumand maximum reactive-power limits (dashed lines) Qgeni,min and Qgeni,max for bus21 (blue), bus 57 (red), and bus 108 (yellow). . . . . . . . . . . . . . . . . . . 86Figure 5.8 (a) Bus voltages Vi and (b) Reactive-power generations Qgeni for Method 3and their minimum and maximum reactive-power limits (dashed lines) Qgeni,minand Qgeni,max; (c) Bus voltages Vi and (d) Reactive-power generations Qgeni forMethod 4 and their minimum and maximum reactive-power limits (dashedlines) Qgeni,min and Qgeni,max; (e) Bus voltages Vi and (f) Reactive-power gener-ations Qgeni for Method 5 and their minimum and maximum reactive-powerlimits (dashed lines) Qgeni,min and Qgeni,max for bus 21 (blue), bus 57 (red), andbus 108 (yellow). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87xviFigure 5.9 Comparison of Method 3 (Centralized control, blue), Method 4 (Two-levelcontrol with VV droop controllers, yellow) and Method 5 (Two-level controlwith local PI controllers, red). (a) Bus voltages Vi and (b) Reactive-powergeneration Qgeni for low intermittency period between 08:00 AM and 08:12AM at bus 21. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Figure 5.10 Comparison of Method 3 (Centralized control, blue), Method 4 (Two-levelcontrol with VV droop controllers, yellow) and Method 5 (Two-level controlwith local PI controllers, red). (a) Bus voltages Vi and (b) Reactive-powerQgeni for high intermittency period between 11:18 AM and 11:30 AM at bus21. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Figure 5.11 Evaluation of tap position of the substation transformer in the simulationscenario with intermittent renewable generation for the 24-hour period with-out DER reactive-power control (Method 0). (a) Bus voltages Vi at buses21, 57, and 108; and (b) Tap position of the substation transformer. . . . . . 91Figure 5.12 Phase a (blue), phase b (red), and phase c voltages at bus 83, the bus with thehighest unbalance in the system over the 24-hour simulation period. (a) NoDER reactive-power control; (b) DER reactive-power control with local PIcontrollers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Figure 5.13 Voltage profile at all buses for phase a (blue), phase b (red), and phase c(yellow). (a) No DER reactive-power control; (b) DER reactive-power controlwith local PI controllers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Figure 6.1 IEEE 33-bus test system with DER systems at buses 6, 12, 18, 25, and 33.Switches SW1 and SW2 are normally open, and switches SW3 and SW4 arenormally closed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102Figure 6.2 Network loss resulting from the measurement-based DER reactive-power dis-patch method compared to the model-based method. . . . . . . . . . . . . . . 102Figure 6.3 Bus-voltage magnitudes Vi and DER reactive-power generations Qgeni att = 4.9 s prior to changes in network topology and system load obtainedvia (i) model-based method and (ii) proposed measurement-based method. . 103xviiFigure 6.4 Bus-voltage magnitudes Vi and DER reactive-power generations Qgeni at t =7 s after changes in network topology and system load obtained via (i) model-based method and (ii) proposed measurement-based method. . . . . . . . . . 104Figure 6.5 Time-domain simulation via proposed measurement-based optimal DER reactive-power dispatch. DER reactive-power dispatch is activated at time t = 2 s.Top pane: bus voltages Vi at buses with DERs; Bottom pane: substationreactive-power injection Q1 and DER reactive-power outputs Qgeni at con-trollable buses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Figure 6.6 Evaluation of cost function value over time for the measurement-based DERreactive-power dispatch method considering DER costs and voltage con-straints, compared to the model-based method. . . . . . . . . . . . . . . . . . 105xviiiGlossaryAC Alternative currentADMM Alternating direction method of multipliersANSI American National Standard InstituteD-PMU Distribution-level phasor measurement unitDC Direct currentDER Distributed energy resourceGHG Greenhouse gasOLS Ordinary least-squaresOLTC On load tap changingPI Proportional-integralPLS Partial least-squaresPV PhotovoltaicR/X Resistance-to-reactanceRMSD Root-mean square deviationRWPLS Recursive weighted partial least-squaresSCADA Supervisory control and data aquisitionSDP Semidefinite programmingxixSOCP Second order cone programmingVV Volt-varVW Volt-wattxxAcknowledgmentsI feel so fortunate to have received endless support, encouragement and advice throughout thecourse of my Ph.D.First and foremost, I would like to thank my supervisors Dr. Yu Christine Chen and Dr. LiweiWang for their guidance and support during my Ph.D. at UBC. Under their supervision, Ifelt supported, trusted, and valued, and they taught me so much about good work ethics.Specifically, they have taught me countless relevant technical details, how to work professionallyand patiently in an organized manner, to pay attention to details, how to present scientifictopics, and how to write scientific papers. In addition, I enjoyed the fact that they were alwayswilling to listen to my needs and opinions and available to discuss them patiently. And mostimportantly, they helped me to learn and grow and never hesitated to encourage me in takingthe next step in my academic carrier.Furthermore, I would like to thank my supervisory committee members Dr. Wilson Eberle,and Dr. Jonathan Holzman for their valuable guidance in my research. I would like to highlightthat Dr. Wilson Eberle enabled me to come to Canada, start my graduate studies at UBC andwas supervising my research during the first few years of my Ph.D. program. I would also liketo thank Dr. Rudolf Seethaler for providing access to his laboratory facilities.Next, I would like to express my sincere gratitude for all the support and guidance through-out my graduate studies to Malcolm S. Metcalfe, the founder of Enbala Power Networks. Hesupported me from day one until the end and beyond, no matter the circumstances. He hastruly opened my eyes with his pioneering approach and vision for the future. He has taught meso many things not only specific to technical aspects but also about life and is a true mentorand good friend. Moreover, I would like to thank the entire team at Enbala Power Networks forthe great support and interaction throughout the graduate studies, including but not limitedxxito Jeff MacAulay, Bud Vos, Michael Ruth, Gavin Hume, Nima Tehrani, Eric Young, AndyGassner, Duncan Brown, Sahand Behboodi, Lana Gonoratsky, and David Bopp.I would also like to acknowledge the support of my great laboratory group members, Sepehr,Peyman, Yuri, Alireza, Toby, Brittney, Kevin, Yanqi, Brad, Amin, and Milad, and my friendsat the UBC Vancouver campus Abdullah, Bo, Rubinder, and Shuan. We share so many funnymoments and stories as well as ups and downs which made my grad studies an immenselyunforgettable and enjoyable journey. I wish them all the best in the future.My life as a graduate student would not have been the same without my Canadian “family”Lo¨ıc & Tess with Taio and Mica, Peter & Halyna, Jeff & Sarah with Isla, Woody & Stacey, andHannah & Jared. They made us feel so welcome and at home in Kelowna. I will never forgetour countless days biking, skiing, and adventuring across beautiful British Columbia, our earlymorning work outs and runs, wing nights, dinners, Halloween parties, etc. But they were alsoalways available to help out in any circumstances. I miss them already now and cannot waituntil the next visit to Kelowna. Special thanks also go to Okanagan Grocery, their galettes areunrivalled.Only now I start realizing how lucky I am with my upbringing. My parents Silvia and Stefanhave fostered so many of my diverse interests and unconditionally supported my decisions inlife, even if it involved being far away for a long time. Some of my best memories with themare from our trips in the back country in Canada during their visits. I am forever gratefulfor this. Even though physically far apart, my siblings Mario, Benjamin, and Laura, my other“brothers” Lu¨psi, Guaina, and Ma¨ni, as well as the Flu¨ck family always made me feel close tohome.Happiness is only real when shared. I am so grateful to share the Canadian adventure andmy life with Daniela. Her love and support help me weather any storm. Our journey togetherhas evolved so much during these years and I am beyond proud of our son Henry and our futurelittle one.Lastly, I would like to acknowledge the funding support from Enbala Power Networks Inc.,MITACS Canada, NSERC and the Eminence Program of UBC.Everyone involved made these past 5 years such a memorable period of my life that I aminfinitely thankful for and proud of.xxiiDedicationTo Daniela, Henry & ?xxiiiChapter 1IntroductionIn this chapter, we motivate the need to develop techniques for control and optimization of mod-ern power distribution systems which leverage real-time measurements. We provide backgroundinformation and review existing literature related to research in this thesis to contextualize theproposed work with existing state of the art. Finally, we state the research objectives andprovide an outline of this thesis.1.1 MotivationEver-increasing electric energy demand along with environmental concerns of conventional,mostly fossil fuel-based power generation motivates an imminent shift toward clean power gen-eration to prevent further climate change. Particularly solar photovoltaic (PV) and wind gen-eration are promising technologies to reduce emissions of power generation and to strive fora sustainable energy future. In the last decades, these renewable energy sources have seena tremendous growth globally and are forecasted to grow even more significantly [2]. Small-scale renewable generation units typically supply power close to end-customer within the dis-tribution systems [3]. In addition to the rapid rise of renewable generation in distributionsystems, increasing numbers of demand-side management systems and energy storage systemswithin distribution systems lead to highly complex power systems. The variety of small-scale,grid-connected generation, demand-side management, and energy storage devices close to end-customers are collectively referred to as distributed energy resources (DERs) [4]. Integrationof these relatively new technologies into modern power grids presents notable challenges to the1reliable and efficient operation. For example, in contrast to conventional, dispatchable genera-tion from fossil-fuel-based plants, renewable resources such as solar and wind are intermittent,variable, and uncertain [5]. Systematic DER management is essential to enable power availabil-ity and quality, ensure reliability, and avoid costly expansions of power delivery infrastructure.Major benefits for grid operations are enabled when DERs are suitably dispatched via online op-timization routines and penetration levels of renewables may be significantly increased withoutviolating regulatory limits of system voltages and frequency.As a result, system operators are eager to find new ways to incorporate DERs to reducegrid stresses, avoid costly grid upgrades, shift peak loads, and manage disruptions causedby increasing levels of intermittent generation. Consequently, there has been a tremendousresearch interest in developing tools and techniques to more effectively manage and operatepower grids with significant penetration levels of DERs. The majority of this research relieson accurate and up-to-date network parameters, such as line impedances and system topologyinformation, and current operating conditions of power distribution systems. These attributesare typically leveraged in optimization tasks to determine optimal DER outputs which optimizesome operational objective, meet physical and regulatory constraints of power injections andflows, as well as voltage magnitudes while satisfying the physics governing the power grids (i.e.,the power-flow equations). However, in practice accurate and up-to-date network attributes,and current operating conditions are rarely available in real-time.On the other hand, the recent development and deployment of advanced sensors, such assmart meters and distribution-level phasor measurement units (D-PMUs) provide vast amountsof data from distribution systems and end-customer sites previously unavailable to systemoperators. The availability of new data streams enables ample ways to observe, monitor, control,and optimize the electric power grid.Capitalizing on these trends, in this thesis, the overarching goal is to develop new techniquesto optimize and control modern power distribution systems with high penetrations of DERsbased on real-time measurements. Specifically, the research presented in this thesis focuses onoptimally dispatching DER active- and reactive-power injections to manage voltages, to reg-ulate supply and demand at specific locations, to minimize network losses, and to minimizecost of DER injections within the distribution systems. Figure 1.1 offers an illustration of the2Figure 1.1: Massive deployment of DERs, such as solar PV, wind, and storage systems iscrucial in the transition to a more sustainable energy future. Additional benefits for gridoperations are enabled when DERs are suitably dispatched via online optimization routines.However, optimal DER dispatch may be compromised by inaccurate offline network models dueto incorrect topology information and fast-changing operating points, resulting in suboptimalor undesired system behaviours. The research in this thesis is motivated by the use of high-precision measurements from D-PMUs to develop measurement-based optimal DER dispatchmethods.overarching research goals in this thesis. Central to the proposed methods within this thesisis the real-time estimation of system attributes such as power-to-voltage and voltage-to-powersensitivity models of the full or partial network, and network loss sensitivities with respect toDER injections. Furthermore, D-PMU measurements are leveraged at locations with DERs toaccurately control system voltages to optimal levels in case of significant local disturbances fromintermittent generation. The key benefits of the research presented in this thesis are (i) elimi-nating the need of accurate network model information in determining optimal DER setpoints,(ii) the accurate optimal DER dispatch compared to network model-based approaches, (iii) themodest computational complexity by using linearized models in the optimization problemsand implementing distributed optimization frameworks, and (iv) the robustness against distur-bances caused by intermittent renewable generation and delays attributed to communication,3computation, and actuation.Overall, the proposed methods advance the efficient and reliable operation of modern dis-tribution systems, and facilitate the widespread integration of DERs based on measurements.Consequently, this research helps to enable higher penetration levels of renewable generationand contributes to the transition to more environmentally-friendly electric power supply.1.2 BackgroundThis section provides background information on the electric power system, the transition tosustainable electricity supply, traditional distribution system control, DERs in power systems,challenges and opportunities integrating high penetrations of DERs in distribution systems,and monitoring of distribution systems.1.2.1 The Electric Power SystemThe electric power system has its origin in 1882, when Thomas A. Edison introduced the firstcentral power station, the Pearl Street Station, in New York City. The steam-powered directcurrent (DC) generator provided a total power of 30 kW to 59 customers for 110 V incandescentlighting over DC transmission lines. Through the introduction of alternative current (AC)transformers and alternators in the mid 1880s, the first three-phase AC power systems wereenabled to transport power over long distances at high voltages to reduce transmission losses atthe end of the 19th century (i.e., 179 km at 12 kV in Germany 1891). From this beginning, theelectric power system has experienced a tremendous growth at a high pace over the years [6].Throughout the 20th century, the electric power system has been operated based on large-scale power generators at remote locations that fed a more and more interconnected three-phaseAC power grid at frequencies of 60 Hz in North America and 50 Hz in Europe [7]. The powergenerators were mainly turbine-driven, rotating synchronous generators at central locationsfeeding widely distributed loads.To deliver the power of the generators to load centers, step-up transformers are used toboost the voltage for transmission. With the transmission system, large interconnections oper-ating at high voltage levels (e.g., 230, 345, 765 kV) have been established to connect multiplegenerators. Large transmission systems increase the reliability in case of failures of single gen-4erators or transmission lines, and enable trading of electric energy between utilities. Step-downtransformers located at substations are used to reduce the voltage levels from the transmissionsystem and feed the distribution system. Finally, the distribution system spreads across urbanand rural areas to deliver electric power to end-consumers.1.2.2 Transition to Sustainable Electricity SupplyThe traditional power grids were largely fed by fossil fuel-based generators from coal, oil orgas, and nuclear power. However, significant environmental concerns and the contribution toclimate change of greenhouse gas (GHG) emitting resources drive the transition to a moresustainable electricity supply. In fact, a fast and fundamental shift toward low-emission energysupply away from fossil fuels is needed to mitigate climate change. In 2015, 195 countries signedthe Paris climate agreement to limit global warming to 2°C relative to pre-industrial levels [8].To reach this ambitious climate target, the single largest consumer, the electricity sector whichaccounts for 37% of the total primary energy demand in the US [9], must undergo a significantchange [10]. Today the global electricity sector is powered by 74% from non-renewable energysources [2] from which 63.9% are GHG emitting fuels and 10.1% are nuclear powered [11]. Inaddition to overall cleaner electricity supply, the transition of the transportation sector fromfossil-fuels to electricity by the use of electric vehicles will further increase the total electricenergy demand.Particularly wind and solar PV generation are considered to be viable resources to econom-ically address the environmental concerns of conventional fossil fuel-based generation. Indeed,in every year since 2016, solar PV generation was the largest capacity to be added to the electricgrid worldwide, while also being the most economic resource [2]. Small-scale solar PV and windgeneration units typically supply power close to end-users within the distribution systems [11].The effects of significant amounts of distributed generation within the power system will befurther discussed in Section 1.2.4.1.2.3 Traditional Distribution System ControlPower system operators are required to provide regulatory-compliant electric power throughthe distribution system. The American National Standard Institute (ANSI) for electric power5systems regulates specific acceptable levels for both voltage and frequency [12]. In order toprovide high-quality electric power, balance between generation and demand needs to be main-tained at all times, as the electrical power can not be stored within the power system. Thesystem frequency is directly linked to the rotating speed of the generators and is closely coupledfor an entire interconnection. The large inertia of rotating generators prevents the frequencyfrom sudden changes that arise from imbalance in generation and demand. Therefore effectivefrequency control is comparatively simple by adjusting the output power of large generators.In large interconnections, the frequency can be considered to be relatively stable in steady-state [13].As opposed to frequency, the system voltages vary from location to location. Furthermore,there is very little inertia to slow local voltage variations, which makes it inherently more difficultto control the system voltage [14]. Traditionally, devices such as switched capacitors, on-load tapchanging (OLTC) transformers, and voltage regulators were introduced to adjust distributionsystem voltage levels [15]. In the so-called volt-var control, the voltage control equipment wasmainly controlled by rules, based on local measurements such as voltage level, load level orpower factor [14, 16]. For radial distribution systems, the voltage at the end of a feeder couldbe controlled by acting on the substation OLTC transformer. A good estimate of the worstcase voltage within the feeder could be determined by measuring the total power demand of thefeeder and the voltage at the substation. However, the traditional voltage control mechanismsbecome impracticable for regulatory-compliant power delivery in cases with high penetrationsof intermittent distributed generation supplying directly into distribution systems [14].1.2.4 Distributed Energy Resources in Power SystemsThe supply of electric power connected directly to the distribution system or on the customerside of the meter is known as distributed generation [17]. The energy sources of distributedgeneration are various, both renewable (e.g., solar PV, wind, geothermal, mini-hydro, andbiomass) or non-renewable (e.g., reciprocating engines, fuel cells, and micro gas turbines).Particularly, the number of renewable energy plants powered by solar and wind have drasticallyincreased over the last decades. Consequently, the power generation within the distributionsystem has progressively been growing [5].6The growth and nature of renewable distributed generation significantly complicates theintegration of these resources. Power from solar and wind plants is difficult to predict andcan have a variable, uncertain, and intermittent behavior due to its dependency on regionalweather [5]. In the case of solar power generation, cloud transients can cause ramp rates ofgeneration in the order of 15% per second for a site, and approximately 15% per minute foran entire distribution system [18]. The uncertainty and the non-dispatchability of renewablegeneration further complicates its integration in distribution systems. Despite significant re-newable power capacity, energy needs may not be met if the weather conditions are not suitable.In addition, the peak power generation from renewables around noon typically does not occurduring peak demand hours in the evening. As a consequence, the load/generation-balance needsto be maintained by the generators from remote locations [19].In recent years, research and industry efforts have focused on these power balance issues.An enabling technology is energy storage, which is widely seen as a critical component in thetransition of the power system to low-emission resources [19]. As electrical power can not bestored directly, various transformations can be employed to store energy such as mechanicaltransformation (e.g., flywheels, pumped hydro, and compressed air), chemical transformation(e.g., hydrogen and batteries), and thermal transformation (e.g., water heaters). The prevalentenergy storage technology today is pumped hydro storage accounting for 95% of the grid-connected storage [2], while small-battery storage systems have seen an exponential growthsince the early 2000s [2].Battery energy storage systems at sites with distributed generation can help to shift loadand generation peaks, increase self-consumption and smoothen the net power profile of an end-customer. The continued developments in battery technology and their falling prices promotewide battery deployment in distribution systems. However, there are several issues related towidespread battery deployment such as longevity, economics, recycling or disposal, efficiency,and the time horizon of storage [19]. The strength of battery energy storage systems lies indaily storage, however seasonal storage (summer/winter) is difficult to achieve. In contrast,hydro power storage is very well suited for seasonal storage, but it is often at remote locationsand slow in response.Another area of interest for grid balancing activities is the management of loads - the7so-called demand-side management or demand response. Demand-side management systemsintend to manage a number of loads such as water heaters, heat pumps, heating, venting andair-conditioning devices, and electric vehicle chargers to follow a load profile. These systemscan shift loads, without need of additional storage systems, which makes them economic andhighly efficient [20].In this thesis, we refer to the term DERs as the ensemble of controllable, small-scale,grid-connected generation, demand-side management, and energy storage devices close to end-customers [4].1.2.5 Challenges and Opportunities Integrating High Penetrations ofDERs in Distribution SystemsTraditional power systems were designed to carry power uni-directionally from generation sitesthrough substations to highly distributed loads. With the integration of widespread DERs atcustomer-sites, this paradigm is no longer valid. As a result, significant DER integration indistribution systems leads to new operating situations that the distribution systems were notoriginally designed for. The effects of the new operating situations in distribution systems maylead to potential damage of system and customer equipment, decrease of power quality andreliability, extended time for restoration after outage, and potential safety risks [14].One of the major challenges when integrating DERs into the distribution systems, is themanagement of the system voltage [14]. The voltage drop along distribution lines is directlyrelated to the active- and reactive power flows they carry [15], and consequently also dependon active- and reactive power injections at downstream buses. As a result, fluctuations of DERinjections directly affect the system voltage profile.These facts suggest that conventional voltage control mechanisms are no longer able to ef-fectively adjust voltage levels at DER buses across the distribution systems as power generationof DERs may impact the voltage locally. Particularly the uncertain and intermittent generationof renewable sources such as solar PV and wind may cause significant voltage fluctuations. Inworst case scenarios, DER fluctuations result in voltage violations from acceptable levels andundesired switching operations of conventional voltage control devices such as OLTCs, voltageregulators, and switched capacitors. This could ultimately lead to their wear-out [14]. In addi-8tion to the voltage problems, the bi-directional power flows caused by DER generation can leadto new protection and stability challenges, increased network losses in distribution systems, andreliability concerns.An additional challenge for system operators are the variations of renewable generation overthe course of a day. For example, the peak of solar generation around noon typically does notcoincide with demand peak in the evening, the effect leading to the so-called duck curve with in-creasing solar penetration [21]. This imbalance between supply and demand requires the legacygenerators to compensate for large swings in renewable generation, which (i) decreases theircapacity factor and (ii) requires high ramp rates once solar irradiation decreases in the evenings.A remedy to this, as for example implemented in Germany, is the systematic curtailment ofsolar generation, which in turn, limits harvesting the full potential of renewables. Other meth-ods include the appropriate control of storage systems and the management of flexible loadsthrough demand-side management in distribution systems.At the same time, the advent of DERs in distribution systems creates ample opportunitiesto improve operation and control of power distribution systems. Many modern DERs suchas e.g., solar PV, wind, and battery systems are typically connected to the grid through DCto AC power electronic converters to allow for grid-compliant power injection. The advancedfunctionalities of these converter-based DERs such as remote control of active- and reactive-power injections enable new ways of controlling the power system from the grid edge. Theircapability of adjusting active- and reactive-power operating points with very little inertia maybe the solution to high ramp-rate voltage fluctuation issues. Moreover, the communicationinterfaces of DERs can provide near real-time voltage, current, and power generation data tosystem operators to improve system observability.Ultimately, the key to increased penetration of renewables in the power distribution systemswill be an effective means of managing DERs [22]. As a result, there has been a tremendousresearch interest in the operation of distribution systems with high penetration levels of DERsto mitigate the aforementioned challenges.91.2.6 Monitoring of Distribution SystemsHistorically, electric grid operators had very limited information for understanding the currentstatus and behavior of the grid. But more extensive real-time monitoring and control is neededfor effective operation of distribution systems with regard to the massive deployment of DERs.As a consequence, utilities have continuously upgraded their distribution systems by installingsupervisory control and data acquisition (SCADA) systems to provide more real-time measure-ments in 2 to 4 second intervals [23]. The SCADA systems typically provide measurementsfrom substations but does not encompass real-time measurements of customer loads. Via theoffline network model-based on equipment ratings and specifications, estimation of nodal volt-ages was possible. However, given the growing complexities and uncertainties in distributionsystem operations, these methods are increasingly becoming unsuitable in precision, accuracy,and resolution.Recently, utilities have invested in advanced sensors, such as e.g., smart meters, for moreaccurate monitoring of customer demand and generation. But neither time resolution of smartmeters (which is typically around 15 minutes, dispatched once or twice a day), nor the datareadings (mainly active power measurements) is sufficient for real-time control of distributionsystems [24, 25].A more promising technology for real-time monitoring of distribution systems are D-PMUs,as they provide high-accuracy time-synchronized voltage and current phasor measurements en-abled by GPS technology with phase angle accuracy up to 0.01 degrees [26]. D-PMUs streamphasor measurements through a standardized communication interface (e.g., IEEE C37.118)in intervals in the sub-second range [26]. Recent research has focused on integrating the syn-chronized D-PMU measurements with existing non-synchronized SCADA data (see, e.g., [27])to provide overall more accurate monitoring of distribution systems. Another active researcharea has been the optimal placement of D-PMUs in distribution system in order to achieve thebest real-time observability while minimizing the number of D-PMUs installed (see [28] andreferences therein). Furthermore, the precise phasor measurements are suitable for the onlineestimation of desirable system attributes and greatly benefit the real-time control of DERs,which will be reviewed in Section 1.3.101.3 Literature reviewIn this section, we review relevant literature to contextualize the proposed research presentedin this thesis with existing state of the art. We focus our literature review on optimal DERdispatch methods, voltage control methods, and measurement-based applications in power sys-tems. Through the literature review, we identify shortcomings which this thesis aims to address.1.3.1 Optimal DER Dispatch MethodsThe optimal DER dispatch problem is a fundamental problem in power system studies.1 Itcan be used for a variety of applications in the planning and the operation of power systems,such as e.g., optimal sizing and sitting of DERs, volt-var control, demand response, DERmanagement, etc. The optimal DER dispatch problem aims to determine optimal DER outputswhich optimize some operational objective, and meet physical and regulatory constraints ofpower injections and flows, as well as voltage magnitudes while satisfying the physics governingthe power grids (i.e., the power-flow equations). Depending on the devices included in theoptimal DER dispatch problem, it may also include discrete decision variables for DERs withdiscrete setpoints, capacitor banks, and voltage regulators. Due to the nonlinear power-flowequations, it is a non-convex, nonlinear optimization problem and therefore generally NP-hardto solve [29], i.e., there is no algorithm to directly solve the problem and find an exact, uniquesolution. Existing methods for solving the optimal DER dispatch problem can be divided basedon their communication network:• Centralized: All necessary parameters are collected in a centralized computing unitwhich solves the optimal DER dispatch problem centrally for the full network and dis-patches setpoint updates to DERs.• Distributed: The full network is partitioned into multiple areas. Within each area, areacontrollers collect necessary parameters specific to their area and solve the optimal DERdispatch problem in a distributed fashion by exchanging information with adjacent areas.1The optimal DER dispatch problem is a form of the optimal power-flow problem. The work in this thesisfocuses on optimally dispatching DERs, hence the choice of our terminology.11Centralized Optimal DER Dispatch MethodsMultiple centralized methods intend to solve the non-convex optimal DER dispatch problem di-rectly via interior-point [30], and heuristic algorithms [31–33] relying on accurate network modelinformation. However, these methods cannot guarantee to converge toward a global minima.As a result, significant research efforts have gone into developing linearizations and convex re-laxations to recover a (sufficiently) exact solution to the optimal DER dispatch problem in acentralized fashion.In transmission systems, the most prevalent linearization is the DC optimal power flow [30].However, the DC assumptions are not applicable to distribution networks due to higher lineresistance-to-reactance (R/X) ratio. As a result, methods using linear power-flow models specif-ically tailored to distribution systems have been developed, such as the LinDistFlow power-flowmodel [34] (which was extended to multi-phase systems in [35]), and more recently linear power-flow models accounting for network losses [36–38]. While the linearizations of the optimal DERdispatch problem lead to tractable convex optimization problems, their solution is affected byan approximation error stemming from the linear power-flow models.On the other hand, with the goal to recover an exact, globally optimal solution to theoriginal non-convex optimal DER dispatch problem, convex relaxation techniques have beenproposed (see [39, 40] for a tutorial on formulations, equivalence and exactness). The semidefi-nite programming (SDP) relaxation (see, e.g., [41, 42]) and the second-order cone programming(SOCP) relaxation (see, e.g., [43]) have been employed in centralized fashion. In case of radial,balanced distribution systems, it was shown that both SDP and SOCP relaxations recover theexact solution to the optimal DER dispatch problem [39, 40].While these centralized methods are suitable to obtain sufficiently accurate DER setpoints,there are some concerns regarding their scalability for large-scale distribution networks due tolatency of communication channels and computational complexity, and they all rely on accuratenetwork-model data.Distributed Optimal DER Dispatch MethodsTo mitigate scalability concerns of centralized optimizations, the optimal DER dispatch problemhas been solved in distributed fashion. For the most part, existing distributed methods rely on12accurate network-model information and consider single- and multi-phase distribution systems(see [44] for a thorough review of distributed optimization methods for DER dispatch). Forexample, [45] solves the original optimal DER dispatch problem in distributed fashion via thealternating direction method of multipliers (ADMM) algorithm, however it does not addressthe non-convexity of the optimal DER dispatch problem, which may result in finding a localminimum. Interestingly, both the linearizations and convex relaxations of the optimal DERdispatch problem are amenable to decomposition into distributed optimization frameworks.For example, [46] uses the consensus-based ADMM algorithm and employs the LinDistFlowmodel to obtain optimal DER reactive-power setpoints. Extending the LinDistFlow model togeneral multi-phase networks, [47] employs a hierarchical, distributed implementation of theprimal-dual gradient algorithm, and [48] uses ADMM. Further, distributed implementationsapply the SDP relaxations (see, e.g., [49, 50] using dual decomposition, and [42, 51] usingconsensus-based ADMM), and the SOCP relaxations (see, e.g., [52] using consensus-basedADMM). Several formulations require a controller per bus, requiring considerable communi-cation infrastructure (e.g., [52]). On the other hand, other methods separate the distributionsystem in sub-areas, each managed by an area controller (see, e.g., [42]). Consequently, fewercommunication links between area controllers are required.The distributed optimal DER dispatch methods offer better scalability and require lesscomplex communication networks for large distribution systems than centralized implementa-tions, nevertheless they still require accurate network-model data. Moreover, their convergence,optimality and stability may be affected by deteriorated communication channels [53].Feedback-based Optimal DER Dispatch MethodsRecently, feedback-based optimization methods have been developed which leverage measure-ments and adjust DER setpoints recursively to converge to the optimal operating point. Usingthe feedback of measurements, they tend to be robust against uncertainties and disturbances dueto fluctuating loads and renewables [44], and can cope with inaccuracies in the network model.For example, feedback optimization is used to synthesize controllers that achieve distribution-level voltage regulation in centralized fashion [54, 55] and distributed manner [56–58]. Combinedcontrol of system voltages and substation power setpoints is accomplished via primal-dual gradi-13ent methods to solve a saddle-point problem in a distributed fashion in [59], and the frameworkis extended to include aggregations of DERs, multi-phase systems, and discrete DER setpointsthrough a bilevel optimization formulation in [60]. Additionally, [61] solves the optimal powerflow problem via quasi-Newton methods aided by real-time measurements of values and con-straints of decision variables. However, the aforementioned feedback optimization methods stillall rely on some (albeit possibly limited) knowledge of the underlying power network structureto compute relevant gradients and Hessians.1.3.2 Voltage Control MethodsSince regulating voltage levels in distribution systems is of particular importance in case ofpossibly intermittent renewable generation, developing effective frameworks to manage busvoltages by active- and reactive power outputs of DERs has been a very active research area.Although the more general optimal DER dispatch can be leveraged to optimize distributionsystem voltages [43], both the centralized and the distributed optimization tasks could facedifficulties to respond timely to local disturbances due to latency of communication channelsand the computational burden of the optimization algorithms [62]. Therefore, an important lineof research has focused on (i) decentralized/local voltage control and (ii) hierarchical voltagecontrol methods.Decentralized / Local Voltage Control MethodsIn contrast to the system-wide optimizations, decentralized/local controllers forgo the use ofcommunication channels and enable fast primary control at multiple DER locations simultane-ously to regulate voltage levels by leveraging measurements of local operating points [53, 63–70].Typical implementations are the volt-var (VV) and volt-watt (VW) droop controllers, whichhave been adopted in the IEEE standards and are currently widely available in commercialDERs [63]. However, droop controllers can lead to undesired voltage oscillations if the droopslope is too steep [64, 65]. Conversely, if the droop slope is small to ensure voltage stability,the full potential of DERs to regulate voltages may not be utilized.To mitigate the voltage oscillations of droop controllers, several variations of the standarddroop control have been studied and stability conditions considering network voltage sensitivi-14ties have been established to design stable local controllers. For example, delayed droop controlwith prolonged response time to improve voltage stability is proposed in [64]. In [66], a droop-based local control scheme is proposed, which considers the previous reactive-power output anda linear droop curve with conditions for a stable design. Furthermore, a network-cognizantapproach is presented to calculate optimal, stable droop coefficients in [69].Similarly, in [53, 67, 68] scaled versions of the local controller in [66] are proposed, with theobjective to operate the system at an optimized operating point. Generally, the improved droop-based methods enhance both voltage stability and response time compared to the standarddroop control. Nevertheless, a voltage mismatch between the actual bus voltage and a referencelevel may remain [53].The autonomous local controllers provide fast primary control at the locations of DERswithout the need of communication infrastructure, and they can mitigate effects of intermittentrenewable generation. However, they may not achieve a system-wide optimized operation, whichcould potentially increase network losses [67] and lead to suboptimal voltages [56].Hierarchical Voltage Control MethodsWith the objective of enabling system-wide optimization and coordination of multiple local con-trollers, further research has focused on hierarchical control systems [71–75]. The upper-levelcontrol layer enabling system-wide optimization can either be implemented in a centralized ordecentralized fashion. In [71], a decentralized upper-level control through integral controllers forvoltage regulation and reactive-power sharing is implemented while using droop controllers forprimary control locally. In [72], a hierarchical and decentralized voltage management method isproposed. In a decentralized upper-level, the voltage optimization is separated in independentcontrol areas. In the lower-level, local DER controllers adjust reactive-power supply propor-tional to DER active-power output and a sensitivity coefficient. Further, a combined centralizedand local control system is proposed in [73]. Centrally a multi-time step optimization problemcalculates optimal operating points for local controllers. The local controllers are piece-wise lin-ear droop controllers. Similarly, a combined control system is proposed in [74] with a centralizedcontroller which uses power injection predictions for the next hours to update parameters ofthe local droop controllers. In [75], an online hierarchical method is proposed for voltage con-15trol. In the lower-level, DERs independently optimize bus voltages based on a gradient descentalgorithm. If the lower-level does not succeed in eliminating voltage violations, the upper-levelcontrols the transformer tap at the substation to effectively regulate system voltages.The hierarchical control architecture has the advantage of fast response to local disturbancesthrough effective primary control at DER locations while the upper-level controller is capable ofcoordinating multiple local controllers and obtain system-wide optimized DER operating points,even if measurement-data availability at the optimization stage is limited. Nevertheless, thereis a potential to improve upon existing hierarchical voltage control methods by implementinglocal controllers which are capable of adjusting DER outputs to (i) specific values similar tofeed-forward centralized controller when new setpoints become available from the upper-leveloptimization and (ii) adjust DER outputs according to local measurements between the setpointupdates from the upper-level to specific voltage setpoints.1.3.3 Measurement-based ApplicationsThe vast data volume provided by advanced sensors such as e.g., D-PMUs installed in distribu-tion systems, enables ample ways to observe, monitor, control, and optimize the electric powergrid. We focus our review of relevant literature squarely on measurement-based methods indistribution systems for (i) estimation of system attributes and (ii) optimal DER dispatch. Areview of other applications, such as state estimation, event detection, and phase identification,can be found in [76].Estimation of System AttributesThere is growing interest in inferring the distribution network topology from measurementsby recovering the admittance matrix with graph theoretical approaches [77], using maximum-likelihood estimation with error-in-variable models [78], actively probing smart inverters toestimate the grid Laplacian [79], developing low-complexity algorithms [80], and leveragingconvex-optimization techniques [81]. Prior art has also tackled online estimation of other sys-tem attributes, including injection shift factors via weighted and recursive least-squares estima-tion [82], the power-flow Jacobian through linear total least-squares estimation. [83], linearizedpower-flow models via partial least-squares and Bayesian linear regression-based algorithms [84],16and power-to-voltage sensitivities using sparse-recovery via compressive sensing [85] and viageneralized least-squares [86].It is worth noting that the resulting estimated models intend to explicitly recover the gridtopology or other system attributes. However, with the optimal DER dispatch in mind, thesemethods may not be directly applicable to obtain optimal DER setpoints effectively.Measurement-based Optimal DER DispatchShifting attention to measurement-based methods which combine the tasks of estimating net-work attributes and directly embed them in optimal DER dispatch methods, [87] leveragesestimated power-to-voltage sensitivities to determine DER setpoints that regulate the voltageprofile across a distribution network. In order to reduce the number of samples needed to esti-mate the sensitivities, [87] assumes prior knowledge of feasible topology configurations and lineR/X ratios. Voltage regulation is also the objective in [88], where rule- and optimization-basedcontrol schemes are deployed in conjunction with an estimated second-order power-flow sensi-tivity model. Furthermore, [89] optimizes DER setpoints to regulate the active-power injectionfrom the distribution feeder into the transmission grid while relying on prior knowledge of net-work topology. Finally, model-free combined voltage and active-power control via an extremumseeking approach is developed in [90], and it is extended to unbalanced systems to regulate lineflows in [91].While above literature provides promising directions for measurement-based DER dispatch,there are opportunities to improve upon these works, namely by (i) formulating more gen-eral objective functions to simultaneously optimize bus voltages and active- and reactive-powerinjections as well as costs of DER outputs, (ii) not assuming any prior knowledge of the net-work topology, (iii) implementing distributed methods to address scalability and data privacyconcerns, and (iv) including network loss minimization as an operational objective.1.4 Research ObjectivesAs identified in Section 1.3, there are several gaps in existing literature on measurement-basedmethods to optimally dispatch DERs in distribution systems enabled by high-precision measure-ments of D-PMUs. Thus, this thesis formulates four research objectives, targeting at addressing17Research ObjectivesIIIIIIIVFigure 1.2: Overview of research objectives.the shortcomings of previous research. Specifically, an overview of the research objectives ofthis thesis is illustrated in Fig. 1.2 and listed below:1.4.1 Objective I: Centralized Method with a Recursively EstimatedSensitivity ModelFirst, we aim to develop a measurement-based optimal DER dispatch method to determine DERactive- and reactive-power setpoints. Distinct from existing model- and feedback-optimizationbased methods, we obtain optimal DER setpoints without any offline knowledge of the underly-18DispatchMeasurementModel EstimationOptimal DER Dispatch SubstationDERSynchrophasorFigure 1.3: Centralized measurement-based optimal DER dispatch method with (i) modelestimation and (ii) optimal DER dispatch.ing network. Moreover, compared to contemporary measurement-based methods, we formulatea more general objective function which facilitates the optimization of bus voltages and powerinjections, as well as minimization of DER costs. We will tackle two challenges in solving theoptimal DER problem simultaneously: (i) eliminating the reliance on an accurate offline modeland (ii) address its non-convexity. This is achieved by using online measurements of nodalvoltages and injections to estimate linear sensitivities and construct a linear voltage-to-powersensitivity model. We will then incorporate the estimated linear sensitivity model as a proxy forthe nonlinear power-balance constraint in a modified optimal DER dispatch problem. Further-more, we place a special focus on removing the requirement of a fully observable distributionsystem by only considering measurements obtained at a subset of buses, and on reducing thecomputational burden in solving the optimal DER setpoints with a tractable algorithm us-ing ADMM. We will illustrate key benefits of the proposed method via numerical simulationsinvolving the IEEE 33-bus distribution test system [1].1.4.2 Objective II: Distributed Method with Estimated Per-areaSensitivity ModelsNext, we take aim at potential data privacy and scalability concerns of centralized optimizationmethods in light of projections of vast deployments of D-PMUs in large scale distributionnetworks in the near future [92] via a distributed optimization approach. Existing distributed19SubstationDERSynchrophasorInter-areadata exchangeModelEstimationOptimal DERDispatchModelEstimationOptimal DERDispatchBoundarybus voltagesModelEstimationOptimal DERDispatchModelEstimationOptimal DERDispatchModelEstimationOptimal DERDispatchFigure 1.4: Distributed measurement-based optimal DER dispatch method with (i) modelestimation and (ii) optimal DER dispatch for each area.optimal DER dispatch methods rely on an accurate network model, and measurement-basedmethods are typically implemented in a centralized fashion. We fill this important gap in theliterature by capitalizing on the decomposability of the full network and propose to estimateequivalent per-area sensitivity models from real-time measurements, each representing a portionof the full network. As a result, both the estimation and optimization tasks decompose into lowerdimension subproblems, require only the exchange of information at boundary buses amongadjacent areas, and do not rely on any prior knowledge of the network model. Furthermore,we will develop a tractable consensus-based ADMM solution approach to effectively obtainoptimal DER setpoints which is amenable to decomposition into per-area subproblems. We willrun numerical simulations involving the IEEE 33-bus distribution test system to illustrate keybenefits of the proposed method compared to centralized counterparts. In addition, we will usethe IEEE 123-bus multi-phase test system with realistic load and renewable generation profilesin order to demonstrate the practical applicability and scalability of the proposed method.201.4.3 Objective III: Mitigating Effects of Highly Intermittent RenewableGenerationPotential communication delays or lower time granularity of system-wide measurements fromdistribution systems could compromise the availability of up-to-date optimal DER setpointsfrom methods introduced in objectives I and II. This could particularly cause suboptimal andineffective operation in case of highly intermittent renewable generation with rapid fluctuationsof operating points [76]. To this end, we will develop a two-level control system which pairsthe system-wide optimization (it could be implemented in a centralized or distributed fashion)with local controllers at DER locations for voltage control. In contrast to existing hierarchicalsystems, we propose to dispatch both the optimal voltage and reactive-power setpoints tothe local controllers. Through this approach, the proposed local controllers are capable ofadjusting DER outputs to (i) specific setpoints similar to feed-forward centralized controllerat intervals of system-wide optimizations and (ii) leverage local measurements to continuouslyadjust DER outputs between the setpoint updates from the upper-level as local disturbancesoccur. Our objective is to develop local controllers with faster response time to regulate busvoltages to reference setpoints without steady-state voltage mismatch compared to existing VVdroop-based local controllers by leveraging real-time measurements locally. Furthermore, wewill define design criteria of the local controllers to ensure stability of the primary control andmitigate integrator windup. We will demonstrate the performance of the proposed method viasimulation studies of the IEEE 123-bus test system with historical load and solar generationdata for a 24-hour period, and by comparing it to prior art. Furthermore, the applicability inscenarios with OLTC transformers and unbalanced loading conditions will be demonstrated.1.4.4 Objective IV: Gradient-descent Method for Optimal Reactive-PowerDispatchLastly, a desirable operational objective in distribution systems is to regulate bus voltageswhile minimizing network losses through DER reactive-power generation. Existing literature onmeasurement-based methods do not achieve effective loss minimization without network modelinformation. Therefore, we aim to develop a fully measurement-based approach that does notrely on any prior knowledge of the distribution-network topology and parameters to optimally21dispatch DER reactive-power outputs via feedback-based optimization. We will (i) formulatethe objective function to minimize network loss and cost of DER outputs, (ii) estimate relevantsensitivities of bus voltages and network loss with respect to DER reactive-power outputs fromonline measurements, and (iii) embed these estimated sensitivities in a measurement-basedgradient-descent method. Numerical simulations involving the IEEE 33-bus distribution testsystem [1] will reveal that the proposed measurement-based method yields sufficiently accuratereactive-power setpoints compared to those obtained from a model-based benchmark solutionto minimize network loss while regulating bus voltages2.1.5 Thesis OutlineThe remainder of this thesis is organized as follows. In Chapter 2, we describe the networkand power-flow model, and we present the formulation of the optimal DER dispatch problem.In Chapters 3 and 4, we propose a centralized and a distributed measurement-based optimalDER dispatch method using recursively estimated sensitivity models, respectively. Then, inChapter 5, we propose a two-level centralized and local voltage control method mitigating effectsof highly intermittent renewable generation. Chapter 6 presents a measurement-based gradient-descent method to optimally dispatch DER reactive power. Finally, Chapter 7 summarizes thecontributions, and provides concluding remarks and promising directions for future work.2Within this thesis, we refer to model-based methods for the conventional methods which require offlinenetwork model information such as topology and line impedances. On the other hand, we refer to measurement-based methods for the once which do not rely on network model information, instead they use measurements toinfer relevant system attributes22Chapter 2PreliminariesMoving to the mathematical formulation in this thesis, in this chapter, we first establish no-tation and the network model. We then formulate the nonlinear and linear power-flow modelrepresentation. Finally, we define the standard optimal DER dispatch problem, and introducea linearized version of it.2.1 NotationIn the remainder of this thesis, (·)T denotes transposition, (·)∗ denotes complex-conjugate;j =√−1 denotes the imaginary unit, | · | denotes the absolute value or the cardinality of a set.For a given length-N column vector x ∈ RN , ||x||2 =√xTx denotes the `2-norm, min(x) andmax(x) return the minimum and maximum values of vector x, and diag(x) returns a N × Nmatrix with the elements of x in its diagonal. Further, projY(x) denotes the projection of vectorx onto the set Y. For the function f(x) : N → 1, ∇f(x0) is the vector differential operatorwhose values at point x0 is a vector with partial derivatives of function f with respect to x0evaluated at x = x0. 0N and 1N denote the length-N vectors of all 0s and 1s, respectively.For a given N ×N matrix X ∈ RN×N , ||X||F returns the Frobenius norm, λ(X) returns theeigenvalues of matrix X. Similarly, λmin(X) and λmax(X), respectively, return the minimumand the maximum eigenvalue of matrix X. Finally, IN denotes a N ×N identity matrix.2.2 Network- and Power-flow Model RepresentationConsider a multi-phase distribution system with buses collected in the set N = {1, . . . , |N |},and power delivery elements such as distribution lines and transformers collected in the set of23edges T = {(i, j)} ⊂ N ×N . Suppose pertinent system variables are sampled at time t = k∆t,k = 0, 1, . . . , where ∆t is the sampling interval. We collect phases found at bus i ∈ N inPi ⊆ {a, b, c} and at power delivery element (i, j) ∈ T in Pij ⊆ {aij , bij , cij}. Let V φi,[k] and θφi,[k]denote, respectively, the magnitude and phase-angle of the line-to-neutral1 voltage at bus i ∈ Nin phase φ ∈ Pi, and at discrete time step k. Also let P φi,[k] and Qφi,[k] denote, respectively, thenet active- and reactive-power injections at bus i ∈ N in phase φ ∈ Pi, and time step k. Denoteby P φ,geni,[k] and Qφ,geni,[k] , respectively, the controllable components of the active- and reactive-power injections at phase φ ∈ Pi, bus i ∈ N and time step k. Also denote P φ,loadi,[k] and Qφ,loadi,[k] ,respectively, the uncontrollable active- and reactive-power injections (arising from e.g., loads)so that for bus i ∈ D and phase φ with controllable component of active- and reactive-powerinjectionsP φi,[k] = Pφ,geni,[k] − P φ,loadi,[k] , (2.1a)Qφi,[k] = Qφ,geni,[k] −Qφ,loadi,[k] , (2.1b)and for bus i ∈ N \ D and phase φ with uncontrollable injectionsP φi,[k] = −P φ,loadi,[k] , (2.2a)Qφi,[k] = −Qφ,loadi,[k] . (2.2b)Further collect multi-phase bus voltage phase-angles at all buses and time step k in vector Θ[k] =[{θφi,[k]}φ∈Pi,i∈N ]T and V[k] = [{V φi,[k]}φ∈Pi,i∈N ]T, and collect the multi-phase complex bus voltagephasor in V[k] = V[k]jΘ[k] . Analogously, collect bus net active- and reactive-power injections atall buses and time step k in vectors P[k] = [{P φi,[k]}φ∈Pi,i∈N ]T and Q[k] = [{Qφi,[k]}φ∈Pi,i∈N ]T,respectively. The relation between bus voltages and injections abides Ohm’s law, such thatP[k] + jQ[k] = diag(V[k]) (YV[k])∗, (2.3)1Although the notation is established for line-to-neutral quantities, delta connections can be converted to wyeconnections in accordance with our notation.24where Y ∈ C∑i∈N |Pi|×∑i∈N |Pi| is the complex network admittance matrix containing networktopology information. Comprehensive modeling details to construct the network admittancematrix Y for multi-phase networks are provided in [93] considering different characteristicsof loads, transmission lines, step-voltage regulators, and transformers. The model in (2.3) isgenerally referred to as the complex power-flow equation or the complex bus-injection model.To compactly express the power-flow equations (2.3), we collect multi-phase bus voltagephase-angles and magnitudes at all buses and at time step k in vector x[k] = [{θφi,[k]}φ∈Pi,i∈N ,{V φi,[k]}φ∈Pi,i∈N ]T. Analogously, we collect bus net active- and reactive-power injections at allbuses and at time step k in vector y[k] = [{P φi,[k]}φ∈Pi,i∈N , {Qφi,[k]}φ∈Pi,i∈N ]T. With the notationestablished above, we express power-flow equations (2.3) at time step k asy[k] = g(x[k]), (2.4)where g : R2Nφ → R2Nφ , with Nφ = ∑i∈N |Pi|. The dependence on network parameters (suchas topology, series line impedances and shunt admittances, and circuit breakers status etc.) isimplicitly considered in the function g(·).The nonlinear power-flow formulation is fundamental to all aspects of modelling, analysis,operation, and control of power systems. In spite of this, the inherent nonlinearities in thepower-flow model complicate the computation of solutions in many power system applicationssince iterative numerical methods are required to solve the power-flow problem [94].2.3 Linearized Power-flow Model RepresentationTo alleviate computational complexity in solving the nonlinear power-flow problem, oftentimesit is desirable to employ linearized power-flow models which are sufficiently accurate in theproximity of the system operating points. One such linear approximation of the power-flowequations is obtained by the first-order Taylor series expansions [95]. To this end, let x[k] =x[k−1] + ∆x[k], where x[k−1] collects bus voltage phase-angles and magnitudes at the previoustime step k − 1. Then, according to (2.4),y[k] = g(x[k−1] + ∆x[k]). (2.5)25Suppose g(·) is continuously differentiable with respect to x or a specific setting of discretedevices such as voltage regulators and circuit breakers, and assume that ∆x[k] is sufficientlysmall. Then we can approximate y[k] as [95]y[k] ≈ g(x[k−1]) + J[k]∆x[k], (2.6)whereJ[k] =∂g∂x∣∣∣∣x[k−1](2.7)is the Jacobian matrix of the power-flow equations evaluated at the operating point x[k−1].Recall that ∆x[k] = x[k] − x[k−1] and further substitute this into (2.6) to yieldy[k] ≈ J[k]x[k] + c[k], (2.8)where c[k] = g(x[k−1]) − J[k]x[k−1]. We note that the evaluation of both c[k] and J[k] relies onthe up-to-date operating point at time step k − 1 as well as an accurate network model of thedistribution system which are implicitly considered in g(·). The underlying assumption is thatJ[k] and c[k] at time step k− 1 model the relationship between y[k] and x[k] at time step k withsufficient accuracy.2.4 Optimal DER Dispatch ProblemOptimal DER dispatch methods typically search for the optimizer which minimizes networklosses, cost of DER injections, and voltage deviations or maximizes economic benefits to utilityand end-users while satisfying constraints on voltage magnitudes, power flows and injections,and DER operational regions. A solution to the optimal DER dispatch problem is also subjectto the nonlinear power-flow equations (2.4) which govern the physics of power systems. Tocompactly express the optimal DER dispatch problem, we collect multi-phase bus active- andreactive-power generations at buses with DERs (i ∈ D) in vector ygen[k] = [{P φ,geni,[k] }φ∈Pi,i∈D,{Qφ,geni,[k] }φ∈Pi,i∈D]T. With these definitions in place, the optimal DER dispatch problem can be26expressed asminimizex[k], y[k], ygen[k]f(x[k], y[k], ygen[k] ), (2.9a)subject to y[k] = g(x[k]) = Cygen[k] − yload[k] , (2.9b)xmin ≤ x[k] ≤ xmax, (2.9c)ygen[k] ∈ Ygen, (2.9d)where f : R2Nφ × R2Nφ × R2Dφ → R, with Dφ = ∑i∈D |Pi| is the objective function to be min-imized; xmin and xmax denote, respectively, minimum and maximum limits of voltage phase-angles and magnitudes; and Ygen represents the (typically convex) space of allowable DERactive- and reactive-power outputs [96]. In addition to minimum and maximum active- andreactive-power limits, typically DER outputs are constrained by the inverter apparent-powerrating (more details are provided in [59]). Finally, C ∈ R2Nφ×2Dφ is the matrix that maps theDER bus/phase indices to buses/phases collected in the entire distribution network. Particu-larly, the entry in the ith row and jth column of C is 1 if the jth element of ygen[k] represents thesame phase and bus as the ith element of y[k], and it is 0 otherwise. The optimization problemin (2.9) is, in general, non-convex and NP-hard (see, e.g., [29]).2.5 Linearized Optimal DER Dispatch ProblemA common way to circumvent the nonconvexity of the optimal DER dispatch problem is toembed a linearized power-flow model in (2.9b) instead of the nonlinear power-flow model (seerelevant literature in Section 1.3.1). This yields an optimization problem with linear equalityand inequality constraints, and the convex-set constraint in (2.9d). For both linear or quadraticobjective functions f(·), many tractable algorithms are available to obtain a sufficiently accuratesolution to the original non-convex optimal DER dispatch problem. The linearized optimal DERdispatch problem is given byminimizex[k],y[k],ygen[k]f(x[k], y[k], ygen[k] ), (2.10a)subject to y[k] = J[k]x[k] + c[k], (2.10b)27y[k] = y[k−1] + C(ygen[k] − ygen[k−1]), (2.10c)xmin ≤ x[k] ≤ xmax, (2.10d)ygen[k] ∈ Ygen. (2.10e)We note that the assumption for our linear power-flow model is that ∆x[k] is sufficiently small.Considering that the linearized power-flow model is frequently updated in our applications (≈ 1 sintervals), the linear power-flow model obtained at time step k − 1 represents the operatingconditions around the operating point with sufficient accuracy to model the relation of busvoltages to bus injections at time step k. The constraint in (2.10c) results by recognizing that,at time steps k − 1 and k, respectively,y[k−1] = Cygen[k−1] − yload[k−1], (2.11)y[k] = Cygen[k] − yload[k] . (2.12)Assuming that the loads do not change significantly between time steps k− 1 and k, (2.12) canbe expressed asy[k] = Cygen[k] − yload[k−1]. (2.13)Suitably rearranging (2.11) and substituting the resultant into the above, we get the constraintin (2.10c). The entries of J[k] and c[k] may be computed from a model if one is at hand, or theycan be estimated from online measurements, as will be proposed in this thesis.28Chapter 3Centralized Method with a RecursivelyEstimated Sensitivity ModelOur literature review has revealed some gaps in prior art on measurement-based DER dis-patch methods, such as the lack of generalized objective functions for combined optimizationof bus voltages and injections as well as cost of DER outputs, and the assumption of priorknowledge about the network topology. In an effort to close these gaps, in this chapter, wepropose a measurement-based approach, which does not rely on any prior knowledge of oreven regard for the underlying network topology, to determine DER active- and reactive-powersetpoints aimed at regulating bus voltages and injections as well as minimizing DER costs.1The proposed framework consists of two successive and iterative stages: (i) sensitivity modelestimation and (ii) optimal DER dispatch. For the estimation stage, we assume the installa-tion of D-PMUs installed at a subset of the buses in the distribution system for estimatingthe linear voltage-to-power sensitivity model with online measurements of bus voltage pha-sors and active- and reactive-power injections. We also leverage the recursive weighted partialleast-squares (RWPLS) algorithm to improve (i) the practicality of the proposed approach byperforming recursive updates and (ii) its adaptability to network-topology and operating-pointchanges by placing more weight on recent measurements and less on past ones. The resultingestimated model contains key sensitivity characteristics amongst measured quantities withoutexplicitly recovering the grid topology or power-flow model as in [77–81, 84]. It is worth not-1A version of this chapter has been published in: S. Nowak, Y. C. Chen, and L. Wang, “Measurement-basedoptimal DER dispatch with a recursively estimated sensitivity model,” IEEE Trans. Power Syst., vol. 35, no.6, pp. 4792–4802, Nov. 2020.29ing that while non-recursive variants of least-squares estimation have been used to computelinearized power-flow models [84], the power-flow Jacobian matrix [83], and power-to-voltagesensitivities [85], to the best of our knowledge, the RWPLS algorithm has not been utilized forapplications in the power systems domain.In the DER dispatch stage, we incorporate the estimated linear sensitivity model into a con-vex quadratic optimization problem. As a direct consequence, the optimal dispatch is solvedwithout any offline knowledge of the underlying network, distinct from methods in [54–60].Also, unlike [54–58, 87–89], our proposed formulation achieves combined objectives of minimiz-ing (i) DER active- and reactive-power costs and (ii) deviations of bus voltages and injectionsfrom their respective reference values. Moreover, compared with [90, 91], our proposed methoduses lower temporal measurement resolution and smaller injection perturbations. Extensive nu-merical simulations demonstrate that the proposed measurement-based optimal DER dispatchadapts to unexpected operating-point and network-topology changes, and closely matches themodel-based optimal dispatch solved with accurate system model. Also via numerical simula-tions, we evaluate the impact of DER dynamics, delays potentially attributed to communica-tion, computation, and actuation, and RWPLS parameters on the performance of the proposedmethod. Furthermore, we illustrate the flexibility of the proposed framework by accommodat-ing typical constraints on DER active- and reactive-power outputs. Finally, we report executiontimes involved with the model estimation and optimal dispatch stages of the proposed frame-work.The remainder of this chapter is organized as follows. In Section 3.1, we motivate the need fora measurement-based approach. Section 3.2 outlines the estimation of the measurement-basedsensitivity model. Section 3.3 formulates the optimal DER dispatch problem with the estimatedsensitivity model and presents the ADMM solution approach. In Section 3.4, we offer numericalcase studies to demonstrate the effectiveness of the proposed measurement-based optimal DERdispatch framework. Finally, we provide a summary in Section 3.5.3.1 Problem StatementRepeatedly solving the optimization problem in (2.9) may pose significant computational hur-dles for the utility operator in real time, especially with rapidly varying operating point. Fur-30thermore, accurate solutions of (2.9) rely on an offline system model that represents the up-to-date network topology, parameters, and operating point, which may not be available due toinsufficient telemetry from, e.g., all circuit breakers and load variations. Thus, aimed at prac-tical field deployment, we use online measurements of nodal voltages and injections obtainedat a subset of buses to estimate linear sensitivities and construct a linear voltage-to-powersensitivity model. We then incorporate the estimated linear sensitivity model as a proxy forthe nonlinear power-balance constraint in a modified optimal DER dispatch problem basedon (2.9). The proposed method accomplishes the objectives of (i) eliminating the reliance onan accurate offline model, (ii) removing requirement of a fully observable distribution system,and (iii) reducing the computational burden in solving the optimal DER setpoints.3.2 Estimation of Sensitivity ModelIn this section, we estimate the linear sensitivity model relating measured bus voltages andpower injections via the RWPLS method without relying on an offline system model.3.2.1 Problem FormulationLet E ⊆ N represent the set of |E| buses equipped with D-PMUs, including the substationbus (set as bus 1 without loss of generality), where measurements of voltage phasors andpower injections are collected. Assume that the substation-bus voltage is fixed and known,and let E− = E \ {1}. Also recall that D denotes the set of |D| buses that are connectedto DERs, whose setpoints can be updated with each optimal dispatch solution. We assumethat D ⊆ E−, i.e., measurements are available at all buses with controllable DERs. Denotethe measured voltage phase-angle and magnitude at bus i ∈ E− and time step k as θ̂i,[k] andV̂i,[k], respectively.2 Similarly, let P̂i,[k] and Q̂i,[k] denote the measured active- and reactive-power injections at bus i ∈ E and time step k. Accordingly, collect these quantities in vectorsx̂[k] = [{θ̂i,[k]}i∈E− , {V̂i,[k]}i∈E− ]T and ŷ[k] = [{P̂i,[k]}i∈E , {Q̂i,[k]}i∈E ]T. We hypothesize thatthe active- and reactive-power injections are linearly related to bus-voltage phase-angles and2Note that in this chapter we consider single-phase or single-phase equivalent distribution networks, conse-quently, with respect to the notation established in Chapter 2, Pi = {a},∀i ∈ N and |Pi| = 1, ∀i ∈ N . To easenotation, we remove the label of phase φ ∈ Pi in quantities, equivalently representing (·)i = (·)φi and (·) = (·)φ.Furthermore, since |Pi| = 1, ∀i ∈ N , |N | =∑i∈N |Pi|, |E| =∑i∈E |Pi|, and |D| =∑i∈D |Pi|.31magnitudes, as follows:ŷ[k] = J[k]x̂[k] + c[k], (3.1)where J[k] and c[k] form the sensitivity model relating measured voltages to power injections atthe same buses. We highlight that above formulation is similar to (2.8). However, in (3.1) weform a linear sensitivity model which only retains the subset of buses from which measurementsare available, i.e., buses in E ⊆ N . Reformulating (3.1), there exists H[k] ∈ R(2|E|−1)×2|E| thatsatisfies the relationshipŷT[k] =[x̂T[k] 1]H[k], (3.2)which is equivalent to (3.1) with H[k] = [J[k], c[k]]T. In order to eliminate the reliance on a knownnetwork model, we propose to estimate the entries of H[k] using only online measurementsobtained from buses in E . To this end, suppose that M samples of bus-voltage angles andmagnitudes, x̂[k−M+1], . . . , x̂[k], and active- and reactive-power injections, ŷ[k−M+1], . . . , ŷ[k],are available. Further suppose that the operating point remains approximately constant overthe M measurement samples (we will remove this assumption later). Then, with M > 2|E|, westack up M instances of (3.2) to yield the following over-determined system:Y[k] = X[k]H[k], (3.3)where X[k] ∈ RM×(2|E|−1) and Y[k] ∈ RM×2|E| are given byX[k] =x̂T[k−M+1] 1......x̂T[k] 1 , Y[k] =ŷT[k−M+1]...ŷT[k] . (3.4)Since (3.3) is over-determined, we can obtain the ordinary least-squares (OLS) estimate for H[k]asĤ[k] ≈ (XT[k]X[k])−1XT[k]Y[k]. (3.5)However, in practice, OLS estimation is challenged by the observation that voltage phase-anglesand magnitudes at different buses may be highly correlated because they evolve similarly with32variations in operating point [97], which results in an ill-conditioned regressor matrix XT[k]X[k]in (3.5). Furthermore, in order to capture changing operating conditions, it is desirable toupdate Ĥ[k] recursively while reducing computational burden. Also advantageous is to placegreater emphasis on more recent measurements than earlier ones that may become out of dateover time. In light of these circumstances, we propose to use the RWPLS algorithm to computeĤ[k], as it offers several benefits over OLS estimation.3.2.2 RWPLS-based EstimationThe potentially high collinearity in X[k] makes matrix inversion in (3.5) numerically challenging.A suitable solution approach in such a setting is the partial least-squares (PLS) method. Thecentral idea in PLS is to project so-called latent features or key components in X[k] and Y[k] ontolower-dimensional latent matrices, which best model the relationship in (3.3). Then the least-squares regression is performed on the latent matrices. Particularly, given matrices constructedfrom online measurements X[k] and Y[k], the PLS algorithm returns{X[k], Y[k]} PLS−−→ {Γ[k], G[k], L[k]}, (3.6)where G[k] and L[k] are lower-dimensional loading matrices corresponding to X[k] and Y[k], re-spectively, and Γ[k] is a diagonal matrix with regression coefficients from the PLS decomposition.Then, the PLS estimate is given byĤ[k] = (G[k]GT[k])−1G[k]Γ[k]LT[k]. (3.7)The PLS method is summarized in Appendix A, and more details can be found in [98].Now, suppose that at time step k+1, a new set of measurements, x̂[k+1] and ŷ[k+1], becomesavailable, so thatX[k+1] = X[k]x̂T[k+1] 1 , Y[k+1] = Y[k]ŷT[k+1] . (3.8)Then, following (3.6), the PLS method yields{X[k+1], Y[k+1]} PLS−−→{Γ[k+1], G[k+1], L[k+1]}. (3.9)33In other words, the PLS method solves the full regression problem by decomposing X[k+1] andY[k+1] at the next time step k + 1, but these repeated decompositions may be computationallyburdensome in online implementation. A less computationally expensive, and indeed equivalent,alternative is to recursively update the estimated sensitivities by performing the PLS algorithmin (3.9) with (see, e.g., [98, Th. 1])X[k+1] = GT[k]x̂T[k+1] 1 , Y[k+1] =Γ[k]LT[k]ŷT[k+1] , (3.10)leading to a lower-dimensional PLS decomposition problem. Furthermore, we can embed weightfactors that prioritize more recent measurements over earlier ones. Particularly, the RWPLSmethod updates Ĥ[k] with the so-called forgetting factor σ ∈ (0, 1] by making use of the proce-dure represented by (3.9), except with [98]X[k+1] = σGT[k]x̂T[k+1] 1 , Y[k+1] =σΓ[k]LT[k]ŷT[k+1] . (3.11)In the above, if σ < 1 , then earlier measurements would not contribute as much to the finalestimate as more recent ones, i.e., weights of earlier measurements decrease exponentially asmore data is acquired. This is especially useful for the case where the system experiences achange in operating point during the time window in which measurements are obtained. Foronline implementation, the initial estimate of the sensitivity model Ĥ[k] can be obtained usingthe standard PLS estimation (3.6)–(3.7) once a sufficient number of samples has been acquired.Then at each subsequent time step when a new set of measurements becomes available, theestimated model can be updated via (3.9) in conjunction with (3.11).Example 3.1 (Comparing OLS and PLS). We present the numerical advantages of the PLSalgorithm over OLS estimation via an example involving the IEEE 33-bus test system (see,e.g., [1]). The single-line diagram of the test system is shown in Fig. 3.1. We collect 25 sets ofmeasurements of bus-voltage phase-angles and magnitudes as well as active- and reactive-powerinjections from D-PMUs installed at buses in the set E = {1, 6, 12, 18, 25, 33}. Suppose thatthe active- and reactive-power components of loads at buses in E− = E \ {1} vary randomly341 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1823 24 25 26 27 28 29 30 31 32 3319 20 21 22 SW1SW2SW3 SW4DERDERDERDERDERFigure 3.1: IEEE 33-bus test system with DERs at buses 6, 12, 18, 25, and 33; measurementsare collected at bus 1 and all DER buses. Switches SW1 and SW2 are normally open, andswitches SW3 and SW4 are normally closed.around their nominal values as Gaussian distributed random variables with zero mean and0.1% standard deviation relative to the respective nominal load values. We compare conditionnumbers of matrices that need to be inverted to obtain OLS and PLS estimates in (3.5) and (3.7),respectively. In OLS, the condition number of matrix XT[k]X[k] is 1.2245× 1015, indicating thatit is nearly singular. In contrast, the condition number of matrix G[k]GT[k] is 6.5973, whichleads to a numerically meaningful inverse and consequently an accurate linear sensitivity modelrelating measured voltages and injections. 3.3 DER Dispatch Problem with Estimated Sensitivity ModelInstead of relying on an accurate offline model to solve the problem in (2.9), we leverage onlinemeasurements to estimate linear sensitivities and construct the sensitivity-based power-balancemodel in (3.1). In this section, we formulate an optimal DER dispatch problem that incorporatesthe estimated linear sensitivity model as a constraint and outline a tractable solution approachusing the ADMM.3.3.1 Problem FormulationLet x[k+1] = [{θi,[k+1]}i∈E− , {Vi,[k+1]}i∈E− ]T comprise nodal voltage phase-angles and mag-nitudes at buses equipped with D-PMUs and time step k + 1. Note that the substationvoltage is assumed to be fixed to its known reference value, so it is not included as a vari-able. In our problem, we determine optimal DER active- and reactive-power outputs ygen[k+1] =[{P geni,[k+1]}i∈D, {Qgeni,[k+1]}i∈D]T and, in turn, net injections y[k+1] = [{Pi,[k+1]}i∈E , {Qi,[k+1]}i∈E ]Twhile satisfying operational constraints on x[k+1], y[k+1], and ygen[k+1]. Note that y[k+1] includes35the substation injections so that we can achieve feeder-level active- and reactive-power regula-tion. With the above notation in place, we propose to solve the following optimization problem:minimizex[k+1],y[k+1],ygen[k+1]f(x[k+1], y[k+1], ygen[k+1]), (3.12a)subject to y[k+1] = J[k]x[k+1] + c[k], (3.12b)y[k+1] = y[k] + C(ygen[k+1] − ygen[k] ), (3.12c)xmin ≤ x[k+1] ≤ xmax, (3.12d)ygen[k+1] ∈ Ygen, (3.12e)where xmin and xmax represent, respectively, the minimum and maximum limits of voltagephase-angles and magnitudes at buses that are measured, and Ygen represents the allowablespace of DER outputs. Under typical DER control schemes (including, e.g., active-power con-trol, reactive-power control, and joint active- and reactive-power control), Ygen is convex [59].Furthermore, in (3.12c), C ∈ R2|E|×2|D| is the matrix that maps the DER bus indices to busesfrom which measurements are obtained. Particularly, the entry in the ith row and jth column ofC is 1 if the jth element of ygen[k+1] is measured as the ith element of y[k+1], and it is 0 otherwise.The constraint in (3.12c) results by recognizing that, at time steps k and k + 1, respectively,y[k] = Cygen[k] − yload[k] , (3.13)y[k+1] = Cygen[k+1] − yload[k+1]. (3.14)Assuming that the loads at the measured buses do not change significantly between time steps kand k + 1, (3.14) can be expressed asy[k+1] = Cygen[k+1] − yload[k] . (3.15)Suitably rearranging (3.13) and substituting the resultant into the above, we get the constraintin (3.12c). The entries of J[k] and c[k] may be computed from a model if one is at hand, orthey can be estimated from online measurements, as discussed in Section 3.2. In either case,36the underlying assumption is that J[k] and c[k] computed or estimated at time step k model therelationship between y[k+1] and x[k+1] at time step k+ 1 with sufficient accuracy. We highlightthat the formulation in (3.12) is similar to (2.10). However, in (3.12) we consider the linearizedoptimal DER dispatch problem which only retains the subset of buses from which measurementsare available, i.e., buses in E ⊆ N .In (3.12), we utilize a quadratic cost function that comprises a weighted combination of(i) deviations of voltage phase-angles and magnitudes from their reference values, (ii) deviationsof nodal active- and reactive-power injections from their setpoints, and (iii) cost of DER active-and reactive-power outputs. Thus, the cost function in (3.12) can be expressed asf(x[k+1], y[k+1], ygen[k+1]) = (x[k+1] − x◦)TΨ(x[k+1] − x◦)+ (y[k+1] − y◦)TΦ(y[k+1] − y◦)+ (ygen[k+1])TΥygen[k+1], (3.16)where x◦ and y◦, respectively, denote the desired voltage setpoints for DER buses and desiredsetpoints for power injections at measured buses. In (3.16), Ψ = diag(ψ1, . . . , ψ2|E|−2), Φ =diag(ϕ1, . . . , ϕ2|E|), and Υ = diag(υ1, . . . , υ2|D|) are diagonal matrices with non-negative entries,i.e., ψi, ϕi, υi ≥ 0. The cost function in (3.16) is general in the sense that weight matrices Ψ,Φ, and Υ, respectively, enforce minimization in the cost of voltage-phasor deviations, power-injection deviations, and DER outputs.With positive semidefinite weight matrices, (3.12) is a convex quadratic optimization prob-lem that can be solved in polynomial time [99]. Particularly, by collecting decision variablesin χ[k+1] = [xT[k+1], yT[k+1], (ygen[k+1])T]T ∈ R4|E|+2|D|−2, the problem in (3.12) can be rewritten inthe standard form of a convex quadratic program as follows:minimizeχ[k+1]12χT[k+1]Πχ[k+1] + piTχ[k+1] + κ, (3.17a)subject to A[k]χ[k+1] = b[k], (3.17b)χ[k+1] ∈ X . (3.17c)Entries in and structures of Π, pi, κ, A[k], b[k], and X are detailed in Appendix B. Subsequently,37we remove the constant term κ from (3.17a) as it does not affect the solution of the optimizationproblem.3.3.2 ADMM-based Solution ApproachThe problem in (3.17) can be solved using ADMM, which is a tractable method as each subprob-lem in the ADMM admits a closed-form solution in our setting. Also, the ADMM convergesin linear time in our problem setting [100]. To begin, define an auxiliary variable ω[k+1] sothat the equality constraint and the bound constraint in (3.17b) and (3.17c), respectively, areassociated with different variables. Then the problem in (3.17) can be reformulated as follows:minimizeχ[k+1], ω[k+1]12χT[k+1]Πχ[k+1] + piTχ[k+1], (3.18a)subject to A[k]χ[k+1] = b[k], (3.18b)ω[k+1] ∈ X , (3.18c)χ[k+1] = ω[k+1]. (3.18d)In (3.18), the equality and inequality constraints involve different variables, coupled by theconstraint in (3.18d). The ADMM alternates between solving an equality-constrained quadraticprogram with decision variable χ[k+1] and a projection onto bound constraints with decisionvariable ω[k+1]. The augmented Lagrangian associated with (3.18) is given by [99]Lρ(χ[k+1], ω[k+1], µ[k+1]) =12χT[k+1]Πχ[k+1] + piTχ[k+1]+ρ[k]2∥∥χ[k+1]−ω[k+1]−µ[k+1]∥∥22− ρ[k]2∥∥µ[k+1]∥∥22, (3.19)where ρ[k] is a positive scalar and µ[k+1] is the scaled dual variable for the coupling constraintin (3.18d). The value of the ADMM parameter ρ[k] for optimal convergence is given byρ[k] =√λmin(ZT[k]ΠZ[k])λmax(ZT[k]ΠZ[k]), (3.20)38where λmin(·) and λmax(·), respectively, represent the minimum and the maximum eigenvaluesof their arguments, and Z[k] an orthonormal basis for the null space of A[k] [99]. Interestedreaders are referred to [99] for further background on ADMM and its convergence properties.Below, we briefly summarize the iterative solution procedure used in numerical case studiespresented in Section 3.4.The ADMM uses a sequence of iterations indexed by ` to search for the minimizer of (3.18),as follows:χ`+1[k+1] = arg minimizeχ[k+1]Lρ(χ[k+1], ω`[k+1], µ`[k+1]),subject to A[k]χ[k+1] = b[k], (3.21)ω`+1[k+1] = arg minimizeω[k+1]Lρ(χ`+1[k+1], ω[k+1], µ`[k+1]),subject to ω[k+1] ∈ X , (3.22)µ`+1[k+1] = µ`[k+1] + ω`+1[k+1] − χ`+1[k+1], (3.23)until ||χ`[k+1] − ω`[k+1]||2 <  and ||ρ[k](ω`[k+1] − ω`−1[k+1])||2 < , for some predefined tolerance > 0, along with initial conditions χ0[k+1] = 0, ω0[k+1] = 0, and µ0[k+1] = 0. The first step in eachADMM iteration is to update χ[k+1] by solving (3.21), which has a closed-form solution (see,e.g., [99]). Next, with the updated χ`+1[k+1], ω`+1[k+1] is obtained by projecting χ`+1[k+1] − µ`[k+1] ontothe feasible set X . Hence, the ω`+1[k+1]-update in (3.22) equivalently recovers the following:ω`+1[k+1] =arg minimizeω[k+1]∈X12∥∥∥χ`+1[k+1] − ω[k+1] − µ`[k+1]∥∥∥22, (3.24)which also admits a closed-form solution for voltage bounds and typical constraints on DERoutputs [59]. As a specific example, projections onto box constraints can be obtained as follows:ω`+1[k+1] =min(χmax,max(χmin, χ`+1[k+1] − µ`[k+1])), (3.25)where χmin and χmax comprise, respectively, the entry-wise lower and upper bounds of χ[k+1].The update in (3.25) can be computed for each entry of ω`+1[k+1] independently. We refer interestedreaders to [59] for closed-form solutions to projections onto other typical allowable spaces of39DER outputs. Finally, in each iteration, µ`+1[k+1] is obtained via (3.23) with updated values ofχ`+1[k+1] and ω`+1[k+1]. The ADMM iterations continue until the stopping criteria are satisfied withoptimizers given by χ?[k+1], ω?[k+1], and µ?[k+1], from which optimal DER dispatch ygen?[k+1] (i.e.,P gen?i,[k+1] and Qgen?i,[k+1], i ∈ D) can be extracted.3.4 Case StudiesIn this section, we demonstrate the effectiveness of the measurement-based optimal DER dis-patch framework proposed in Sections 3.2–3.3 via numerical case studies involving the IEEE33-bus test system introduced in Example 3.1 (see Fig. 3.1 for its one-line diagram), in which thepower base is set to 10 MVA. Through the case studies, we show that the proposed measurement-based optimal DER dispatch (i) adapts to changes in topology and operating point, (ii) closelymatches results obtained from the model-based optimization approach with accurate systemmodel, and (iii) outperforms the model-based approach with outdated system model. We fur-ther evaluate the impact of DER dynamics, dispatch delays, and RWPLS parameters on theperformance of the proposed method via simulations. In addition, we illustrate the flexibilityof the proposed framework by considering typical constraints on DER outputs. Finally, wereport execution times involved with the model estimation and optimal dispatch stages of theproposed framework.3.4.1 Simulation SetupAssume that DERs are connected to |D| = 5 buses in the setD = {6, 12, 18, 25, 33}, as annotatedin Fig. 3.1. Their active- and reactive-power outputs are, respectively, constrained withinP geni ∈ [−0.25, 0.25] p.u. and Qgeni ∈ [−0.25, 0.25] p.u. Although our simulations (for now) focuson box constraints for the DER active- and reactive-power outputs, the problem formulation andthe ADMM solution method outlined in Section 3.3 can, in general, accommodate constraintsdelineated by typical DER operational limits (as we will demonstrate in Section 3.4.5). Assumethat voltage phasors and active- and reactive-power injections at |E| = 6 buses in |E| = D∪{1}are sampled at 1-second intervals. With each new set of measurements, the sensitivity modelis updated via RWPLS by performing PLS with (3.11). Optimal dispatch is solved at 1-secondintervals when DERs are active. In the proposed measurement-based method, the optimal40dispatch is obtained via the ADMM solution to (3.18) immediately after an updated sensitivitymodel is estimated.We dispatch DER active- and reactive-power setpoints to simultaneously (i) minimizevoltage-magnitude deviations from reference levels (1 p.u. in our case studies), (ii) regulate theactive-power injections at the substation bus 1 and at bus 33, and (iii) minimize cost of DERactive- and reactive-power outputs. To realize the above objectives, we set the diagonal entriesin weight matrix Ψ as ψi = 0 for i = 1, . . . , |D| (corresponding to voltage phase-angles) andψi = 10 for i = |D| + 1, . . . , 2|D| (corresponding to voltage magnitudes). Furthermore, we setappropriate entries in x◦ to reference voltage magnitude 1 p.u., i.e., x◦ = [0T|D|, 1T|D|]T, where 0|D|and 1|D| denote the length-|D| vectors of all 0s and 1s, respectively. Diagonal entries in weightmatrix Υ corresponding to costs of DER outputs are set to various values between 0.04 and 0.1to reflect potentially different costs of distinct DER technologies. Finally, to demonstrate theability for the proposed method to regulate the net power injection at a measured bus, the firstentry (corresponding to bus 1) in y◦ is set to the substation active-power reference of 0.3 p.u. andthe |E|th entry (corresponding to bus 33) is set to the active-power reference of 0.1 p.u.; accord-ingly, in Φ, ϕ1 and ϕ|E| are set to 10 while ϕi = 0 for i = 2, . . . , |E| − 1, |E| + 1, . . . 2|E|. Theparameter values for our simulations are chosen in a fairly arbitrary manner, and the proposedoptimal DER dispatch method can easily accommodate a different cost function of the formin (3.16).3.4.2 Adaptability to Topology and Operating-point ChangesSuppose that the system initially operates without DER contribution, and the DERs are ac-tivated at time tstart = 5 s (time step kstart = 5). Subsequently, at time t∆ = 10 s (time stepk∆ = 10), the system topology is reconfigured by closing switches SW1 and SW2 and openingswitches SW3 and SW4, and simultaneously, active- and reactive-power loads at all buses inthe system grow by 25%. Simulations run until time tend = 16 s (time step kend = 16). TheRWPLS forgetting factor is set to σ = 0.6.410.90.95100.10.20.30.40 5 10 15-0.200.2Figure 3.2: Time-domain simulation via proposed measurement-based optimal DER dispatch.DERs are activated at time tstart = 5 s and the network topology and operating point arechanged at time t∆ = 10 s. Top pane: bus voltage magnitudes Vi at measured buses; Middlepane: bus active-power injections Pi at measured buses; Bottom pane: substation reactive-powerinjection Q1 and DER reactive-power outputs Qgeni at controllable buses.Simulation ResultsTime evolution of bus voltage magnitudes, as well as bus active-power injections and DERreactive-power outputs resulting from simulating the scenario described above with the pro-posed measurement-based DER dispatch method are plotted in Fig. 3.2. Indeed, once DERs areactivated at time tstart = 5 s, we observe that the proposed measurement-based dispatch effec-tively achieves the weighted objectives of minimizing voltage-magnitude deviations from 1 p.u.,and regulating the active-power injections at the substation bus 1 and at bus 33 to 0.3 p.u. and0.1 p.u., respectively. Upon activation of DERs, the substation reactive-power injection de-creases sharply since DERs collectively provide voltage support by injecting reactive power, asshown in the bottom pane of Fig. 3.2. Also, after network-topology and operating-point changesat time t∆ = 10 s, the proposed method updates the linear sensitivity model and adjusts the421 5 10 15 20 25 30 330.9811.02Figure 3.3: Bus voltage magnitudes Vi at t = 16 s obtained via (i) model-based method withaccurate system model, (ii) model-based method with inaccurate system model, and (iii) pro-posed measurement-based method. Voltage profile resulting from the proposed measurement-based method matches that from model-based benchmark with accurate system model, and itoutperforms the model-based approach when the system model is not updated.Table 3.1: Comparison of relative errors in cost function, voltage magnitudes, and active- andreactive-power dispatch among (i) model-based DER dispatch with accurate system model,(ii) model-based DER dispatch with inaccurate system model, and (iii) proposed measurement-based DER dispatch.Cost Cost Voltage- Active- Reactive-function function magnitude power power(p.u.) error (%) error (%) error (%) error (%)Model-based (acc.) 0.0044 — — — —Model-based (inacc.) 0.0668 1872.75 0.94 15.47 12.63Meas-based 0.0047 6.17 0.17 3.22 7.34DER dispatch to achieve the same objectives. Our choices for Υ, Φ, and Ψ as described earlieroptimize for a weighted objective. With different settings of Υ, Φ, and Ψ, other objectives,such as voltage control only, can be easily realized.Comparisons to Model-based DER DispatchWe benchmark the measurement-based optimal dispatch results presented above against the oneobtained from the model-based problem in (2.9), which is solved with the MATPOWER InteriorPoint Solver [30]. We record the voltage profiles and bus active- and reactive-power injections,at tend = 16 s, resulting from optimal DER dispatch setpoints obtained via (i) model-basedapproach assuming that the topology and operating-point changes are accurately captured inthe model, (ii) model-based approach assuming that the model is not updated after the systemchanges at t∆ = 10 s, and (iii) the proposed measurement-based approach. Figure 3.3 plots busvoltage magnitudes resulting from the three cases, and Table 3.1 summarizes errors in cases(ii) and (iii) as compared to the benchmark case (i). Results from the proposed method indeed43match very closely to those from model-based optimal dispatch with accurate system model,with respect to the cost function value, voltage magnitudes, as well as resulting active- andreactive-power injections. Moreover, the measurement-based approach outperforms the model-based one when the system model is not updated to reflect the changes implemented at timet∆ = 10 s. Although we consider topology and load changes in this case study, the proposedframework is flexible in that the optimal DER dispatch can easily adapt to other operating-point changes resulting from conventional distribution-level control mechanisms, including, e.g.,switched capacitors and OLTC transformers for voltage regulation.3.4.3 Effect of RWPLS Forgetting FactorThe performance of the measurement-based DER dispatch method depends significantly onthe value of the forgetting factor σ in the RWPLS estimation. To see this, we compare cost-function values obtained using (i) model-based dispatch vs. (ii) measurement-based dispatchby evaluating their root-mean square deviation (RMSD), given byRMSD =√√√√ 1kend − k∆kend∑k=k∆+1(fmodel[k] − fmeas[k] )2, (3.26)where k∆ = 10 is the time step when the network-topology and operating-point changes occur,and kend = 16 is the last time step of the simulation. We set the forgetting factor rangingfrom 0.01 to 1, and for each, we conduct 100 repeated simulations set up as described inSections 3.4.1 and 3.4.2 with random Gaussian-distributed load increases of (25± 1)% at timestep t∆ = 10 s and compute the average RMSD over all 100 simulations. The average RMSDvalues for different forgetting factors are plotted in Fig. 3.4a. We find that setting the forgettingfactor σ = 0.6 yields the lowest errors. Furthermore, Fig. 3.4b shows a histogram representingthe distribution of RMSD values for the 100 repeated simulations with forgetting factor σ = 0.6,with most scenarios centered around an RMSD value of 6×10−4. Considering these observations,we set σ = 0.6 for simulations presented in Section 3.4.2.440 0.2 0.4 0.6 0.8 1678910-44 6 8 10-4051015# Scenarios(a) (b)Figure 3.4: Effect of forgetting factor in RWPLS algorithm on dispatch accuracy. (a) Meancost-function RMSDs among model-based dispatch and measurement-based dispatch with dif-ferent forgetting factors σ for 100 repeated simulations. (b) Histogram of RMSD values for therepeated simulations, with best parameter choice (σ = 0.6).0 5 10 1500.10.20.30.4Figure 3.5: Evaluation of cost function value over time for different DER time constants τwith the measurement-based DER dispatch method, compared to the model-based method withideal DERs.3.4.4 Effect of DER Dynamics and DelaysIn practice, DER outputs do not change immediately after a new setpoint is dispatched. Weadopt a first-order model for the DERs, similar to [101]. Simulations are performed with thesetup described in Sections 3.4.1 and 3.4.2 along with DER time constants of τ = 0.1, 0.5, 1 s.The evaluated cost function at each time step are plotted in Fig. 3.5. We see that the proposedmethod is robust against slow-responding DERs, and the combined estimation/optimizationroutine converges to the optimal value achieved by the model-based benchmark.Furthermore, we evaluate the robustness of the proposed method against delays betweeninstants that measurements are acquired and DER setpoints are actuated. In practice, suchdelays may be attributed to communication, computation, and actuation. We adopt a sim-450.90.9510 5 10 15 20 2500.10.20.30.4Figure 3.6: Effect of delays between measurement acquisition and DER actuation. DERs areactivated at time tstart = 5 s and the network topology and operating point are changed at timet∆ = 10 s. Top pane: evaluation of cost function value over time for different DER dispatchdelays ξ with the measurement-based DER dispatch method, compared to the model-basedmethod without delay; Bottom pane: bus voltage magnitudes Vi at measured buses in caseDER actuation is delayed by ξ = 3 s.ilar simulation setup as in Sections 3.4.1 and 3.4.2, except the DER dispatch is artificiallydelayed and simulations run until time tend = 25 s. We evaluate the cost function value ateach time step and plot the resulting data points in the top pane of Fig. 3.6, where we use themodel-based optimal DER dispatch without delay as a benchmark and compare it to the pro-posed measurement-based dispatch with delay of ξ = 1, 2, 3 s. We observe that the combinedestimation and optimization routine converges to the optimal value achieved by the model-based benchmark, albeit with a delay as expected. In the bottom pane of Fig. 3.6, we plottime evolution of bus-voltage magnitudes for ξ = 3 s. We note that the significant networkreconfiguration and network-wide load changes do not cause voltage constraint violations inthe simulation. Moreover, the proposed method converges to the new optimal operating pointwithin a reasonable amount of time even in the presence of delays.46-0.1 -0.05 0 0.05 0.1 -0.1 -0.050 0.05 0.1 Bus 6Bus 12Bus 18Bus 25Bus 33(a)-0.1 -0.05 0 0.05 0.1 -0.1 -0.050 0.05 0.1 Bus 6Bus 12Bus 18Bus 25Bus 33(b)Figure 3.7: DER active- and reactive-power outputs obtained via the measurement-basedmethod when each DER is constrained by (a) independent active- and reactive-power limits,(b) its maximum apparent power.3.4.5 Consideration of Typical DER ConstraintsSome DER technologies may have different operational constraints than the independent active-and reactive-power limits considered in the case studies thus far. In this case study, we considerthe DER dispatch resulting from scenarios in which each DER i has (i) independent active-and reactive-power limits, constrained within P geni ∈ [−0.1, 0.1] p.u. and Qgeni ∈ [−0.1, 0.1] p.u.,respectively and (ii) joint active- and reactive-power limits, constrained by the apparent-powerlimit, where√(P geni )2 + (Qgeni )2 ≤ 0.1 p.u. The projection step (3.24) in ADMM yields closed-form solutions under both of these DER output limits. Specifically, we utilize (3.25) to computethe projection onto box constraints, and we leverage a closed-form solution similar to the onein [59] for the projection onto the allowable region delineated by apparent-power limits. Asshown in Fig. 3.7a, the DER outputs are indeed within the independent active- and reactive-power limits, whereas in Fig. 3.7b, DER outputs are constrained by the apparent-power limit of0.1 p.u. We note that the more constrained scenario of having apparent-power limits increasesthe resulting steady-state cost function value from 0.0052 to 0.0095. This simulation scenarioillustrates that the proposed method using the ADMM can accommodate typical constraintson DER outputs.3.4.6 Execution TimesTo assess the computational burden and the scalability of the proposed measurement-basedframework, we report execution times involving the 874-bus test system from [102], in which47Table 3.2: Computation times for (i) sensitivity model estimation (test), (ii) optimal DERdispatch via ADMM (tADMM), and (iii) optimal DER dispatch via MATLAB’s standard solverquadprog (tIP)# DERs 5 10 20 40 60 80 100test(s) 0.0015 0.0033 0.0086 0.0389 0.2130 0.3643 0.5570tADMM(s) 0.0177 0.0263 0.0904 0.2025 0.4016 0.5954 1.0548tIP(s) 0.0182 0.0252 0.0457 0.0609 0.1231 0.3711 1.2234up to 100 DERs are installed at random locations. The 874-bus is a single-phase equivalent ofa balanced 13.8 kV distribution system with 7 feeders and 12400 MVA total load. In Table 3.2,we report the execution times required to (i) update the linear sensitivity model (test), (ii) solvethe convex quadratic DER dispatch problem using ADMM (tADMM), and (iii) solve the sameproblem using the interior-point solver from MATLAB quadprog (tIP). All algorithms are runusing MATLAB R2018b on a personal computer with Intel Core i5-8250U processor at 1.60 GHz,and 8 GB RAM. We find that the method can be executed in near real time for up to 100 DERswith the relatively limited computational resources available on a personal computer.3.5 SummaryIn this chapter, we present a measurement-based framework to dispatch optimal DER active-and reactive-power outputs without relying on a system model. Optimal DER setpoints areobtained by embedding a recursively estimated sensitivity model as an equality constraint in aconvex quadratic optimization problem, which is solved via ADMM. As demonstrated by numer-ical case studies, the main advantages of the proposed method are (i) combined minimization ofbus voltage and injection deviations from prescribed references, as well as cost of DER outputs,(ii) accuracy of resulting DER setpoints without relying on a system model as compared to amodel-based benchmark, (iii) adaptability to operating-point and network-topology changes,and (iv) modest computational burden.48Chapter 4Distributed Method with EstimatedPer-area Sensitivity ModelsThe centralized measurement-based method proposed in Chapter 3 may not scale well in caseof significant deployments of D-PMUs and widespread installations of DERs. Furthermore,system-wide sharing of relevant data may pose data privacy concerns. At the same time, exist-ing distributed methods which are suitable to mitigate these issues require accurate knowledgeabout the network topology. With the aim to fill this key gap in the literature, this chapter1 pro-poses a distributed measurement-based approach, to determine DER active- and reactive-powersetpoints aimed at regulating bus voltages and power injections as well as minimizing DERcosts, unlike [56–58, 87–89]. Also, distinct from the centralized approach presented in Chap-ter 3 and [103] that optimizes over the entire distribution system, we partition the networkinto multiple areas, and each area performs its own sensitivity model estimation and optimalDER dispatch in parallel. For the estimation of voltage-to-power sensitivities in an area, we usevoltage phasor and power injection measurements obtained at buses equipped with D-PMUswithin the area as well as the one-hop adjacent bus in each adjacent area. The key advantage isthat the number of unknown sensitivities is greatly reduced in the per-area estimation problemas compared with the full-network analogue. In the DER dispatch stage, we incorporate allthe estimated sensitivity models as equality constraints into a convex quadratic optimizationproblem. As a direct consequence, the optimal dispatch is solved without any offline knowledge1A version of this chapter has been submitted: S. Nowak, Y. C. Chen, and L. Wang, “Distributedmeasurement-based optimal DER dispatch with estimated sensitivity models,” IEEE Trans. Power Syst., sub-mitted.49of the underlying network, distinct from the methods in [56–60]. Moreover, per-area sensitivitymodels enable decomposition of the optimization into per-area subproblems that can be solvedvia consensus-based ADMM in parallel fashion. The proposed distributed ADMM-based solu-tion algorithm affords closed-form solutions in each subproblem, and it requires only voltagephasor information at the boundary buses among adjacent areas to be exchanged. Moreover,compared with [91], our proposed method requires lower temporal measurement resolutionand smaller injection perturbations. Numerical simulations involving the IEEE 33-bus systemdemonstrate the effectiveness and adaptability of the proposed distributed measurement-basedframework. Further simulations of the multi-phase IEEE 123-bus system with realistic loadand renewable generation profiles demonstrate practical applicability and scalability.The remainder of this chapter is organized as follows. Section 4.1 motivates the needfor a distributed measurement-based approach. Section 4.2 presents the estimation of per-area sensitivity models. In Section 4.3, we formulate the distributed optimal DER dispatchproblem with the estimated sensitivity models and subsequently solve it via consensus-basedADMM. Numerical simulations in Section 4.4 demonstrate the effectiveness and scalability ofthe proposed distributed DER dispatch framework. Finally, we provide concluding remarks inSection 4.5.4.1 Problem StatementIn the previous Chapter 3 (published in [103]), we developed a measurement-based methodthat leverages D-PMU measurements at a subset of buses to estimate a reduced version of thesensitivity model in (3.1), and both the model estimation and DER dispatch steps are solvedin centralized fashion. Numerical case studies demonstrate that execution times are feasible forreal-time implementation with relatively few measurements and DERs in the network. How-ever, with vast deployments of DERs and measurement equipment projected in multi-phasedistribution systems [92], the centralized method may involve large-scale estimation and opti-mization problems that are computationally burdensome for real-time field implementations.To avoid potential scalability issues, in this chapter, we partition the distribution system intomultiple areas, each consisting of a portion of the full network. We estimate a linear sensitivitymodel for each area and further solve the optimal DER dispatch problem in a distributed man-50ner, enabled by consensus-based ADMM. The proposed distributed measurement-based methodhas several advantages: (i) it eliminates the reliance on an accurate offline network model byestimating per-area sensitivity models online, (ii) it reduces the computational burden in boththe model estimation and optimal DER dispatch stages, and (iii) it affords greater data privacyas measurements, cost function, and operational constraints are only shared within the area.4.2 Per-area Sensitivity Model EstimationIn this section, we estimate the linear sensitivity model relating measured bus voltages andpower injections for each area in the distribution network.4.2.1 Problem FormulationWe partition the distribution network into non-overlapping areas collected in the set A. Withinarea α ∈ A, buses equipped with D-PMUs are collected in the set Eα, and denote by Dα the setof buses that are connected to DERs, whose setpoints can be remotely updated. For distinctareas α, γ ∈ A, Eα ∩ Eγ = ∅. Furthermore, let Eα+ represent the extended set of buses in eacharea α ∈ A, which also collects the one-hop adjacent buses equipped with D-PMUs belongingto areas adjacent to area α. Denote the set of adjacent areas of area α by Aα, and boundarybuses for area α are collected in Bα = ∪γ∈Aα(Eα+ ∩ Eγ+). Let Ai = {α, i ∈ Eα+} compriseareas for which the extended set of buses contains bus i.Denote the measured voltage phase-angle and magnitude at bus i in phase φ at time stepk as θ̂φi,[k] and V̂φi,[k], respectively. Similarly, let P̂φi,[k] and Q̂φi,[k] denote the measured active-and reactive-power injections at bus i in phase φ at time step k. Collect measured quantitiespertinent to area α ∈ A in vectors x̂α[k] = [{θ̂φi,[k]}φ∈Pi,i∈Eα+ , {V̂ φi,[k]}φ∈Pi,i∈Eα+ ]T and ŷα[k] =[{P̂ φi,[k]}φ∈Pi,i∈Eα , {Q̂φi,[k]}φ∈Pi,i∈Eα ]T. Further hypothesizing that, for area α ∈ A, the active-and reactive-power injections are linearly related to bus-voltage phase-angles and magnitudes,we getŷα[k] = Jα[k]x̂α[k] + cα[k], α ∈ A, (4.1)where Jα[k] and cα[k] form the sensitivity model relating measured voltages to power injectionsin area α ∈ A. With the voltage measurements at the one-hop adjacent buses in adjacentareas included in (3.1), the structure of the distribution network admittance matrix enables the51estimation of per-area sensitivities without compromising the accuracy. Then, there exists Hα[k]that satisfies the relationship(ŷα[k])T =[(x̂α[k])T 1]Hα[k], α ∈ A, (4.2)where Hα[k] = [Jα[k], cα[k]]T. In the per-area estimation stage, our goal is to evaluate the entries ofHα[k] for each area α ∈ A based on per-area measurements.4.2.2 Solution AlgorithmIn order to eliminate the reliance on a known network model and to reduce computationalburden, we estimate the entries of Hα[k] using only online measurements collected from D-PMUs pertinent to area α. To this end, suppose that M samples of voltage phase-angles andmagnitudes, x̂α[k−M+1], . . . , x̂α[k], and active- and reactive-power injections, ŷα[k−M+1], . . . , ŷα[k], areavailable. Further suppose that the operating point remains approximately constant over theM measurement samples (we will remove this assumption later). Then, with M > 2|Eα|, westack up M instances of (4.2) to yield the following over-determined system of linear equations:Y α[k] = Xα[k]Hα[k], α ∈ A, (4.3)where Xα[k] and Yα[k] are given byXα[k] =(x̂α[k−M+1])T 1......(x̂α[k])T 1 , Y α[k] =(ŷα[k−M+1])T...(ŷα[k])T . (4.4)Since (4.3) is over-determined, we can obtain the OLS estimate for Hα[k] asĤα[k] ≈ ((Xα[k])TXα[k])−1(Xα[k])TY α[k]. (4.5)However, in practice, OLS estimation is challenged by voltage phase-angles and magnitudes atdifferent buses evolving similarly with variations in operating point [97], which results in anill-conditioned regressor matrix in (4.5). Also, recursive updates of Ĥα[k] are desirable to capture52changing operating points while reducing computational burden, and greater emphasis oughtto be placed on more recent measurements than earlier ones that may become out of date. Inlight of these circumstances, similar to [103], we use the RWPLS algorithm to compute Ĥα[k], asit offers several benefits over OLS estimation.Assuming that the per-area sensitivity models are updated every R samples, the RWPLSmethod employs the PLS decomposition recursively to project key components in Xα[k−R] andY α[k−R] onto lower-dimensional latent matrices Γα[k−R], Gα[k−R], Lα[k−R]. New design and responsematrices Xα[k] and Yα[k] at time step k can equivalently be represented by combining the PLSmodel of the previous estimation step at time step k − R with newly collected measurementsbetween time steps k −R+ 1 and k such thatXα[k] =σGα[k−R]Tx̂α[k−R+1]T 1......x̂α[k]T 1, Y α[k] =σΓα[k−R]Lα[k−R]Tŷα[k−R+1]T...ŷα[k]T, (4.6)where the forgetting factor σ ∈ (0, 1] prioritizes recent measurements over earlier ones. Finally,the estimated per-area sensitivity model Ĥα[k] at time step k is obtained after decomposing Xα[k]and Y α[k] into Γα[k], Gα[k], Lα[k] via PLS byĤα[k] = (Gα[k] Gα[k]T)−1Gα[k]Γα[k] Lα[k]T . (4.7)Details on the RWPLS method can be found in [98, 103].Example 4.1 (Sensitivity Model Estimation). To illustrate ideas introduced above, considerthe IEEE 33-bus test system (see, e.g., [1]), the one-line diagram for which is shown in Fig. 4.1.As a benchmark for comparison, we obtain the full-network sensitivity model H by evaluat-ing the model-based power-flow Jacobian matrix at the operating point assuming an accuratenetwork model is available. Next, we estimate the full-network sensitivity model Ĥ computedusing measurements obtained from all buses in the system. We also evaluate entries of theper-area sensitivity models as outlined in this section. For both full-network and per-area es-timates, we collect 100 sets of measurements of bus voltage phase-angles and magnitudes as531 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1823 24 25 26 27 28 29 30 31 32 3319 20 21 22Figure 4.1: IEEE 33-bus test system [1] with DERs connected at buses collected inD = {6, 12, 18, 22, 25, 33}. The network is partitioned into five areas collected in the setA = {1, 2, . . . , 5}. Assume D-PMUs are installed at all buses. This test system is single phase, soPi = {a}, ∀ i ∈ N . As an example of the notation used throughout this chapter, consider area 1.Within area 1, buses equipped with D-PMUs are collected in E1 = {1, 2, 3, 4, 5, 6, 23, 24, 25}and buses connected to DERs are collected in D1 = {6, 25}. The extended set of buses isE1+ = E1 ∪ {7, 19, 26}. Adjacent areas are collected in A1 = {2, 3, 4}. Boundary buses arecollected in B1 = (E1+ ∩ E2+) ∪ (E1+ ∩ E3+) ∪ (E1+ ∩ E4+) = {2, 6, 7, 19, 26}. Finally, considerboundary bus 6, which belongs to the extended set of buses in areas collected in A6 = {1, 3, 4}.well as active- and reactive-power injections from D-PMUs assumed to be installed at all buses.Also assume that the active- and reactive-power components of loads at buses in N \ {1} varyrandomly around their nominal values as Gaussian distributed random variables with 0 meanand 0.1% standard deviation relative to the respective nominal load values.To assess the accuracy of the per-area sensitivity models, we reorder and stack the entriestherein appropriately to form a matrix comparable to the full-network sensitivity model. Usingthe relative error of the Frobenius norm as a metric, i.e., ||Ĥ −H||F /||H||F , we find that thefull-network estimate has an error of 1.995 × 10−4, compared to an error of 6.224 × 10−5 forthe per-area estimate. The per-area estimation leads to a lower estimation error comparedto the full-network counterpart, because we made use of network connectivity information inmassaging the per-area sensitivity models into the full-network analogue. 4.3 Distributed Optimal DER DispatchIn this section, we formulate the distributed optimal DER dispatch problem that incorporatesthe per-area estimated sensitivity model as a constraint in the subproblem for each area. Wealso outline a tractable solution approach using the consensus-based ADMM algorithm.544.3.1 Problem FormulationIn our problem, we aim to determine the controllable DER active- and reactive-power setpointsin area α ∈ A at time step k+1. Namely, we optimize over variables collected in vector yα,gen[k+1] =[{P φ,geni,[k+1]}φ∈Pi,i∈Dα , {Qφ,geni,[k+1]}φ∈Pi,i∈Dα ]T, where Dα denotes the set of buses with controllableDERs in area α. In turn, decision variables also include net injections and voltage phase-angles and magnitudes respectively collected in yα[k+1]=[{P φi,[k+1]}φ∈Pi,i∈Eα , {Qφi,[k+1]}φ∈Pi,i∈Eα ]Tand xα[k+1]=[{θφi,[k+1]}φ∈Pi,i∈Eα+ , {V φi,[k+1]}φ∈Pi,i∈Eα+ ]T. We impose constraints on xα[k+1], yα[k+1],and yα,gen[k+1] to account for various operating ranges such as minimum and maximum allowablevoltage limits and DER operating regions. Without loss of generalization, we assume thatarea 1 contains the substation bus 1. Note that y1[k+1] includes the substation injections so thatwe can achieve feeder-level active- and reactive-power regulation. With the above notation inplace, we formulate the following optimization problem:minimizexα,yα,yα,gen,α∈A∑α∈Afα(xα, yα, yα,gen) (4.8a)subject to yα= Jα[k]xα + cα[k], ∀α∈A, (4.8b)xαmin ≤ xα ≤ xαmax, ∀α∈A, (4.8c)yα,gen ∈ Yα,gen, ∀α∈A, (4.8d)yα= yα[k] + Cα(yα,gen− yα,gen[k] ), ∀α∈A, (4.8e)zαi = Fαi xα = zi, ∀ i∈Bα, α∈A, (4.8f)where the parameterization of decision variables with respect to k + 1 is dropped to containnotational burden. In (4.8e), Cα, α ∈ A, is the matrix that maps the DER bus/phase indices tobuses/phases from which measurements are obtained in area α. Furthermore, (4.8f) representconsensus constraints that enforce voltage phase-angles and magnitudes at boundary busesshared among adjacent areas to be equal. To this end, we use matrix Fαi , α ∈ A, to mapvoltage phase-angles and magnitudes in all phases of bus i ∈ Bα in xα to auxiliary variable zi,which is common for all areas with bus i as a boundary bus. In (4.8), we utilize a quadratic costfunction that is fully separable among different areas. It comprises a weighted combination of(i) deviations of voltage phase-angles and magnitudes from their reference values, (ii) deviations55of nodal active- and reactive-power injections from their setpoints, and (iii) cost of DER active-and reactive-power outputs. Thus, the cost function in (4.8) takes the following form:fα(xα, yα, yα,gen) = (xα − xα◦)TΨα(xα − xα◦)+ (yα − yα◦)TΦα(yα − yα◦)+ (yα,gen)TΥαyα,gen, (4.9)where xα◦ and yα◦, respectively, denote the desired voltage setpoints and desired setpointsfor power injections at measured buses in area α ∈ A. In (4.9), Ψα = diag(ψ1, . . . , ψ2|Eα+|),Φα = diag(ϕ1, . . . , ϕ2|Eα|), and Υα = diag(υ1, . . . , υ2|Dα|) are diagonal matrices with non-negativeentries, i.e., ψi, ϕi, υi ≥ 0.We collect decision variables pertinent to area α in χα = [(xα)T, (yα)T, (yα,gen)T]T andrewrite (4.8) as follows:minimizeχα, α∈A,zαi ,zi, i∈Bα, α∈A∑α∈A12(χα)TΠαχα + (piα)Tχα + κα (4.10a)subject to Aα[k]χα = bα[k],∀α ∈ A, (4.10b)χα ∈ Xα, ∀α ∈ A, (4.10c)zαi = Kαi χα = zi,∀ i ∈ Bα, α ∈ A, (4.10d)where Πα and piα appropriately collect the quadratic and linear weights of the cost functionin (4.9), and κα is the constant term therein. Subsequently, we remove κα from (4.10a) asit does not affect the solution of the optimization problem. For area α ∈ A, the equalityconstraints in (4.8b), and (4.8e), are stacked up to form the constraint in (4.10b) with matrixAα[k] and vector bα[k] appropriately defined. Similarly, the set constraints in (4.8c) and (4.8d) arecollectively rewritten as (4.10c), where Xα, α ∈ A, denotes the typically convex set of allowableDER operating regions as well as bus voltage bounds in area α. Finally, (4.10d) correspondsto consensus constraints in (4.8f), where Kαi , α ∈ A, is a matrix containing 0s and 1s thatmaps voltage phase-angles and magnitudes of bus i in χα to zαi . Also, voltage phase-angle andmagnitude of each boundary bus i ∈ Bα are associated with auxiliary consensus variable zi,56coupled across areas that share bus i as a boundary bus, i.e., areas in the set Ai.With positive semidefinite weights in Πα, (4.10) can be decomposed into |A| per-area convexquadratic subproblems coupled with equality constraints pertinent to adjacent areas. Due tothe consensus constraints among adjacent areas in (4.10d), standard solvers for the subproblemsare not immediately applicable. In the next section, we present a tractable consensus-basedADMM algorithm to solve the DER dispatch problem in (4.10) in a distributed fashion, whereonly voltage phase-angles and magnitudes of boundary buses need to be exchanged amongadjacent areas.4.3.2 ADMM-based Solution ApproachThe general idea of the ADMM-based approach is to treat each per-area subproblem as a convexquadratic program. In addition, using the auxiliary variables defined for the voltage phase-angleand magnitude for the boundary buses, consensus among adjacent areas can be achieved. Wedecompose the problem in (4.10) for each area α ∈ A:minimizeχα,ωα,zγi ,zi,i∈Bα,γ∈Ai12(χα)TΠαχα + (piα)Tχα (4.11a)subject to Aα[k]χα = bα[k], (4.11b)ωα ∈ Xα, (4.11c)ωα = χα, (4.11d)zγi − zi = 0, ∀ i ∈ Bα, γ ∈ Ai (4.11e)zαi = Kαi χα, ∀ i ∈ Bα, (4.11f)where we define auxiliary variables ωα and tie it to decision variables χα with the equalityconstraint in (4.11d), and γ ∈ Ai is a dummy index. Notably, in (4.11), the per-area equalityconstraints on χα, per-area bound constraints on χα, and consensus constraints among adjacentareas are each associated with different decision variables.For the per-area subproblem, the ADMM algorithm alternates between solving an equality-constrained quadratic program with a projection onto feasible bound constraints, while achiev-ing consensus among areas via (4.11e). The augmented Lagrangian associated with (4.11) is57given byLαρ (χα, ωα, µα, {zγi , zi, sγi }i∈Bα,γ∈Ai)=(12(χα)TΠαχα + (piα)Tχα+ρ2‖χα − ωα + µα‖22 −ρ2‖µα‖22)+∑i∈Bα∑γ∈Aiρ2(||zγi − zi + sγi ||22 − ||sγi ||22), (4.12)where ρ is a positive scalar ADMM parameter, µα, and sγi are scaled dual variables for the cou-pling constraints (4.11d) and (4.11e). Note that we do not include constraints (4.11b), (4.11c)and (4.11f) in the augmented Lagrangian as they will be considered in the respective subprob-lems. Below, we provide a brief summary of the iterative solution procedure, which is usedin numerical case studies presented in Section 4.4. The ADMM algorithm uses a sequence ofiterations indexed by ` to search for the minimizer of (4.11).The primal variable update for area α ∈ A solves for (χα)`+1 and (zαi )`+1 from the followingstandard equality-constrained convex quadratic program:minimizeχα,zαi , i∈Bα12(χα)TΠαχα + (piα)Tχα+ρ2∥∥∥χα − (ωα)` + (µα)`∥∥∥22+∑i∈Bαρ2∥∥∥zαi − (zi)` + (sαi )`∥∥∥22(4.13a)subject to Aα[k]χα = bα[k], (4.13b)zαi = Kαi χα, ∀ i ∈ Bα. (4.13c)The solution to (4.13) can be readily obtained via a closed-form solution [99]. Next, armed withthe primal updates for iteration `+ 1, the updates for auxiliary variables ωα and zi, i ∈ Bα, arerespectively given by(ωα)`+1 = arg minimizeωα∈Xαρ2||(χα)`+1 − ωα + (µα)`||22, (4.14)(zi)`+1 = arg minimizezi∑γ∈Aiρ2||(zγi )`+1 − zi + (sγi )`||22. (4.15)58The projection problem in (4.14) admits closed-form solution for voltage bounds and typicalconstraints on DER outputs [99]. For example, the projection onto box constraints yields(ωα)`+1 = min(χαmax,max(χαmin, (χα)`+1 − (µα)`)), (4.16)where χαmin and χαmax comprise, respectively, the entry-wise lower and upper bounds of χα. Forthe auxiliary variable zi, i ∈ Bα, the problem in (4.15) solves as the average of (zγi )`+1, γ ∈ Ai,i.e., [104](zi)`+1 =1|Ai|∑γ∈Ai(zγi )`+1. (4.17)Finally, the scaled dual variables admit straightforward recursive updates given by(µα)`+1 = (µα)` + (ωα)`+1 − (χα)`+1, (4.18)(sαi )`+1 = (sαi )` + (zi)`+1 − (zαi )`+1. (4.19)In the ADMM algorithm outlined above, we repeatedly update primal, auxiliary and dualvariables until ||(χα)` − (ωα)`||2 + ||{(zαi )` − (zi)`}i∈Bα ||2 <  and ||ρ((ωα)` − (ωα)`−1)||2 +||ρ({(zi)` − (zi)`−1}i∈Bα)||2 < , for some predefined tolerance  > 0. We initialize auxiliaryvariables (ωα)0 and (zi)0 with values corresponding to the operating point at time step k justbefore optimization, and set initial conditions for primal and scaled dual variables to 0. TheADMM iterations continue until the stopping criteria are satisfied with optimizer given by χα?from which optimal DER dispatch yα,gen? (i.e., Pα,gen?i and Qα,gen?i , i ∈ Dα) can be extracted.We note that all the updates of primal and dual variables can be computed in parallel for eacharea α ∈ A. The only step that requires exchange among the areas is the auxiliary variableupdate in (4.17), in which updated voltage phase-angles and magnitudes at boundary buses areshared among adjacent areas.Example 4.2 (Distributed Optimal DER Dispatch). We present simulation results of the dis-tributed optimal DER dispatch method outlined above as compared to the centralized versionin [103]. Adopting the same simulation scenario as presented in [103] for the IEEE 33-bus testsystem, we assume that DERs are connected to buses in the set D = {6, 12, 18, 22, 25, 33}, as590.90.95100.20.40 5 10 15-0.200.2Figure 4.2: Time-domain simulation via proposed distributed measurement-based optimal DERdispatch when DERs are activated at time tstart = 5 s, and at time t∆ = 10 s, the system op-erating point is changed. Top pane: bus voltages Vi at buses with DERs; Middle pane: busactive-power injections Pi at buses with DERs; Bottom pane: substation reactive-power injec-tion Q1 and DER reactive-power injections Qgeni at buses with DERs.annotated in Fig. 4.1. The DER active- and reactive-power setpoints are dispatched to simul-taneously (i) minimize voltage-magnitude deviations from reference level of 1 p.u., (ii) regulateactive-power injections at buses 1 and 33 to reference values of 0.3 p.u. and 0.1 p.u., respectively,and (iii) minimize cost of DER active- and reactive-power outputs.In Fig. 4.2, we plot time evolution of voltage magnitudes and active-power injections atDER buses, as well as DER and substation reactive-power outputs, resulting from simulatingthe scenario described above. The system initially operates without DER participation. TheDERs are activated at time tstart = 5 s. Subsequently, at time t∆ = 10 s, the system operatingpoint changes as active- and reactive-power loads at all buses in the system increase by 25%.Simulations run until time tend = 15 s. Once DERs are activated at time tstart = 5 s, we observethat the proposed measurement-based dispatch effectively achieves the weighted objectives ofminimizing voltage-magnitude deviations from 1 p.u. and regulating the active-power injectionsat buses 1 and 33 to their respective setpoints. After the operating-point change at timet∆ = 10 s, the proposed method updates the linear sensitivity models and the DER dispatch toachieve the same objectives.We benchmark the proposed distributed measurement-based optimal DER dispatch method60Table 4.1: Comparison of cost-function and relative-errors among (i) the centralized model-based nonlinear DER dispatch, (ii) the centralized measurement-based DER dispatch, and(iii) the proposed distributed measurement-based DER dispatch.Cost Cost Voltage- Active- Reactive-function function magnitude power power(p.u.) error (%) error (%) error (%) error (%)Model-based (Central) 0.0077 — — — —Meas-based (Central) 0.0077 0.054 0.002 5.800 0.939Meas-based (Dist) 0.0077 0.054 0.002 5.849 0.067against centralized model- and measurement-based analogues, where all buses in the networkare assumed to belong to one area. The nonlinear model-based optimal dispatch problem issolved with the MATPOWER Interior Point Solver [30]. We record voltage profiles and busactive- and reactive-power injections, at tend = 15 s for (i) the centralized model-based nonlinearDER dispatch, (ii) the centralized measurement-based DER dispatch, and (iii) the proposeddistributed measurement-based DER dispatch. Table 4.1 summarizes errors in (ii) and (iii)as compared to the model-based benchmark (i). Results from the proposed method indeedmatch very closely to those from model-based optimal dispatch with accurate system modeland the centralized measurement-based dispatch, with respect to the cost function value, voltagemagnitudes, and active- and reactive-power injections. 4.4 Application to Multi-phase SystemIn this section, we demonstrate the scalability of the proposed distributed measurement-basedoptimal DER dispatch framework via numerical case studies involving the IEEE 123-bus testsystem shown in Fig. 4.3. We refer interested readers to [105] for details on system parameters.4.4.1 Simulation SetupWe use the residential loading conditions for a 24-hour period with 1-min resolution [81], inwhich a random number of customers (integer between 5 and 10) are connected to load buses,resulting in non-uniform distribution of loads across different phases. We assume a constantpower factor of 95% for each load. Furthermore, we connect uncontrolled three-phase solarPV systems at buses 13, 18, 35, 52, 60, 67, 72, single-phase solar at buses 32c, 94a, and 99b, andtwo-phase solar at bus 36 between phases a and b, all operating at unity power factor. The solarPV output is normalized from actual measurements with 1-second resolution [106], and each61Figure 4.3: IEEE 123-bus multi-phase test system partitioned into 6 areas.solar PV system uses a different profile from the data set, as shown in Fig. 4.4. We use solarPV profiles from a day with intermittent generation before noon, which challenge the modelestimation and optimal dispatch due to high ramp rates.We partition the test system into 6 areas, with each containing several single- and three-phase DERs, as indicated in Fig. 4.3. The DERs operate with independent active- and reactive-power limits within P geni ∈ [−0.05, 0.05] p.u. and Qgeni ∈ [−0.05, 0.05] p.u. Three-phase DERsoperate in balanced manner, where active- and reactive-power outputs are equal across allphases. In the simulations, we deactivate existing voltage regulators as DERs are used toregulate voltages and we limit our studies to the default switch configurations.Assume that measurements of voltage phasors and active- and reactive-power injectionssampled at 1-second intervals are available from all buses, except those with zero injections.The sensitivity model for each area is updated every R = 10 samples via RWPLS in (4.7). Opti-620 2 4 6 8 10 12 14 16 18 20 2200.5124Figure 4.4: Normalized daily solar PV output for different locations with intermittent solarirradiation.Table 4.2: Comparison of execution times.Model Optimal DER DispatchEstimation (s) Per iteration (s) # iterations Total (s)Centralized 3.7697 3.1001× 10−4 3127 0.9694Distributed 0.0346 8.9803× 10−5 1167 0.1048mal DER dispatch is similarly solved every 10 seconds to allow ample time for the ADMM-basedsolution to converge to the new setpoints. We dispatch DER active- and reactive-power set-points to simultaneously (i) regulate the active- and reactive-power injections at the substationbus 1 to 0.15 and 0.03 p.u., respectively, and (ii) minimize cost of DER active- and reactive-power outputs while keeping voltages within acceptable bounds of 0.95 and 1.05 p.u. We setentries in the diagonal weight matrices Ψ,Φ, and Υ accordingly to achieve the objective above.4.4.2 Execution TimeIn Table 4.2, we highlight the computational benefits of the distributed method by reportingexecution times required for model estimation via RWPLS and DER dispatch via distributedADMM-based optimization. We compare these to the centralized implementation from [103].We run the algorithms in MATLAB R2018b on a personal computer with Intel Core i5-8250Uprocessor at 1.60 GHz, and 8 GB RAM. For the distributed method, we report execution timesfor area 6, as it is the largest area. We set the ADMM parameter to ρ = 0.7 and set thetolerance for convergence to  = 10−6.634.4.3 Simulation ResultsIn Figs. 4.5 and 4.6 for the cases without and with DER control via the distributed measurement-based framework, respectively, we plot time evolution of bus voltage magnitudes at all monitoredbuses, as well as active- and reactive-power injections at the substation and at buses withDERs. In Fig. 4.6, we observe that the substation active- and reactive-power injections areindeed regulated to their respective references and voltages are regulated within the acceptablebounds of 0.95 and 1.05 p.u. The proposed framework is very effective during times withrelatively smooth solar PV output, but it is challenged under highly intermittent conditions,where the operational objectives are not met with the 10-second DER setpoint updates. Thisissue may be addressed by pairing the proposed method with local DER controllers or byreducing the time between consecutive DER setpoint updates.4.5 SummaryIn this chapter, we present a distributed measurement-based framework to dispatch optimalDER active- and reactive-power outputs without relying on an offline model. Optimal DER set-points are obtained by embedding recursively estimated per-area sensitivity models as equalityconstraints in an optimization problem, which is solved in distributed fashion via the consensus-based ADMM method. The decentralized estimation and distributed optimization require littleinformation exchange among adjacent areas (only voltages at boundary buses) and yield closed-form solutions. As demonstrated by numerical case studies, the main advantages of the proposedmethod include (i) accurate optimal DER setpoints for changing operating points without rely-ing on an offline system model and (ii) computational savings over centralized implementationsin both the model estimation and optimal DER dispatch stages.640.90.9511.05-0.100.10.20 2 4 6 8 10 12 14 16 18 20 22-0.0500.0524Figure 4.5: Time-domain simulation of the scenario without active DER control. Top pane: busvoltages Vi at monitored buses; Middle pane: bus active-power injections Pi at substation andbuses with DERs; Bottom pane: bus reactive-power injections Qi at substation and buses withDERs.0.90.9511.05-0.100.10.20 2 4 6 8 10 12 14 16 18 20 22-0.0500.0524Figure 4.6: Time-domain simulation via proposed estimation and optimization framework.Top pane: bus voltages Vi at monitored buses; Middle pane: bus active-power injections Pi atsubstation and buses with DERs; Bottom pane: bus reactive-power injections Qi at substationand buses with DERs.65Chapter 5Two-level Voltage Control MethodMitigating Effects of Highly IntermittentRenewable GenerationThe methods in Chapters 3 and 4 could face challenges in cases with communication delays orlow time granularity of available system-wide measurements to obtain accurate and up-to-dateDER setpoints. Highly intermittent renewable generation with rapid fluctuations of operatingpoints could further deteriorate their performance, and potentially lead to suboptimal andineffective operation [76]. As a remedy, in this chapter,1 we propose a two-level voltage controlmethod which leverages real-time measurements specifically designed to mitigate the effects ofhighly intermittent renewable generation. The proposed method adopts a hierarchical controlarchitecture consisting of an upper-level centralized controller and lower-level local controllersfor primary control. It is considered that up-to-date measurements at a centralized location arepotentially not available at the same intervals as locally at DER locations. Thus the centralizedoptimization could be compromised by out-of-date measurements. Using the proposed two-levelarchitecture this is mitigated by implementing local controllers which are able to act locally toregulate voltage levels.First, a new lower-level local reactive-power controller to regulate bus voltages to a fixedsetpoint using a proportional-integral (PI) scheme is proposed. The PI control scheme, as1A version of this chapter has been published in: S. Nowak, L. Wang, and M. S. Metcalfe, “Two-levelcentralized and local voltage control in distribution systems mitigating effects of highly intermittent renewablegeneration,” Int. J. Electr. Power Energy Syst., vol. 119, p. 105858, Jul. 2020.66an alternative to the predominant droop-based local controllers, is motivated by the recentamendments in the IEEE 1547 standard in which the DERs are allowed to participate involtage regulation through local reactive-power generation and the use of voltage setpoints.Stability conditions via a discrete-time state-space model are formulated to design the localcontrollers based on local voltage sensitivities. Then, the discrete-time state-space model isextended for the full system in order to ensure a globally stable control framework. In addition,within the upper-level centralized controller, a system-wide, nonlinear optimal DER dispatchoptimization program is implemented. We note that, in this chapter, we use a model-basedapproach to compute optimal DER setpoints. However, the concept could easily be paired withthe measurement-based versions in Chapters 3 and 4. The primary purpose of the two-levelcontrol system is to regulate bus voltages to regulatory-compliant levels within ANSI limits.Furthermore, by using centralized coordination of the local controller, the system networklosses are reduced, which provides additional benefits to the system operation. The controlsystem is designed to consider DER reactive-power operational limits, to mitigate effects ofhighly intermittent renewable generation and to ensure system voltage stability. Additionally,the local PI control framework is applied to an unbalanced test case, in which it achievesbetter voltage symmetry and voltage levels closer to the nominal voltage level. Overall, thecontributions are summarized as follows:• The proposed local controllers enable faster response time to regulate bus voltages toreference levels without steady-state voltage mismatch compared to existing VV droop-based local controllers [53, 63, 64].• The design criteria of the local controllers to ensure stability of the primary control andthe mitigation of integrator windup are proposed. Conditions for a stable design of thelocal controllers based on bus voltage sensitivities to reactive-power generation using adiscrete-time state-space model are formulated. The use of the state-space model forthe assessment of distribution system voltage stability generalizes the existing methods(i.e., [53, 65]) for the proposed local controllers.• In contrast to existing literature, this chapter proposes to dispatch both the optimalvoltage and reactive-power setpoints to the local PI controllers from the central optimal67DER dispatch optimization. The use of optimal reactive-power setpoints ensures minimalnetwork losses during large operating-point changes while the optimal voltage setpointsmaintain accurate voltage regulation during times of highly intermittent renewable gen-eration.The remainder of this chapter is organized as follows. Section 2 states the preliminaries.Section 3 describes the proposed lower-level local controllers and their stable design. Section4 presents the upper-level centralized optimization. Section 5 provides results from the casestudies and finally, the chapter is summarized in Section 6.5.1 PreliminariesIn this section, we establish the branch flow model and its linearized version, and describe theDER model.5.1.1 Branch Flow Model FormulationAs an alternative to the complex power-flow model (2.3), several simplifications can be madefor balanced distribution networks, which can be represented with a single-phase equivalentusing the branch-flow model [107]. In the branch-flow model, the power flow and voltage dropin edges between buses (i, j) ∈ T of the distribution system are modeled by2Pij,[k] =∑l:(j,l)∈TPjl,[k] + rijP 2ij,[k] +Q2ij,[k]V 2i,[k]− Pj,[k],∀ (i, j) ∈ T (5.1a)Qij,[k] =∑l:(j,l)∈TQjl,[k] + xijP 2ij,[k] +Q2ij,[k]V 2i,[k]−Qj,[k],∀ (i, j) ∈ T , (5.1b)V 2i,[k] − V 2j,[k] = 2(rijPij,[k] + xijQij,[k])− (r2ij + x2ij)P 2ij,[k] +Q2ij,[k]V 2i,[k],∀ (i, j) ∈ T , (5.1c)where Pij,[k] and Qij,[k] are active- and reactive-power flows for each line (i, j) ∈ T with seriesimpedance zij = rij + jxij . Since short lines of distribution systems are typically modelledwithout shunt admittances as they are considered negligible [6, Chap. 5.1].2Note that in this chapter we consider single-phase or single-phase equivalent distribution networks, conse-quently, with respect to the notation established in Chapter 2, Pi = {a},∀i ∈ N and |Pi| = 1, ∀i ∈ N . To easenotation, we remove the label of phase φ ∈ Pi in quantities, equivalently representing (·)i = (·)φi and (·) = (·)φ.Furthermore, since |Pi| = 1, ∀i ∈ N , |N | =∑i∈N |Pi| and |D| =∑i∈D |Pi|.685.1.2 Linearized Branch Flow Model FormulationA linearized version of the branch flow model (5.1a)–(5.1c) has been extensively used in theliterature [53, 65]. The linearized branch flow model is formed under two assumptions: (i) theloss term (r2ij + x2ij)P 2ij,[k]+Q2ij,[k]V 2i,[k]is considered negligible compared to the branch power flow,(ii) under a relatively flat voltage profile, the difference of the squared voltage magnitudesV 2i,[k] − V 2j,[k] can be approximated by 2(Vi,[k] − Vj,[k]), resulting in the expressions of the powerflow and voltage drop in line segments between buses i and j asPij,[k] =∑l:(j,l)∈TPjl,[k] − Pj,[k],∀ (i, j) ∈ T (5.2a)Qij,[k] =∑l:(j,l)∈TQjl,[k] −Qj,[k],∀ (i, j) ∈ T , (5.2b)Vi,[k] − Vj,[k] = rijPij,[k] + xijQij,[k], ∀ (i, j) ∈ T . (5.2c)Assuming that DERs are connected to all buses, i.e., D = N (we will remove this assumptionlater), the linearized branch flow model (5.2) can be compactly rewritten in matrix form as [65]V[k] = V0 + RPgen[K] + XQgen[k] , (5.3)where V0 denotes the operating point bus voltage vector without additional active or reactive-power generation, R ∈ R|N |×|N | and X ∈ R|N |×|N | denote the sensitivity of bus voltages toactive-power generation and bus voltages to reactive-power generation, respectively. A detailedprocedure to calculate the sensitivity matrices R and X is provided in [108].5.1.3 DER ModelModern DERs such as i.e., smart solar PV inverters typically use maximum power point track-ers to maximize active-power generation from available renewable energy of DC-side solar gen-erators. Furthermore, the power electronic converters can generate reactive power, in caseactive-power generation does not fully utilize the available inverter apparent-power capacity.In this chapter, it is assumed that DERs have the functionality of adjustable reactive-powergeneration/absorption. The DER reactive-power generation/absorption Qgeni,[k] is limited by the69(a) (b)Figure 5.1: (a) Single-phase distribution system network model representation with line seg-ments between buses i, j, l ∈ N ; (b) Operating zone of DERs (shaded area).maximum and minimum reactive-power operating limits Qgeni,max and Qgeni,min of the DER for lowactive-power generation for each bus i ∈ D. For high active-power generation P geni,[k], the maxi-mum apparent-power rating of the DERs Sgeni,max limits the available DER reactive power Qgeni,[k].The active-power generation will not be controlled or curtailed to maximize renewable genera-tion. Furthermore, DERs are capable of generating reactive-power without active-power (i.e.,during night). The operating zone of DERs is depicted in Fig. 5.1 (b) by the shaded area.5.2 Lower-Level Local Control Scheme and StabilityAssessmentIn this section, we first state the problem of the lower-level local control scheme, review relevantstate-of-the-art local controllers, propose a new local controller, and compare it with the existingVV controllers.5.2.1 Problem StatementThe objective of the proposed two-level control scheme is to obtain DER reactive-power set-points Qgen[k] such that bus voltages V[k] are regulated within acceptable voltage limits, generallybetween 0.95 and 1.05 p.u. [109]. In this section, the focus is on the design of a local reactive-power control scheme capable of adjusting bus voltages to a specific voltage setpoint Vi,set ∈[0.95, 1.05] p.u. The objective of the proposed local controller is to reduce the steady-statevoltage mismatch between the voltage setpoint and the actual bus voltage to zero and improvethe response time over existing controllers by using a PI control scheme. In addition, the stable70design of the local PI controllers is proposed and a system-wide stability assessment of theproposed control framework is established. Thereafter, optimal voltage setpoints depending onthe evolving operating point are calculated in Section 5.3, with the aim of bringing the systemto an globally optimal operating point.5.2.2 State-of-the-Art Local ControllersThe discrete-time dynamic system that models the voltage control problem in distributionsystems can be derived from the LinDistFlow formulation in (5.3) by [65]V[k+1] = V0 + XQgen[k+1], (5.4a)Qgen[k+1] = fctrl(V[k], Qgen[k]), (5.4b)where f ctrl (·) defines the control function, Qgen[k+1] denotes reactive-power generation at timestep k+ 1, which depend on bus voltages V[k] at time step k, assuming that active-power loadsand generations P load[k+1], Pgen[k+1], and reactive-power loads Qgen[k+1] are approximately steady. Thedynamic system in (5.4) requires full system observability to find appropriate control functionsf ctrl (·) which ensure regulatory-compliant voltage regulation.When system-wide observability is not given, local control functions f ctrli at buses i ∈ Dcan be derived, which require only locally observable data (e.g., voltage measurements) toadjust reactive-power generations. Similar to the system-wide formulation in (5.4), the controlfunctions of local controllers are expressed byVi,[k+1] = Vi,0 + xiQgeni,[k+1], (5.5a)Qgeni,[k+1] = fctrli(Vi,[k], Qgeni,[k]). (5.5b)where Vi,0 is the operating-point voltage at bus i without additional reactive-power generation,the local voltage sensitivity to reactive-power generation is denoted by xi = [Xi,i] and can beextracted from the sensitivity matrix X in (5.2), i.e., the i-th column and i-th row element ofthe sensitivity matrix X.A common local control scheme is the linear VV droop control, where a DER absorbs or71supplies reactive-power proportional to the voltage mismatch between the bus voltage and avoltage setpoint Vi,setQgeni,[k+1] = KP,i(Vi,set − Vi,[k]), (5.6)where KP,i is the proportional gain of the droop controller at bus i. We note that the voltagesetpoint Vi,set can be constant or time-varying, depending on how the local controllers areconfigured. Several studies have shown that the VV droop control [63] may lead to systemvoltage instability, if the droop slope is too steep (i.e., large KP,i) [53, 64]. However, using flatterdroop slopes (i.e., small KP,i) is not desirable as the voltage control may be less effective. As aremedy for a more stable implementation, delaying the droop control action has been proposedin [64]. The local control scheme of the delayed droop controller is given byQgeni,[k+1] =(1− ai)Qgeni,[k] + aiKP,i(Vi,set − Vi,[k]), (5.7)where ai ∈ [0, 1] is a delay coefficient. A limitation of the delayed droop control is the slowerconvergence, which is undesirable in cases of highly intermittent renewable generation. As analternative, the scaled droop-based local control was proposed in [53], which increases conver-gence rate and reduces voltage mismatch over the standard and the delayed droop control. Thecontrol function of the scaled droop-based local control is given byQgeni,[k+1] =(1− ai)Qgeni,[k] + ai((1−KP,i ci)Qgeni,[k] +KP,i(Vi,set − Vi,[k]) ), (5.8)where ci is the scaling factor. The steady-state voltage mismatch, the response time, and thecontroller stability of the VV droop control are improved. However, the steady-state voltagemismatch is not reduced to zero with the scaled droop-based local voltage control approach.5.2.3 Proposed Local PI ControllerIn the proposed control scheme, an integral term is added to the local controller to eliminatethe steady-state voltage error, forming a PI control scheme given byQgeni,[k+1] = KP,i( (Vi,set − Vi,[k])+KI,ik−1∑`=0(Vi,set − Vi,[`]) ), (5.9)72where KP,i and KI,i are the proportional and the integral gains of the local PI controller, and` is the summation index for the integrator. Hereafter, three critical elements in designing thePI controllers to ensure system stability and enable fast convergence are formulated: (i) thestable design criteria of the local PI controller via a discrete-time state-space model, (ii) theglobal stability assessment of the proposed control framework with multiple PI controllers, and(iii) the anti-windup of the local PI controller.Stable Design of Local PI ControllerIn order to achieve a suitable response time of the local PI controllers and to ensure its stability,it is critical to design the controller gains KP,i and KI,i appropriately. In this section, a methodto obtain the controller gains KP,i and KI,i for a desired response time is presented based on thelocal voltage sensitivity. This is achieved by formulating a discrete-time state-space model ofthe local voltage response to DER reactive-power generation. First, the bus-voltage mismatchat bus i and time step k between the actual bus voltage and the setpoint is denoted by∆Vi,[k] = Vi,set − Vi,[k]. (5.10)The summation of the bus-voltage mismatch ∆Vi,[k] over time is denoted by the auxillaryvariable ζi,[k]ζi,[k] =k−1∑`=0∆Vi,[`]. (5.11)With these definitions in place, the discrete-time state-space model which models the voltagemismatch ∆Vi,[k] compared to a voltage setpoint, is formed. A detailed derivation is providedin Appendix C.∆Vi,[k+1]ζi,[k+1]︸ ︷︷ ︸xi,[k+1]=−xiKP,i −xiKP,iKI,i1 1︸ ︷︷ ︸Ai∆Vi,[k]ζi,[k]︸ ︷︷ ︸xi,[k]+1 −10 0︸ ︷︷ ︸BiVi,setVi,0︸ ︷︷ ︸ui,[k], (5.12)where Vi,0 is the bus voltage at bus i without additional DER reactive-power generationfrom (5.5a). In case of a stable PI control scheme, the voltage mismatch ∆Vi,[k] should convergeto 0. From (5.12), the local PI controller response can be adjusted by placing the eigenvalues in73Ai via selecting the proportional and integral gains KP,i and KI,i. The characteristic polynomialof Ai, denoted by pAi(λ) is given bypAi(λ) = |Ai − λI2| = λ2 + λ(xiKP,i − 1)− xiKP,i + xiKP,iKI,i. (5.13)Considering that the characteristic polynomial of Ai is equal to 0 at the desired eigenvaluesλ1, λ2, (i.e., pAi(λ1, λ2) = 0), the local PI controller gains are derived from the desired eigen-values λ1 and λ2:KP,i = −λ1 + λ2 − 1xi, KI,i =λ1 + λ2 − λ1 λ2 − 1λ1 + λ2 − 1 . (5.14)It is proposed to place the eigenvalues λ1 and λ2 so that the PI controller achieves an dampedstep response. Further details on the eigenvalue placement are reported in the case studies inSection 5.4.2.Global Stability Assessment of Proposed Control FrameworkIn case of multiple local controllers in a distribution system, there could be interference betweenthem, potentially leading to voltage instability. Therefore, in [53, 64, 65], stability conditionsbased on the sensitivity matrix X of the linearized branch flow equations are developed for thestate-of-the-art local controllers. However, these stability conditions can not be directly appliedfor the proposed PI control scheme. Hereafter, a stability criteria using the relation in (5.4) isformed, which extends the previous work for the stability assessment of a control frameworkwith multiple local PI controllers. The local discrete-time state-space model (5.12) is extendedfor the entire distribution system, including all buses. First, the vector of voltage mismatchbetween bus voltages V[k] and voltage setpoints Vset is defined as∆V[k] = Vset − V[k]. (5.15)74Then, the vector collecting the element-wise summation of the bus-voltage mismatch ∆V[k] overtime is denoted with the auxiliary variable ζ[k]:ζ[k] =k−1∑`=0∆V[`]. (5.16)With these definitions, the discrete-time state-space model considering multiple local controllersis given by∆V[k+1]ζ[k+1]︸ ︷︷ ︸x[k+1]=−XKP −XKPKII|N | I|N |︸ ︷︷ ︸A∆V[k]ζ[k]︸ ︷︷ ︸x[k]+I|N | I|N |0 0︸ ︷︷ ︸BVsetV0︸ ︷︷ ︸u[k], (5.17)where X denotes the bus voltage to reactive-power sensitivity matrix, the diagonal matrixKP = diag(KP,1, . . . ,KP,|N |) collects the local controller proportional gains KP,i, and similarlythe diagonal matrix KI = diag(KI,1, . . . ,KI,|N |) collects the local controller integral gains KI,i.In the notation (5.17), it is assumed that all gains in KP and KI are non-zero, meaningthat DERs with local controllers are installed at all buses (we made the assumption previouslythat D = N ). However, in practice some of the diagonal entries in KP and KI may be 0 (whenthere is no local controller installed), resulting in columns with all zeros in the sub-matrices−XKP and −XKPKI . As a consequence, eigenvalue pairs of 0 and 1 are introduced in A,and hence the stability evaluation can not be done with (5.17). Therefore, it is proposed toremove the rows and columns of buses without local controllers from the discrete-time statespace model (5.17).Assume that a limited number of DERs is installed at a subset of all buses, i.e., D ⊆ N .Then, the reduced gain matrices which only retain controller gains at the buses with DERsare denoted by K˜P = diag({KP,i}i∈D) and K˜I = diag({KI,i}i∈D). Futhermore, the reducedsensitivity matrix X˜ ∈ R|D|×|D| only consists of the i-th column and i-th row for all buses i ∈ D.Similarly, the reduced version of vectors ∆V[k], Vset, ζ[k] and V0 only retaining entries at busesin D are denoted by ∆V˜[k], V˜set, ζ˜[k] and V˜0, respectively. Finally, the reduced discrete-timestate-space model representing the dynamics of the proposed control system with multiple local75controllers at the subset of buses D is given by∆V˜[k+1]ζ˜[k+1]︸ ︷︷ ︸x˜[k+1]=−X˜K˜P −X˜K˜P K˜II|D| I|D|︸ ︷︷ ︸A˜∆V˜[k]ζ˜[k]︸ ︷︷ ︸x˜[k]+I|D| I|D|0 0︸ ︷︷ ︸B˜V˜setV˜0︸ ︷︷ ︸u˜[k], (5.18)For a discrete-time dynamic system to be stable, the absolute value of the largest eigenvaluesof matrix A˜ must be strictly smaller than one:|λmax(A˜)| < 1. (5.19)Finally, the stability of the control system with multiple local PI controllers can be assessedwith (5.19), by considering the controller gains obtained from (5.14) in A˜.Remark 5.1. Reducing the proposed PI controller (5.9) to a proportional controller by settingKI,i to zero, yields the standard VV droop control in (5.6). Thus, A˜ reduces toA˜ = −X˜K˜P (5.20)Stability for the proportional controller (standard VV droop control) can be guaranteed if|λmax(−X˜K˜P )| < 1. (5.21)The stability condition in (5.21) is equivalent to the formulations in the literature (i.e., eq. (26)in [53], Theorem 1 in [64], Theorem 2 in [65]) for the standard VV droop control.Anti-Windup of IntegratorThe locally available reactive-power is constrained by the operational limits of DERs as specifiedin Section 5.1.3. In cases where locally available reactive-power is not sufficient to adjust busvoltages to their setpoints (e.g., |Vi,[k→inf]−Vi,set| 6= 0), the integrator can be in windup state. Inorder to prevent integrator windup in these cases, a conditional integrator clamping technique isimplemented in the local controller. The integrator is clamped, if (i) the controller is saturatedand (ii) the bus-voltage mismatch and DER reactive-power generation have the same sign. The760 2 4 6 8 10 12 14 16 18 20Time step 0.000.010.020.030.040.05(d) Proposed PI control(c) Scaled droop control(a) VV droop control(b) Delayed VV droop controlFigure 5.2: Comparison of different local controllers: (a) VV droop control, (b) delayed VVdroop control, (c) scaled droop-based local control, and (d) proposed PI control.integrator clamping condition is given by((Vi,set − Vi,[k])Qgeni,[k] > 0) AND (|Qgeni,[k]| 6= Qgeni,max). (5.22)This methodology is effective in clamping the integrator if the DER reactive-power limit Qgeni,maxis reached [110]. The block diagram of the proposed local PI controller is illustrated in Fig. 5.3.5.2.4 Evaluation of Local Control SchemesIn this subsection, the proposed local PI controller is compared to the state-of-the-art localcontrollers. A two-bus system, with one slack bus, one line segment and a DER bus is used.The sample system has a bus-voltage to reactive-power sensitivity xi = 0.05. Four differentlocal controllers are evaluated: (a) the standard VV droop controller in (5.6), (b) the delayedVV droop controller in (5.7), (c) the scaled droop-based local controller in (5.8), and (d) theproposed local PI controller in (5.9). Figure 5.2 illustrates the step responses of the localcontrollers when the steady-state bus voltage before disturbance Vi,0 is 0.951 p.u. and thevoltage setpoint Vi,set is 1 p.u. At time k = 0, the local controllers are activated. The remainingvoltage mismatch of the different local controllers after reaching steady-state is depicted. It canbe observed that the proposed local PI controller achieves a steady-state voltage mismatch of0 p.u. and the fastest response time without voltage oscillations.5.3 Upper-Level Centralized ControllerIn Section 5.2, local PI controllers that are capable of adjusting bus voltages to their setpointswere proposed. However, the stand-alone, autonomous local PI controllers have no knowledge77Advanced SensorsLocal Controller bus iCentralized controllercalculating optimal DER setpoints at intervals +-x++> 0≠AND++010Local Controller bus jDERbus jDERbus iDistributionSystembus isensitivitybus jsensitivity...Figure 5.3: Overview of proposed two-level control system with the centralized controller ex-ecuted at intervals ∆tτ and the local PI controllers at bus i and bus j executed at intervals∆tk.of system-wide operating conditions, hence the operation within the distribution system maymay be compromised due to increased reactive-power flows in line segments, and voltage limitscould be violated due to limited reactive-power resources of DERs. To prevent this, a two-levelcontrol system is proposed in this section. The upper-level centralized controller coordinatesmultiple local PI controllers so that bus voltages can be adjusted toward optimal levels. Theoptimal voltage levels and DER reactive-power setpoints are calculated to (i) comply withvoltage limits of the ANSI standards, (ii) to minimize system-wide network losses, and (iii) toconsider DER operating limits.In the centralized optimization, a nonlinear optimal DER dispatch problem is solved tofind optimal voltage and reactive-power setpoints at locations with DERs. The optimal DERsetpoints are communicated to the local controllers, which maintain the voltages at setpointslocally through the proposed local control scheme presented in Section 5.2. To compute optimalDER setpoints, it is assumed that the voltage at the substation V0,[τ−1], as well as system-wideactive-power loads P load[τ−1], active-power generations Pgen[τ−1] and reactive-power loads Qload[τ−1] areavailable at the centralized controller from advanced sensors such as i.e., smart meters at timesteps τ = 0, 1 . . . and interval of ∆tτ = tτ+1−tτ . The interval of globally available measurements∆tτ is typically longer than the interval of locally available measurements ∆tk = tk+1 − tk sothat the optimization interval at the centralized location is executed at longer intervals (i.e., 60-78second updates centrally vs. sub-second updates locally). The optimal DER dispatch problemis set up to minimize network losses with the objective function formulated as3minimize∑l:(i,j)∈TrijP2ij,[τ ] +Q2ij,[τ ]V2i,[τ ]. (5.23)A feasible solution of (5.23) is subject to the equality constraints from the branch flow equa-tions (5.1a)–(5.1c) at time step τ . In addition, since the primary purpose is to maintain voltageswithin ANSI-levels, decision variables of bus voltages V i,[τ ] are subject to the voltage boundaryconstraintsVmin ≤ V i,[τ ] ≤ Vmax ∀i ∈ N , (5.24)with the lower and upper voltage bounds Vmin, Vmax, respectively. The voltage limits Vmin andVmax were adjusted to 0.952 p.u. and 1.048 p.u., respectively, to maintain voltages withina small margin from the regulatory limits. The active- and reactive-power loads P loadi andQloadi , and the active-power generation of DERs Pgeni are not intended to be controlled. Hence,they are fixed with equality constraints in the optimal DER dispatch program to represent thecurrent operating point at time step τ assuming that they do not change drastically betweentime step τ − 1 and τPloadi,[τ ] = Ploadi,[τ−1], Qloadi,[τ ] = Qloadi,[τ−1], Pgeni,[τ ] = Pgeni,[τ−1] ∀ i ∈ N , (5.25)where Ploadi,[τ ] , Qloadi,[τ ] , and Pgeni,[τ ] denote the decision variables of active- and reactive-power loads,and active-power generations, respectively. Finally, to determine optimal reactive-power set-points at DER locations at time step τ to achieve minimal network losses, DER reactive-powergeneration decision variables Qgeni,[τ ] are subject to the operational limits of the DERs as0.95 Qgeni,min[τ ] ≤ Qgeni,[τ ] ≤ 0.95 Qgeni,max[τ ] ∀ i ∈ N , (5.26)where Qgeni,min[τ ] and Qgeni,max[τ ] are the lower and upper time-varying bounds of the availablereactive-power generation of DERs at time step τ , as stated in Section 5.1.3. The reactive-3Decision variables are marked with (·).79power constraints in the centralized optimal DER dispatch calculation are set to 95% of theavailable reactive-power capacities of DERs, so that the local controllers have some headroomto adjust reactive-power generation locally, similar to [72].The nonlinear optimal DER dispatch problem can be solved via a convex relaxation [43]. Incase of purely radial distribution systems, an optimal solution can be recovered without loosingoptimality [111]. To formulate the convexified optimal DER dispatach problem, bus-voltagemagnitudes are replaced by their squared value. An additional variable Iij denoting the squareof line current magnitude from bus i to bus j is introduced:Iij =P 2ij +Q2ijV 2i∀(i, j) ∈ T . (5.27)With this definition, the objective function (5.23) and the branch flow equations (5.1a)–(5.1c) are represented by linear functions. Finally, the optimal DER dispatch problem is solvedvia a SOCP optimization, by relaxing the equality constraint for Iij with an inequality constraintas [39]Iij ≥P 2ij +Q2ijV 2i∀(i, j) ∈ T . (5.28)This relaxed optimal DER dispatch problem can effectively be solved using cvx, a Matlabtoolbox for convex optimization [112], from which we can extact optimal DER reactive-powersetpoints Qgen?i,[τ ] and bus voltage setpoints V?i,[τ ].In contrast to existing literature, in the proposed method we aim to dispatch the optimalvoltage setpoint V ?i,[τ ] and the optimal reactive-power setpoint Qgen?i,[τ ] at the time step τ to thelocal PI controller at bus i. This concept combines the advantages of the centralized and thelocal control schemes. The main advantages of this method are as follows:• Updating the reactive-power setpoint enables a DER to respond to large operating-pointchanges similar to a feed-forward centralized controller. As a result, the distributionsystem is brought to an optimal operating point nearly instantly when a new optimalDER setpoint becomes available, so that minimal loss and maintaining optimal voltagesis achieved after every update of the centralized optimization.• Dispatching the optimal voltage setpoint to the local PI controllers enables DERs to80maintain voltage levels as local renewable generation varies. As the local PI controllersare in the proximity of the DERs, control actions occur with very low latency, which iscritical in case of highly intermittent renewable generation. Furthermore, even when thecommunication between the centralized controller is interrupted, the local PI controllersare capable of maintaining local voltage levels within the acceptable (albeit possibly sub-optimal) limits. In the proposed control framework, it is essential that the local controllersmaintain the local voltages at reference setpoints in order to prevent voltage violations.Finally, the complete two-level control system including the centralized and the local controllersfor bus i is expressed byQgeni,[k+1] = Qgen?i,[τ ] +KP,i((V ?i,[τ ] − Vi,[k])+KI,ik−1∑`=τ(V ?i,[τ ] − Vi,[`])). (5.29)The voltage setpoint Vi,set in (5.9) is replaced with the time-varying voltage setpoint fromthe centralized optimization V ?i,[τ ]. The reactive-power setpoint Qgen?i,[τ ] is added to the localPI controller for step changes of local DER reactive-power generation. The block diagram inFig. 5.3 illustrates an overview of the proposed two-level control system.Remark 5.2. The equation (5.29) is a generalized formulation for multiple control systems:(i) a purely centralized controller if the coefficients KI,i and KP,i are equal to zero, (ii) purelylocal controllers if Qgen?i,[τ ] is set to zero and V?i,[τ ] is constant (i.e., 1 p.u.), (iii) the two-level controlsystem using the VV droop local controller if KI,i is equal to zero, and (iv) the proposed two-level control with the local PI controllers system in its full form.5.4 Case StudiesIn order to evaluate the performance of the proposed control system, simulations of the IEEE123-bus test system [105] are performed using OpenDSS in a 24-hour time-series simula-tion [113]. For the case study with balanced loads in Section 5.4.3, the existing test feederis modified as follows to facilitate the case studies: (i) loads are balanced so that a single-phase simulation could be performed, (ii) a total of 10 DER units are installed at busesD = {21, 23, 30, 44, 57, 67, 72, 101, 108, 135}, and (iii) the voltage regulators are set to the rated81134 5 627 8121114102019222118353740135252825048 47495051444546424341363839666564636260160 6757585954535255 5613341516179695949315292 908891 89 878680818283847885727374757779300111 110108109 107112 113 11410510610110210310445010097996869707119715115061 610 92423251195451149350769876273332312629VSub= 4.16 kV 1.0 p.u.TR4.16 kV/4.16 kVDER 21150kVACommercial loadResidential loadDER 23180kVADER 30120kVADER 67150kVADER 44210kVADER 57210kVADER 7290 kVADER 101240kVADER 135300kVADER 108270kVA30DER 2080 kVAPhase aDER 5990 kVAPhase bDER 85120 kVAPhase cDER 83120 kVAPhase bFigure 5.4: IEEE 123-bus test system with 10 three-phase DERs (enclosed with solid lines)used for the case studies. Additional 4 single-phase DERs are added for the unbalanced casestudies (enclosed with dashed lines).ratio of 1:1 since the DERs are used to control the voltage. The total capacity of DERs isadjusted to represent a 100% DER penetration case. The IEEE 123-bus test system used forthe case studies is illustrated in Fig. 5.4, with DER capacities indicated. Commercial and res-idential load profiles based on historical data are assigned to the loads [114]. Solar generationprofiles with highly intermittent generation based on measured power data with 1 second reso-lution are applied to all the DERs [106]. The solar generation and load profiles for the 24-hoursimulation period are illustrated in Fig. 5.5, where the profiles are normalized based on themaximum capacity of the specific load/DER.5.4.1 Different Control MethodsIn order to demonstrate the effectiveness of the proposed control system, multiple controlmethods are compared as shown in Table 5.1. Method 0 represents the base case as reference820 6 12 18 24Time (hours)normalized power00.20.40.60.81Solar generationCommercial load Residential loadFigure 5.5: Normalized load and solar generation profiles for the 24-hour simulation period.Table 5.1: Overview of control methods for case studiesMethod Control system Qgen?i,[τ ]Vi,set KP,i ∀ i ∈ D KI,i ∀ i ∈ D0 no Qgen control - - - -1 autonomous VV droop - 1.0 p.u. {4.38, 5.25, 3.50, 6.13, 4.38, 6.13, 2.63, 7.00, 8.75, 7.88} -2 autonomous local PI - 1.0 p.u. {3.12, 2.82, 1.96, 2.34, 2.15, 3.19, 1.99, 1.87, 3.58, 1.62} {1.04}3 centralized control Qgen?i,[τ ]- - -4 two-level VV droop Qgen?i,[τ ]V ?i,[τ ]{4.38, 5.25, 3.50, 6.13, 4.38, 6.13, 2.63, 7.00, 8.75, 7.88} -5 two-level PI control Qgen?i,[τ ]V ?i,[τ ]{3.12, 2.82, 1.96, 2.34, 2.15, 3.19, 1.99, 1.87, 3.58, 1.62} {1.04}without any reactive-power control implemented. The performance of the local controllers instand-alone, autonomous operation is evaluated for the standard VV droop controller (Method1) and for the proposed local PI controller (Method 2). For the autonomous controllers, busvoltage setpoints Vi,set are fixed at 1.0 p.u. Then, a case with only the upper-level centralizedcontroller is simulated for comparison. Optimal DER reactive-power generation setpoints Qgen?i,[τ ]are directly dispatched to the DERs at intervals ∆tτ without implementing the local controllers(Method 3). Finally, the two-level control systems using the standard VV droop controller(Method 4) and the proposed local PI controller (Method 5) are compared.5.4.2 Configuration of ControllersThe controller parameters for each method are reported in Table 5.1. The centralized opti-mization interval ∆tτ is set to 60 seconds to account for limited availability of system-widemeasurement data at a centralized location. The update interval of local controllers ∆tk is setto 0.1 seconds.For the VV droop controllers, a linear droop function without dead-band is used between83-1 -0.5 0 0.5 1Real Part-1-0.500.51Imaginary PartFigure 5.6: Locations of eigenvalues of A˜ within unity circle.the maximum and the minimum reactive-power available from DERs with KP,i =Qgeni,max0.05 , whereDER reactive-power outputs are saturated at Qgeni,max for bus voltages < 0.95 p.u., and at Qgeni,minfor bus voltages > 1.05 p.u., respectively. This parameter selection intends to maximize the useof DER reactive-power resources.For the local PI controllers, a damped dynamic behavior is configured. To calculate the twoeigenvalues of the local controllers (λ1, λ2), the damping ratios ξ1 = ξ2 = 1 and the naturalfrequencies ωn,1 = 46 rad/s and ωn,2 = 2.2 rad/s were used. The eigenvalues λ are calculatedby [115]λ = exp−ξωn∆tk , (5.30)resulting in the eigenvalues of λ1 = 0.01 and λ2 = 0.8. The local PI controller gains KP,iand KI,i are obtained from (5.14) and reported in Table 5.1. The parameter choice resultsin a damped step response with rise time of 1.9 seconds. To ensure the global stability, theexpression (5.19) is used, resulting in |λmax(A˜)| = 0.99. Furthermore, the eigenvalues areplotted in Fig. 5.6, where it is observed that all eigenvalues are within the unity circle and asa consequence the controller configuration leads to stable operation. For the autonomous andthe two-level control systems (Method 2 and 5), identical parameters KP,i and KI,i are used topreserve the control system dynamics.845.4.3 Simulation Results with Balanced Loading ConditionsThis section presents the simulation results of the case study with balanced loading conditions.In Figs. 5.7 and 5.8, bus voltages Vi, active-power generations Pgeni , and reactive-power gen-erations Qgeni with their minimum and maximum reactive-power limits Qgeni,min and Qgeni,max aredepicted for three DER buses in the test system, i.e., (i) at bus 21 (blue), (ii) at bus 57 (red), and(iii) at bus 108 (yellow) for all control methods. Figure 5.7 (a) illustrates bus voltages Vi withoutreactive-power control of DERs (Method 0), showing a baseline with significant under-voltagesin heavy-load/light-generation periods and over-voltages in heavy-generation/light-load peri-ods. Figure 5.7 (b) shows the DER active-power generation P geni which is identical for allcontrol methods.Autonomous Local ControllersFigure 5.7 (c) and (d) show the simulation results for Method 1. The VV droop control with avoltage setpoint of 1.0 p.u. significantly improves bus voltages Vi in Fig. 5.7 (c), mitigating allvoltage violations at the candidate buses. It can be observed that there is a voltage mismatchbetween the voltage setpoint and the actual bus voltages. Figure 5.7 (d) illustrates the reactive-power generation of DERs Qgeni which is proportional to the voltage mismatch between thevoltage setpoint of 1.0 p.u and the actual bus voltages.Figure 5.7 (e) and (f) show the simulation results for Method 2. The proposed local PIcontroller with a voltage setpoint of 1.0 p.u. maintains the voltages at the 3 candidate busesat the reference level. It is observed that available reactive-power resources are effectively usedto adjust bus voltages to reference levels.Generally, in the autonomous operation of local controllers, voltage violations can be effec-tively mitigated even during periods with highly intermittent generation of renewables. How-ever, network losses are increased as reported in Table 5.2 and also observed in [67].Two-Level Control SystemsFigure 5.8 (a) and (b) show the simulation results for Method 3 with optimal DER reactive-power generation setpoints Qgen?i directly dispatched to the DERs without local controllers.Figure 5.8 (a) indicates that the voltage control works well during times without intermittent850.9511.051.10501001502000.90.9511.051.1-200-10001002000.90.9511.051.1-200-100010020000:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 00:00(a) Method 0(b) Method 0 - 5(c) Method 1(d) Method 1(e) Method 2(f) Method 2Figure 5.7: (a) Bus voltages Vi without reactive-power control (Method 0); (b) Active-powergenerations P geni for Methods 0-5; (c) Bus voltages Vi and (d) Reactive-power generations Qgenifor Method 1 and their minimum and maximum reactive-power limits (dashed lines) Qgeni,minand Qgeni,max; (e) Bus voltages Vi and (f) Reactive-power generations Qgeni for Method 2 andtheir minimum and maximum reactive-power limits (dashed lines) Qgeni,min and Qgeni,max for bus 21(blue), bus 57 (red), and bus 108 (yellow).DER active-power generation (i.e., during the evening peak load hours). However, at timeswith highly intermittent generation of DERs around noon, the relatively slow reactive-powersetpoint updates at intervals of ∆tτ = 60 seconds are insufficient to effectively regulate busvoltages within limits.Figure 5.8 (c) and (d) show the simulation results for Method 4, the two-level control system8600:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 00:000.90.9511.051.1-200-10001002000.90.9511.051.1-200-10001002000.90.9511.051.1-200-1000100200(a) Method 3(b) Method 3(c) Method 4(d) Method 4(e) Method 5(f) Method 5Figure 5.8: (a) Bus voltages Vi and (b) Reactive-power generations Qgeni for Method 3 and theirminimum and maximum reactive-power limits (dashed lines) Qgeni,min and Qgeni,max; (c) Bus voltagesVi and (d) Reactive-power generations Qgeni for Method 4 and their minimum and maximumreactive-power limits (dashed lines) Qgeni,min and Qgeni,max; (e) Bus voltages Vi and (f) Reactive-power generations Qgeni for Method 5 and their minimum and maximum reactive-power limits(dashed lines) Qgeni,min and Qgeni,max for bus 21 (blue), bus 57 (red), and bus 108 (yellow).with local VV droop controllers. The bus voltage violations and the losses are significantlyreduced. However due to the nature of the proportional controller, a small voltage mismatchbetween the optimal voltage setpoint V ?i,[τ ] and the actual bus voltages Vi remains, which canstill cause some voltage violations.Figure 5.8 (e) and (f) show the simulation results for Method 5, the two-level control system8711.011.021.031.0408:00 08:02 08:04 08:06 08:08 08:10 08:12-50510(a)(b)SetpointFigure 5.9: Comparison of Method 3 (Centralized control, blue), Method 4 (Two-level controlwith VV droop controllers, yellow) and Method 5 (Two-level control with local PI controllers,red). (a) Bus voltages Vi and (b) Reactive-power generation Qgeni for low intermittency periodbetween 08:00 AM and 08:12 AM at bus 21.with the proposed local PI controllers. The proposed control system both minimizes networklosses and mitigates all voltage violations. Bus voltages are accurately adjusted to setpointsV ?i,[τ ] compared to the previous methods, even during times with highly intermittent active-power generations of DERs.In Figs. 5.9 and 5.10, the effect of intermittent generation on the performance of Method 3(Centralized control, blue), Method 4 (Two-level control with VV droop controllers, yellow) andMethod 5 (Two-level control with local PI controllers, red) at bus 21 is investigated. During thetime with low intermittency from 8:00 AM until 8:12 AM, all control methods adjust voltagelevels effectively as in Fig. 5.9 (a). The proposed local PI controller tracks the voltage setpoints(black in the background) very accurately, while the other controllers show deviations. TheDER reactive-power generation is depicted in Fig. 5.9 (b). The DER reactive-power generationof the centralized method is relatively steady, whereas the local controllers adjust DER reactive-power generation based on small changes of DER active-power generation.The difference between the actual bus voltage and the setpoint is more significant in caseof intermittent active-power generation between 11:18 AM and 11:30 AM in Fig. 5.10 (a). Thecentralized controller (Method 3) no longer mitigates voltage violations when DER reactive-881.021.051.0811:18 11:20 11:22 11:24 11:26 11:28 11:30-40-20020(a)(b)SetpointFigure 5.10: Comparison of Method 3 (Centralized control, blue), Method 4 (Two-level controlwith VV droop controllers, yellow) and Method 5 (Two-level control with local PI controllers,red). (a) Bus voltages Vi and (b) Reactive-power Qgeni for high intermittency period between11:18 AM and 11:30 AM at bus 21.power setpoints are updated at the interval ∆tτ = 60 seconds. The two-level control systemwith local VV droop controllers (Method 4) also leads to deviations from the setpoints. Thesedeviations arise from the remaining voltage mismatch of the proportional controllers. Using thetwo-level control system with the proposed local PI controllers (Method 5), DER reactive-powergeneration Qgeni is continuously adjusted to maintain the bus voltage at the setpoint as seen inFig. 5.10 (b), mitigating all voltage violations.Table 5.2 summarizes the performance for each control method. The total network energyloss Eloss for the 24-hour simulation period is reported, including the percent change ∆Elosscompared to Method 0. The maximum and minimum bus voltages Vmax and Vmin over thesimulation period are evaluated considering all buses in the system. Finally %tout indicates thepercentage of the total time during which voltage violations outside of 0.95 and 1.05 p.u. occur.The two-level control system (Method 5) effectively mitigates all voltage violations and reducesthe network losses by 11.3%.89Table 5.2: Summary of simulation results for balanced operationMethod Eloss (kWh) ∆Eloss (%) Vmax (p.u.) Vmin (p.u.) %tout0 418.54 - 1.0922 0.9233 15.091 531.94 +27.1% 1.023 0.972 02 652.42 +55.9% 1.011 0.986 03 368.42 -12.0% 1.097 0.951 1.314 369.85 -11.6% 1.060 0.952 0.225 371.22 -11.3% 1.049 0.951 0.00Table 5.3: Summary of simulation results with OLTC transformerMethod Tap operations Vmax (p.u.) Vmin (p.u.) %tout0 95 1.077 0.912 2.2972 0 1.009 0.987 0Effect on the On-load Tap Changing TransformerIn this case study, the effect of the local PI controllers on the OLTC transformer at the sub-station is evaluated. The same scenario with variable renewable generation as in the previouscase study is used, in addition the OLTC transformer is activated. The OLTC monitors thevoltage at bus 95 and adjusts the substation transformer ratio, when the monitored bus voltageis below 0.97 p.u., with a delay of 60 seconds.In Fig. 5.11 (a), the voltages at buses 21, 57, and 108 are illustrated when DER reactivepower is not controlled (Method 0) during the 24-hour simulation period. In Fig. 5.11 (b),the tap position of the OLTC is shown. It is observed, that the intermittent renewable gen-eration around 12:00 PM increases the tap operations. Over the simulation duration, withoutcontrolling DER reactive power, the total number of tap operations is 95.In contrast, when the local PI controllers are used to adjust bus voltages to a fixed setpointof 1 p.u. (Method 3 of the previous case study), all tap operations are mitigated as bus voltagesare effectively adjusted to the reference setpoint.The simulation results of the case study with the OLTC are summarized in Table 5.3. Withthe proposed local PI controllers, all tap operations and voltage violations are mitigated. Thesesimulation results provide evidence that the proposed control framework does not have adverseeffects on the existing voltage regulation mechanism of the OLTC.900.9511.0500:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 00:000.90.9511.051.1(a)(b)Transformer TapFigure 5.11: Evaluation of tap position of the substation transformer in the simulation scenariowith intermittent renewable generation for the 24-hour period without DER reactive-powercontrol (Method 0). (a) Bus voltages Vi at buses 21, 57, and 108; and (b) Tap position of thesubstation transformer.5.4.4 Simulation Results with Unbalanced Loading ConditionsMotivated by the opportunity that voltage unbalance can be improved by local reactive-powergeneration [116], in this section, the proposed local PI controllers are utilized to control three-phase DER units in an unbalanced scenario. The IEEE 123-bus test system with unbalancedloads from [105] is simulated. In addition to the 10 three-phase DERs for the balanced study,4 single-phase DERs are connected to phase a of bus 20, phase b of bus 59, phase c of bus 85,and phase b of bus 83. The test system is illustrated in Fig. 5.4.The DERs are connected in wye configuration, hence their reactive-power generation intoeach phase can be adjusted individually. For each phase, the voltage setpoint Vi,set is adjustedto 1 p.u.In Fig. 5.12, the phase voltages of bus 83 are investigated for the 24-hour simulation period.The bus 83 is selected, as it represents the worst case with the largest voltage unbalance fromthe nominal value compared to all other buses. In Fig. 5.12 phase voltages V ai , Vbi , Vci areplotted for the 24-hour simulation period. In the upper pane (a) there is no DER reactive-powercontrol, and in the lower pane (b) DERs are controlled with the proposed PI control framework.910.9511.0500:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 00:00Time (hh:mm)0.9511.05(a) No control(b) Local PI controlFigure 5.12: Phase a (blue), phase b (red), and phase c voltages at bus 83, the bus with thehighest unbalance in the system over the 24-hour simulation period. (a) No DER reactive-powercontrol; (b) DER reactive-power control with local PI controllers.It is observed, that the phase voltages are generally in a narrower band around the nominalvoltage, when DERs are controlled with the local PI controllers.Furthermore, in Fig. 5.13, the phase voltages at all buses within the system are plotted atthe time 12:00 PM, when renewable generation is highly variable. Figure 5.13 illustrates phasevoltages V ai , Vbi , Vci for all buses without DER reactive-power control in the upper pane (a),and the phase voltages V ai , Vbi , Vci for all buses when DERs are controlled with the proposedlocal PI control framework. Similar to the observations at bus 83, all bus voltages are generallyin a narrower band around the nominal voltage of 1 p.u.Table 5.4 summarizes the simulation results with unbalanced loading conditions betweenthe method without DER reactive-power control, and DER reactive-power control with theproposed PI controllers. The local PI control improves the maximum and minimum voltagelevels Vmax and Vmin. It maintains all voltages within the prescribed voltage limits of 0.95 and1.05 p.u. Additionally the system average voltage unbalance index (SAVUI) is improved from1.28% to 0.73%.The SAVUI is a measure of unbalance considering the maximum voltage deviation of thethree-phases from the average voltage of the three-phases, averaged over the simulation period T9211.05(a) No Control0 20 40 60 80 100 1200.9511.050.95(b) Local PI controlFigure 5.13: Voltage profile at all buses for phase a (blue), phase b (red), and phase c (yellow).(a) No DER reactive-power control; (b) DER reactive-power control with local PI controllers.Table 5.4: Summary of simulation results for unbalanced operationMethod Vmax (p.u.) Vmin (p.u.) %tout SAVUI (%)No Control 1.049 0.938 1.093 1.28%Local PI Control 1.028 0.970 0 0.73%and all the buses in the system N . It is expressed as a percentage and calculated as follows [117]:SAVUI =1N(1TN∑i=1T∑t=1max(V ai , Vbi , Vci )− Vi,avgVi,avg)× 100 [%] (5.31)where Vi,avg = (Vai + Vbi + Vci )/3 is the voltage average of the three phase voltages.5.4.5 DiscussionThe proposed two-level control system addresses the voltage control challenge integrating highlyintermittent DER generation into distribution systems. (i) It combines the advantages of a cen-tralized control system which achieves a system-wide optimization and the local control systemwhich locally adjusts voltages when disturbances occur. (ii) It is capable of mitigating effectsof highly intermittent renewable generation through fast local control to manage bus voltages.(iii) The two-level control architecture can handle potential failures between distribution systemtelemetry, the centralized controller and local controllers. In case of communication failures,93the local controllers can fall back into autonomous local control, which ensures voltage regu-lation. (iv) In case of long communication latency, the centralized optimization may providesuboptimal DER setpoints. In this case, a centralized control system with DER reactive-powersetpoints directly dispatched to the DERs (i.e., Method 3) may not be effective to mitigatebus voltage violations. On the other hand, the proposed two-level control system with localPI controllers ensure voltage regulation within acceptable limits, due to the use of voltage set-points. (v) The proposed local PI controllers also operate in conjunction with existing OLTCvoltage regulators. In the case studies, all tap operations are mitigated, while bus voltagesare adjusted within the acceptable limits. (vi) In an unbalanced scenario, the local PI controlscheme reduced voltage unbalance and deviations of voltages from reference levels.5.5 SummaryIn this chapter, a two-level control system for voltage regulation through local reactive-powercontrol of DERs toward optimal operating points is proposed. The main advantage of the con-trol system is that the optimal voltage setpoints at DER buses can be accurately tracked usinga local PI control scheme. The proposed control system is capable of mitigating effects of highlyintermittent renewable generation through fast local control, achieves voltage stability, and en-sures voltage regulation within acceptable limits. Moreover, a method to design controller gainsfor stable operation of the local controllers via a discrete-time state-space model is proposed.In addition, the assessment of the global stability of the proposed control is established, whichgeneralizes the previous methods for VV droop control. The upper-level centralized controllercoordinates the local controllers to bring the distribution system toward an optimal operatingpoint by solving a nonlinear optimal DER dispatch problem. Numerical simulations of the IEEE123-bus test system using real world input data for load and solar generation profiles demon-strate that the proposed two-level control system effectively mitigates bus voltage violationsand reduces network losses significantly in the test cases with highly intermittent renewablegeneration by leveraging real-time measurement locally. In addition, as demonstrated, the lo-cal PI controllers reduce system voltage unbalance and mitigate increased OLTC operations incase of highly intermittent renewable generation, which highlights the benefits of using real-timemeasurements to optimally dispatch DERs.94Chapter 6Gradient-descent Method for OptimalReactive-Power DispatchA desirable operational objective in distribution systems is to regulate bus voltages while mini-mizing network losses through DER reactive-power generation. However, our literature reviewsuggests that existing measurement-based methods do not achieve effective loss minimizationwithout network model information. To this end, in this chapter,1 we focus on the problemof optimally dispatching DER reactive-power outputs and propose a fully measurement-basedapproach that does not rely on any prior knowledge of the distribution-network topology orparameters. Our proposed method makes several contributions over the state of the art. First,we leverage synchronized measurements of nodal power injections obtained from D-PMUs toestimate linear sensitivities of the network loss and bus-voltage magnitudes with respect toreactive-power injections at DER buses. By repeatedly estimating the loss and voltage sen-sitivities, they readily adapt to operating-point and network-topology changes. Subsequently,we directly embed the estimated sensitivities into gradient-descent updates to enable DERreactive-power outputs to converge to setpoints that minimize an objective function consistingof network loss and DER costs. This obviates the need to formulate nonlinear functions ofloss and voltage magnitudes in the optimization problem, as is typically the case in model-based methods (see, e.g., [42]). Furthermore, unlike the measurement- and sensitivity-basedvoltage control in [87, 88] and DER dispatch [89, 103], we achieve the optimal operating point1A version of this chapter has been published in: S. Nowak, Y. C. Chen, and L. Wang, “A measurement-basedgradient-descent method to optimally dispatch DER reactive power,” in Proc. IEEE Photovoltaics SpecialistConf., Jun. 2020, pp. 1-5. Best Paper Award.95with an iterative gradient-descent approach. As a direct consequence, computational burdenincurred in updating DER setpoints is reduced. With respect to existing feedback optimizationmethods [54–60], our proposed method relies only on measurements, and numerical simulationsdemonstrate that DER reactive-power outputs converge to setpoints that correspond to theoptimal operating point. Finally, compared to [118], the proposed method requires smallerinjection perturbations for sensitivity estimation.Via numerical simulations involving the IEEE 33-bus test system [1], we demonstrate thatthe proposed method effectively achieves the optimal operating point and yields accurate DERreactive-power setpoints compared to a model-based benchmark solution, and it quickly adaptsto operating-point and network-topology changes. Furthermore, by embedding DER costs inthe objective function and considering bus voltage constraints, the proposed measurement-basedapproach effectively solves a general formulation of the volt-var optimization problem withoutrelying on any knowledge of system topology or parameters.The remainder of this chapter is organized as follows. In Section 6.1, we formulate themeasurement-based approach to dispatch DER reactive power via gradient-descent updates.Section 6.2 outlines the proposed measurement-based DER reactive-power dispatch method.In Section 6.3, we offer numerical case studies to demonstrate the effectiveness of the pro-posed measurement-based optimal DER dispatch framework. Finally, we provide a summaryin Section 6.4.6.1 Problem FormulationIn this section, we establish notation, describe the network model, and motivate the need for ameasurement-based approach to dispatch DERs.Based on the definitions in 2.2, compactly express the network loss P loss[k] at time step k asa function of DER reactive-power injections Qgen[k] :2P loss[k] = h[k](Qgen[k] ), (6.1)2Note that in this chapter we consider single-phase or single-phase equivalent distribution networks, conse-quently, with respect to the notation established in Chapter 2, Pi = {a},∀i ∈ N and |Pi| = 1, ∀i ∈ N . To easenotation, we remove the label of phase φ ∈ Pi in quantities, equivalently representing (·)i = (·)φi and (·) = (·)φ.Furthermore, since |Pi| = 1, ∀i ∈ N , |N | =∑i∈N |Pi| and |D| =∑i∈D |Pi|.96where h[k] : R|D| 7→ R. In (6.1), the dependence on active- and reactive-power loads, DERactive-power injections, as well as network topology and associated parameters (such as lineimpedances), at time step k, are implicitly considered in h[k](·). Given the notation establishedabove, and assuming all bus active-power injections Pi,[k] are measured, the total network lossat time step k can be computed byP loss[k] =∑i∈NPi,[k]. (6.2)Typically, DER reactive-power setpoints that optimize operation in the distribution system areobtained by solving a problem that relies on an offline network model with accurate topologyand line parameters. This approach formulates f(·) : R|D| 7→ R as an objective function subjectto network power-flow equations, bus-voltage limits, and DER operational constraints in thefollowing optimization problem:minimizeQgen[k]∈Qf(Qgen[k] ), (6.3a)subject to V[k] = σ[k](Qgen[k] ), (6.3b)Vmin ≤ V[k] ≤ Vmax, (6.3c)where σ[k] : R|D| 7→ R|D| maps DER reactive-power outputs to bus-voltage magnitudes throughthe power flow constraint (the dependence on bus active-power injections is implicitly consideredin σ[k]), Vmin and Vmax respectively denote minimum and maximum bus-voltage limits, and Qdenotes the feasible region of DER reactive-power outputs.The nonlinear optimization problem (6.3) may lead to significant computational burden.Instead of solving (6.3) directly, we leverage a gradient-descent method, in which optimal DERsetpoints are updated iteratively as follows:Qgen[k+1]=projQ(Qgen[k] −α[k](∇f(Qgen[k] )+ΣT[k]∇g(V[k]))), (6.4)where projQ(·) denotes the projection onto Q, Σ[k] is the first-order sensitivity matrix ofσ[k](Qgen[k] ) with respect to the decision variable Qgen[k] , g : R|D| 7→ R|D| is a convex and con-97tinuously differentiable penalty function to ensure constraint (6.3c) is satisfied, and α[k] is thestep size. We can evaluate Σ[k] by linearizing a suitable power-flow model of the system or byestimating it using online measurements (as in our proposed framework).The gradient-based approach in (6.4) reduces the computational burden in obtaining theoptimal DER reactive-power setpoints as compared to solving the offline optimization prob-lem (6.3). In contrast to existing work on feedback optimization methods that rely on model-based gradient updates (see, e.g., [55]), we embed estimated sensitivities of network loss andbus-voltage magnitudes with respect to DER reactive-power injections in updating the DERreactive-power setpoints. This enables the system to reach the optimal operating point withoutany prior knowledge of the system model.6.2 Measurement-based DER Dispatch via Gradient UpdatesIn this section, we estimate sensitivities of the network loss and voltage magnitudes with respectto DER reactive-power outputs. We then incorporate the estimated sensitivities in a gradient-descent method to obtain optimal DER reactive-power setpoints while respecting DER reactive-power output limits and bus voltage constraints.6.2.1 Estimation of Sensitivity ModelsSuppose h[k](·) and σ[k](·) are continuously differentiable with respect to Qgen[k] . We approximatenetwork loss and bus-voltage magnitudes at time step k+1, P loss[k+1] and V[k+1], with the followingfirst-order Taylor series expansions:P loss[k+1] ≈ h[k](Qgen[k] ) +∇hT[k](Qgen[k+1] −Qgen[k] ), (6.5)V[k+1] ≈ V[k] +∇σT[k](Qgen[k+1] −Qgen[k] ), (6.6)where ∇h[k] ∈ R|D| and ∇σ[k] ∈ R|D|×|D| denotes the gradient of h[k](·) and σ[k](·) with respectto DER reactive-power injections evaluated at the operating point Qgen[k] , respectively. Thelinearized models in (6.5) and (6.6) can be rewritten asP loss[k+1] ≈ P loss,c[k] + (P loss,l[k] )TQgen[k+1], (6.7)98V[k+1] ≈ V c[k] + (V l[k])TQgen[k+1], (6.8)where P loss,l[k] = ∇h[k] and P loss,c[k] = h[k](Qgen[k] ) − (P loss,l[k] )TQgen[k] , and similarly V l[k] = ∇σ[k] andV c[k] = σ[k](Qgen[k] ) − (V l[k])TQgen[k] are, respectively, the linear and constant terms of the linearpower loss and bus voltage models linearized at time step k. We note that the linear modelin (6.8) is similar to the LinDistFlow model widely used in the literature (its derivation inmatrix form is provided in [53]). Furthermore, we equivalently express (6.7) and (6.8) asP loss[k+1] ≈[1 (Qgen[k+1])T]P loss,cl[k] , (6.9)V[k+1] ≈[1 (Qgen[k+1])T]V cl[k], (6.10)where P loss,cl[k] = [Ploss,c[k] , (Ploss,l[k] )T]T, and V cl[k] = [Vc[k], (Vl[k])T]T combine constant and linear termsin (6.7) and (6.8) in matrix form. In order to remove the reliance on an accurate network model,we estimate the entries of P loss,cl[k] and Vcl[k] using only online measurements. Suppose that Mmeasurements of bus net active-power injections, P̂[k−M+1], . . . , P̂[k], are available so that thenetwork loss at time steps k −M + 1, . . . , k, P̂ loss[k−M+1], . . . , P̂ loss[k] , can be computed using (6.2).Further suppose measurements of DER reactive-power injections, Q̂gen[k−M+1], . . . , Q̂gen[k] , and DERbus-voltage magnitudes, V̂[k−M+1], . . . , V̂[k], are available. Assuming that active- and reactive-power loads as well as active-power DER injections remain relatively constant over the Mmeasurement samples, we stack up M instances of (6.9) and (6.10) to yieldp[k] = Λ[k]Ploss,cl[k] , (6.11)Ξ[k] = Λ[k]Vcl[k], (6.12)where p[k] ∈ RM and Λ[k] ∈ RM×(|D|+1) are given byp[k] =P̂ loss[k−M+1]...P̂ loss[k] , Λ[k] =1 (Q̂gen[k−M+1])T......1 (Q̂gen[k] )T , (6.13)99and Ξ[k] ∈ RM×|D| is given byΞ[k] =(V̂ gen[k−M+1])T...(V̂ gen[k] )T . (6.14)Assume that M > (|D|+ 1), then (6.11) and (6.12) are over-determined systems of equations,and we can obtain the ordinary least-squares estimates for P loss,cl[k] and Vcl[k] asP̂ loss,cl[k] ≈ (ΛT[k]Λ[k])−1ΛT[k]p[k], (6.15)V̂ cl[k] ≈ (ΛT[k]Λ[k])−1ΛT[k]Ξ[k], (6.16)from which we can extract the estimated P̂ loss,c[k] and P̂loss,l[k] in the approximate network lossfunction (6.7), and similarly the estimated V̂ c[k] and V̂l[k] in the approximate linear voltagefunction (6.8). Note that in the proposed framework, we include voltage magnitudes only atbuses in the set D to build the sensitivity model. However, this could easily be generalized byincluding measurements from all buses in Ξ[k], to incorporate voltage constraints at all busesin the set N in problem (6.3).We note that, similar to the estimation of sensitivity models in Chapter 3, recursive variantsof the least-squares estimation such as the recursive weighted partial least-squares method couldbe applied to (i) place more weight on more recent measurements and (ii) reduce computationalburden through recursive updates of the estimated linear models.6.2.2 Optimal Reactive-power Dispatch with Gradient Descent MethodInstead of solving a potentially computationally burdensome model-based optimal reactive-power dispatch problem (6.3), we opt to embed the estimated loss and voltage sensitivities insuccessive measurement-based gradient-descent updates. We aim to minimize network loss andcost of DER outputs by defining the objective function as follows:f(Qgen[k] ) = h[k](Qgen[k] ) +12(Qgen[k] )TΥQQgen[k] , (6.17)100where ΥQ = diag(υ1, . . . , υ|D|) ∈ R|D|×|D| is a diagonal coefficient matrix with non-negativeentries, i.e., υi ≥ 0, representing cost coefficients proportional to the square of DER reactive-power outputs. We opt to associate quadratic costs for DER reactive-power output, but wenote that proportional costs could easily be accommodated in the proposed framework.In order to solve (6.3) with objective function (6.17) using a measurement-based gradient-descent method, we first derive the derivative of f(·). Recognizing that P̂ loss,l[k] corresponds tothe gradient of the loss function h[k](·) evaluated at the operating point, we have:∇f(Qgen[k] ) = P̂ loss,l[k] + ΥQQgen[k] (6.18)Subsequently, we embed (6.18) and the estimated voltage sensitivities V̂ l[k] into (6.4) to obtainthe following DER reactive-power setpoint updates:Qgen[k+1]=projQ(Qgen[k] −α[k](P̂ loss,l[k] +ΥQQgen[k] +(V̂ l[k])TβS(V[k]))), (6.19)where β is a weight parameter and S(·) is the soft-thresholding operator associated with thepenalty function g(·) given by:S(V[k]) =V[k] − 1, if V[k] ≥ Vmax,0, if Vmin ≤ V[k] ≤ Vmax,V[k] − 1, if V[k] ≤ Vmin,(6.20)We emphasize that in (6.19) all variables and parameters are either directly measured or esti-mated from measurements, which makes the DER updates independent from an offline networkmodel.In the proposed framework, the DER reactive-power setpoints collected in Qgen[k+1] are up-dated every M time steps using (6.19), after the estimated sensitivities of network loss P̂ loss,l[k]and bus voltages V̂ l[k], are computed using (6.15) and (6.16), respectively. Repeated and it-erative estimations of the loss and voltage sensitivities, and dispatch of DER reactive-powerinjections achieve convergence toward the optimal operating point while adapting to changesin the network.1011 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1823 24 25 26 27 28 29 30 31 32 3319 20 21 22 SW1SW2SW3 SW4DERDERDERDERDERFigure 6.1: IEEE 33-bus test system with DER systems at buses 6, 12, 18, 25, and 33. SwitchesSW1 and SW2 are normally open, and switches SW3 and SW4 are normally closed.0 2 5 70.0120.0140.0160.0180.02Figure 6.2: Network loss resulting from the measurement-based DER reactive-power dispatchmethod compared to the model-based method.6.3 Case StudiesWe perform simulations involving the IEEE 33-bus test system [1] in MATPOWER [30] (seeFig. 6.1), in which we set the power base to 10 MVA. Five DER (e.g., solar PV) systemsare connected to buses in D = {6, 12, 18, 25, 33}. Assume that synchronized measurementsof net active-power injections at all buses and DER reactive-power outputs as well as voltagemagnitudes at buses in D are available from D-PMUs at intervals of 1/60 s, which is reasonableas D-PMUs are capable of streaming up to 120 samples per second [76]. Also assume thatthe DER active-power outputs remain approximately constant over the time interval in whichM = 10 sets of measurements are collected. Each DER reactive-power output is constrainedwithin Qi = [−0.1, 0.1] p.u., i ∈ D, and Q = Πi∈DQi. In order to estimate the networksensitivities in (6.7) and (6.8), we perturb the reactive-power outputs of DER systems withrandom Gaussian distributed variations of zero mean and 0.01% standard deviation relative tothe their nominal values.1020.940.960.9811.021 5 10 15 20 25 30 3300.020.040.060.08Figure 6.3: Bus-voltage magnitudes Vi and DER reactive-power generations Qgeni at t = 4.9 sprior to changes in network topology and system load obtained via (i) model-based method and(ii) proposed measurement-based method.6.3.1 Loss MinimizationIn the first case study, we simulate a scenario, in which we focus on the capability of themethod to minimize network loss. To this end, we set entries in ΥQ to 0 and use voltagelimits of 1 ± 0.15 p.u. (We find that voltage constraints are not binding with these limits.)We benchmark the measurement-based DER reactive-power dispatch against the model-basedmethod assuming that an accurate network model is available. In a time-domain simulation,the system initially operates without DER reactive-power contribution. At time t = 2 s, DERreactive-power dispatch capability is activated. Subsequently, at time t = 5 s, the system topol-ogy is reconfigured by closing switches SW1 and SW2 and opening switches SW3 and SW4,and simultaneously, active- and reactive-power loads at all buses in the system grow by 25%.The simulation runs until time t = 7 s. The step size α[k] is set using the Barzilai–Borweinmethod, in order to take steepest-descent steps: [119]α[k] =∣∣∣(Qgen[k] −Qgen[k−1])T(P̂ loss,l[k] − P̂ loss,l[k−1])∣∣∣∣∣∣∣∣∣P̂ loss,l[k] − P̂ loss,l[k−1]∣∣∣∣∣∣22. (6.21)Time evolution of the network loss resulting from simulating the scenario described above isplotted in Fig. 6.2. Once DER reactive-power dispatch is activated at time t = 2 s, the proposedmeasurement-based dispatch effectively converges to the minimum-loss operating point obtained1030.940.960.9811.021 5 10 15 20 25 30 3300.020.040.060.08Figure 6.4: Bus-voltage magnitudes Vi and DER reactive-power generations Qgeni at t = 7 safter changes in network topology and system load obtained via (i) model-based method and(ii) proposed measurement-based method.by the model-based approach. Also, after network-topology and operating-point changes attime t = 5 s, the proposed method updates DER reactive-power setpoints to minimize loss.Figures 6.3 and 6.4 visualize bus voltages Vi and DER reactive-power outputs Qgeni at t = 4.9 sbefore network reconfiguration, and at t = 7 s after network reconfiguration, respectively. Re-sults from the proposed method match very closely to those from the model-based dispatchwith accurate system model.6.3.2 Loss and DER Cost MinimizationHere, we simulate a scenario that minimizes a combined cost function comprising network lossand DER costs. To this end, we set the diagonal entries in ΥQ between 0.22 and 0.3 in order togive different costs to different DERs, and use voltage limits Vmin and Vmax of 1± 0.05 p.u. (Wefind some of the voltage constraints binding with these limits.) Instead of the value in (6.21),we fix the step size α[k] to 0.04. We further set the parameter β = 50 to appropriately weighthe penalty term for voltage constraint violations. We simulate the same scenario as in theprevious case study, except that there are no changes in network topology or operating point.In Fig. 6.5, we plot bus-voltage magnitudes and DER reactive-power outputs resulting fromsimulating the scenario described above with the proposed measurement-based DER dispatchmethod. Indeed, once DER reactive-power dispatch is activated at time t = 2 s, we observe thatthe proposed measurement-based framework effectively achieves the objective of minimizing1040.90.9510 2 4 600.10.20.3Figure 6.5: Time-domain simulation via proposed measurement-based optimal DER reactive-power dispatch. DER reactive-power dispatch is activated at time t = 2 s. Top pane: busvoltages Vi at buses with DERs; Bottom pane: substation reactive-power injection Q1 andDER reactive-power outputs Qgeni at controllable buses.0 2 4 60.0140.0160.0180.02Figure 6.6: Evaluation of cost function value over time for the measurement-based DERreactive-power dispatch method considering DER costs and voltage constraints, compared tothe model-based method.network loss and DER costs. After activation of DERs, the substation reactive-power injectiondecreases sharply since DERs collectively provide voltage support by injecting reactive power,as shown in the bottom pane of Fig. 6.5. Furthermore, the evaluated cost function value ateach time step for the above scenario is plotted in Fig. 6.6. We observe that the measurement-based method achieves a slightly lower value compared to the model-based benchmark at steadystate, which is due to the “soft” voltage constraints implemented in the measurement-basedframework.6.4 SummaryIn this chapter, we present a measurement-based method to optimally dispatch DER reac-tive power. We estimate linear sensitivities of the network loss and bus voltages with respect105to DER reactive-power injections from synchronized bus injection and voltage measurements.Subsequently, we use the estimated sensitivities in gradient-descent iterations to obtain opti-mal DER reactive-power setpoints. Simulations of the IEEE 33-bus test system demonstratethat the proposed measurement-based method yields accurate DER setpoints compared to themodel-based method, and achieves combined objectives of minimizing network loss and DERcosts while considering bus voltage constraints without relying on any knowledge of an offlinesystem model.106Chapter 7Research Contributions and FutureDirectionsIn this chapter, first the research results and contributions presented in Chapters 3–6 of thisthesis are summarized. Subsequently, several future research directions based on the presentedwork are offered.7.1 Contribution SummaryWhile the problem of optimally dispatching DERs in distribution systems has been a very activeresearch area as of late, this thesis aims to shed light on a different angle by using real-timemeasurements provided by D-PMU while foregoing the use of offline network models. Instead,the proposed methods rely on estimated models which equivalently represent key characteris-tics of the underlying physical system estimated from real-time measurements. Collectively,the research presented in this thesis advances the efficient and reliable operation of modern dis-tribution systems and facilitates the widespread integration of DERs based on measurements.As a result, the proposed methods help to enable higher penetration levels of renewable gen-eration, and to progress the transition to more environmentally-friendly electric power supply.The contributions of each chapter are summarized as follows:• Centralized Method with a Recursively Estimated Sensitivity ModelIn contrast to existing literature, we propose a measurement-based approach to determineDER active- and reactive-power setpoints aimed at regulating bus voltages and injections107as well as minimizing DER costs. The proposed framework consists of two successiveand iterative stages: (i) sensitivity model estimation and (ii) optimal DER dispatch. Tothe best of our knowledge, this is the first method achieving these multiple operationalobjectives without relying on any prior knowledge of the underlying network topology viathe estimation of a linear sensitivity model. The key benefits of the proposed methodinclude (i) eliminating the need for an accurate offline system model, (ii) adapting toonline network-topology and operating-point changes, (iii) being robust against delayspotentially attributed to communication, computation, and actuation, and (iv) modestcomputational requirements using tractable algorithms for both estimation and optimiza-tion due to sensitivity models which retain information from a subset of all buses (i.e.,buses from which measurements are available).• Distributed Method with Estimated Per-area Sensitivity ModelsBuilding on the centralized measurement-based optimal DER dispatch method, the dis-tributed version achieves the same operational objectives and similar accuracy of resultingDER setpoints compared to the centralized counterpart. Addressing potential scalabilityand data privacy concerns in case of vast deployments of D-PMUs in large scale distribu-tion networks, we capitalize on the decomposability of the full network sensitivity modeland propose to estimate equivalent per-area sensitivity models, each representing a portionof the full network. This leads to lower dimension subproblems in both the estimation andthe optimization step. The optimization subproblems can be solved via consensus-basedADMM in parallel fashion. The proposed distributed ADMM-based solution algorithmaffords closed-form solutions in each subproblem, and it requires only exchange of voltagephasor information at the boundary buses among adjacent areas. Overall, the distributedmeasurement-based optimal DER dispatch extends the benefits of the centralized versionwith (i) better scalability and lower computational burden due to the distributed imple-mentation and (ii) it affords greater data privacy as measurements, cost function, andoperational constraints are only shared within one area.• Mitigating Effects of Highly Intermittent Renewable GenerationThe measurement-based optimal DER dispatch methods in Chapters 3 and 4 could be108challenged to adjust bus voltages within acceptable limits due to highly intermittentgeneration with rapid fluctuations and limited availability of measurements in the op-timization stage (i.e., lower time granularity). As a remedy, we propose a method tocombine the system-wide optimization (which could be implemented in a centralized ordistributed fashion) with local controllers at DER locations. We propose local PI con-trollers which regulate bus voltages to reference setpoints. Furthermore, design criteriaof the local controllers to ensure stability of the primary control and the mitigation ofintegrator windup are proposed. The upper-level optimization enables to coordinate thelocal controllers so that system-wide voltage levels and DER reactive-power generationscan be optimized. The proposed control framework offers several key benefits to the oper-ation of distribution system in case of highly intermittent renewable generation: (i) Theproposed local controllers enable faster response time to regulate bus voltages to referencesetpoints without steady-state voltage mismatch compared to existing VV droop-basedlocal controllers. (ii) Dispatching optimal voltage and reactive-power setpoints to the lo-cal controllers combines the advantages of the purely centralized and purely local control.(iii) As demonstrated by the case studies, the proposed two-level control system mitigatesall voltage violations, prevents increased switching operations of OLTC transformers, andimproves system unbalance.• Gradient-descent Method for Optimal Reactive-Power DispatchThe minimization of network losses while simultaneously regulating bus voltages throughDER reactive-power dispatch is a desirable operational objective in distribution systems.The previous methods in Chapters 3 and 4 can not easily be leveraged to achieve loss min-imization. Therefore, we additionally propose a fully measurement-based approach thatdoes not rely on any prior knowledge of the distribution-network topology or parametersto optimally dispatch DER reactive-power outputs via feedback-based optimization usinga gradient-descent method. We formulate the objective function to minimize network lossand cost of DER outputs. Our proposed method has several benefits over the state of theart. (i) Compared to existing feedback-based optimization methods, we only rely on mea-surements to calculate relevant sensitivities of bus voltages and network loss (as opposed109to the network model typically employed to compute these sensitivities). (ii) Compared toother measurement-based methods as the ones in Chapters 3 and 4 and others mentionedin Section 1.3.3, our proposed methods achieves operating points which minimize networkloss along with cost of DER outputs.7.2 Concluding Remarks and Future DirectionsThe environmental concerns of conventional electricity supply and emergence of new technolo-gies drives radical changes in electricity generation and power system operation. There is agigantic potential held by many disruptive technologies such as renewable electricity genera-tion, electrified transportation, advances in storage systems, sensors, high-speed communicationnetworks and artificial intelligence to transition to a more sustainable and abundant energysupply. While the increasingly decentralized generation and complex grid-edge poses profoundchallenges for the operation of the legacy electricity grid, the recent technological advances bringnew ways of dealing with these challenges and maximizing the benefits of emerging technologiesto form a smart electric power grid.Within this thesis, we have only scratched the surface in the effort to develop measurement-based schemes to optimize and control modern power distribution systems. As we move forwardin the transition of the electric power grid, we envision several promising directions in whichthe research presented in this thesis could be continued and extended, in order to advance thissignificant field. Specifically, these include the following:• Combining Multiple Input Data SourcesThe methods developed in this thesis predominantly rely on D-PMU measurements frompartially or fully observable distribution systems. While the high-precision phasor mea-surements offer vast opportunities for monitoring, control and optimization purposes (asdemonstrated by the proposed methods), widespread adoption might require significantcapital expenditure. Concurrently, other advanced sensors such as e.g., smart metersare already today widely deployed in the field and may offer additional sources of data.Likewise, modern DERs such as for example smart inverters are also capable of providingreal-time measurements. Furthermore, existing conventional concepts of using network110model information or monitoring systems such as e.g., SCADA supply other useful in-put data to optimize operation of distribution systems. The coupling of these variouscomplementary data streams with different granularity and accuracy may offer additionalinsight on the current operating conditions in distribution systems, and ultimately couldenable fewer additional installations of advanced sensors such as D-PMUs while achievingaccurate and optimization observation of the distribution systems.• Leveraging Machine Learning-based Estimation and Optimization TechniquesIn order to deal with the extensive input data as mentioned in the previous paragraph,machine-learning methods such as deep-learning could be leveraged to learn sophisticatedequivalent models which represent accurate and up-to-date operating conditions of distri-bution systems. Furthermore, we envision that reinforcement learning techniques couldcombine the task of estimating equivalent models and obtaining optimal DER setpointsin one step.• Implementing Cloud-based SoftwareApplication programming interfaces along with high-speed communication channels fa-cilitate the integration of various devices and sensors at different geographical locationswith cloud-based computing units. By implementing software to manage distributionsystems such as the proposed methods in cloud-based computing units, the deploymentin the field would be feasible so that DERs could be simultaneously actuated at variouslocations in a timely manner. Our preliminary work has established a test-bed using areal-time simulation environment which demonstrates the feasibility of using cloud-basedcomputers to implement DER management systems in [120].111Bibliography[1] M. E. Baran and F. F. Wu, “Network reconfiguration in distribution systems for lossreduction and load balancing,” IEEE Trans. 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IEEE Power Energy Soc.General Meeting, Aug. 2018, pp. 1–5. → page 111120Appendix ADeriving Partial Least-squares Estimate in(3.7) of Chapter 3The decompositions of X[k] and Y[k] are performed in a way to satisfy an outer model givenby [98]X[k] = T[k]GT[k], (A.1)Y[k] = U[k]LT[k], (A.2)where T[k] = [t[k],1, . . . , t[k],r] (an orthonormal matrix) and U[k] = [u[k],1, . . . , u[k],r] are latentmatrices, each with r ≤ 2E − 1 score vectors t[k],i and u[k],i, i = 1, . . . , r, extracted frommatrices X[k] and Y[k], G[k] = [g[k],1, . . . , g[k],r] and L[k] = [l[k],1, . . . , l[k],r] are correspondingloading matrices, with r loading vectors g[k],i and l[k],i, i = 1, . . . , r. Using the nonlineariterative partial least squares (NIPALS) algorithm, T[k], U[k], G[k], and L[k] are populated incolumn-wise fashion [98].The NIPALS algorithm maximizes the covariance between the score vectors u[k],i and t[k],i,for all i = 1, . . . , r, thereby forming the following inner model:U[k] = T[k]Γ[k], (A.3)121where Γ[k] = diag(γ[k],1, . . . , γ[k],r), withγ[k],i =uT[k],it[k],itT[k],it[k],i. (A.4)The PLS regression is then performed on the over-determined inner model (A.3) instead of theoriginal data set in X[k] and Y[k]. Specifically, by substituting (A.3) into (A.2), we getY[k] = T[k]Γ[k]LT[k]. (A.5)Then, substitution of (A.1) and (A.5) into (3.5) while recognizing that T[k] is orthonormalyields (3.7), as desired. Interested readers may refer to [98] for background on the PLS methodand details of the NIPALS algorithm.122Appendix BReformulating (3.12) as Standard QuadraticProgram in Chapter 3In (3.17a), Π ∈ R(4|E|+2|D|−2)×(4E|E|+2|D|−2), pi ∈ R4|E|+2|D|−2, and κ ∈ R are obtained by suitablealgebraic manipulations of (3.16); and they are given byΠ = 2Ψ 0 00 Φ 00 0 Υ , pi = −2Ψx◦Φy◦0 , (B.1)κ = (x◦)TΨx◦ + (y◦)TΦy◦, (B.2)where 0s are matrices or vectors of all zeros with appropriate dimension. The linear constraintsin (3.17b) is constructed by combining (3.12b) and (3.12c) withA[k] =−J[k] I2|E| 00 I2|E| −C , b[k] = c[k]y[k] − Cygen[k] , (B.3)where I2|E| denotes the 2|E| × 2|E| identity matrix and 0s represent matrices of all zeros withappropriate dimension. Finally, X represents the composition of set constraints in (3.12d)and (3.12e).123Appendix CDerivation of (5.12) in Chapter 5The voltage at bus i considering reactive power generation Qgeni,[k+1] at time step k + 1 is givenby (5.5a)Vi,[k+1] = Vi,0 + xiQgeni,[k+1], (C.1)From (5.9), the control function of the local PI controller is given byQgeni,[k+1] =KP,i( (Vi,set − Vi,[k])+KI,ik−1∑`=0(Vi,set − Vi,[`]) ), (C.2)Defining the voltage mismatch between Vi,set and Vi,[k] at time step k∆Vi,[k] = Vi,set − Vi,[k]. (C.3)Defining the auxiliary variable ζi,[k] denoting the summation of voltage mismatch over timeζi,[k] =k−1∑`=0∆Vi,[`], and, ζi,[k+1] = ∆Vi,[k] + ζi,[k], (C.4)with summation index `. Then, the expression of the local PI controller becomesQgeni,[k+1] = KP,i(∆Vi,[k] +KI,iζi,[k]), (C.5)124The voltage mismatch at time step k + 1 is given by∆Vi,[k+1] = Vi,set − Vi,[k+1] = Vi,set − Vi,0 − xiKP,i(∆Vi,[k] +KI,iζi,[k])(C.6)From (C.6) and (C.4), the state-space model (5.12) is formed with the input Vi,set − Vi,0Ai =−xiKP,i −xiKP,iKI,i1 1 , Bi =1 −10 0 , xi,[k] =∆Vi,[k]ζi,[k] , ui,[k] =Vi,setVi,0 . (C.7)In a similar way, the global and the reduced state-space models (5.17) and (5.18) are formed.125
Thesis/Dissertation
eng
Electrical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Measurement-based techniques for optimization and control of power distribution systems using distributed energy resources
Text