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Planning and operation of active smart grids Ghasemi Damavandi, Mohammad 2017

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Planning and Operation of ActiveSmart GridsbyMohammad Ghasemi DamavandiB.Sc., University of Tehran, 2008M.Sc., University of Tehran, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)January 2017© Mohammad Ghasemi Damavandi 2017AbstractFuture smart grids will be operated actively in the presence of distributed genera-tors and topological reconfigurations. Distributed Storage Systems (DSS) will alsobecome a viable solution for balancing the load and the intermittent generation ofrenewable energy sources. The DSS can also provide the smart grid operator withvarious other benefits including peak load shaving, resilience enhancement, powerloss reduction, and arbitrage gain.The active nature of future smart grids calls for an accurate state estimationmechanism to serve as a building block for many operational tasks. To that end,the first part of the present thesis leverages the concept of submodularity to solve theproblem of robust meter placement for state estimation in reconfigurable smart grids.Next, the thesis proposes a methodology for optimal planning of DSS in smart gridswith high penetration of renewable sources. The presented methodology accounts forvarious advantages of energy storage in smart grids and seeks the optimal trade-offbetween the investment cost and the expected discounted reward of DSS installation.Finally, the thesis focuses on the problem of Volt-VAR Optimization (VVO) in activesmart grids. The optimal joint operation of reconfiguration switches, energy storageunits, under load tap changers, and shunt capacitors is investigated in the presentedVVO methodology. The proposed methodologies in this thesis have been tested onsample distribution systems and their effectiveness is validated using real data ofsmart meters and renewable energy sources.iiPrefaceThe work presented in this thesis is based on the research conducted by the authorwith assistance and guidance of Prof. Vikram Krishnamurthy and Prof. Jose´ RMart´ı. The problem formulations, algorithmic solutions, numerical simulations, andwriteup of all chapters and papers is done by the author. For all these works, Prof.Krishnamurthy and Prof. Mart´ı guided the author with their useful suggestions,technical discussions, supervisory comments, and editorial feedbacks.The following articles are derived from the research conducted in the presentthesis.ˆ M. Ghasemi Damavandi, V. Krishnamurthy, and J. R. Mart´ı, Robust meterplacement for state estimation in active distribution systems, IEEE Trans.Smart Grid, vol. 6, no. 4, pp. 1972-1982, Jul. 2015. [Related to chapter 2 ofthe thesis]ˆ M. Ghasemi Damavandi, J. R. Mart´ı, V. Krishnamurthy, A methodology foroptimal distributed storage planning in smart distribution grids, submitted.[Related to chapter 3 of the thesis]ˆ M. Ghasemi Damavandi, V. Krishnamurthy, and J. R. Mart´ı, A comprehen-sive methodology for volt-VAR optimization in active smart grids, submitted.[Related to chapter 4 of the thesis]iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Active Smart Grids: An Overview . . . . . . . . . . . . . . . . . . . 11.2 Research Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Related Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.1 Robust Meter Placement . . . . . . . . . . . . . . . . . . . . 71.3.2 Distributed Storage Planning . . . . . . . . . . . . . . . . . . 81.3.3 Volt-VAR Optimization in Active Smart Grids . . . . . . . . 81.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9ivTable of Contents1.4.1 Robust Meter Placement . . . . . . . . . . . . . . . . . . . . 101.4.2 Distributed Storage Planning . . . . . . . . . . . . . . . . . . 101.4.3 Volt-VAR Optimization in Active Smart Grids . . . . . . . . 111.5 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Robust Meter Placement in Reconfigurable Smart Grids . . . . 142.1 State Estimation Using Non-Linear Measurements and the FIM . . 162.2 Meter Placement Using Linearized Power Flow Equations . . . . . . 182.2.1 Linearized Power Flow Equations In Rectangular Coordinates 182.2.2 The Linearized Measurement Model . . . . . . . . . . . . . . 212.3 Meter Placement in Single Configuration Distribution Systems . . . 232.3.1 Formulation of the Meter Placement Problem . . . . . . . . 232.3.2 Submodularity and the Greedy Algorithm . . . . . . . . . . 272.4 Meter Placement in Reconfigurable Distribution Systems . . . . . . 292.4.1 Formulation of the Robust Meter Placement Problem . . . . 292.4.2 The Submodular Saturation Algorithm . . . . . . . . . . . . 322.5 Numerical Results and Discussions . . . . . . . . . . . . . . . . . . . 382.5.1 Meter Placement in a 33-node, 12.66 kV Active DistributionSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.5.2 Meter Placement in a 70-node, 11 kV Active Distribution Sys-tem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.5.3 Meter Placement in an 119-node, 11 kV Active DistributionSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.5.4 Computational Cost . . . . . . . . . . . . . . . . . . . . . . . 512.5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52vTable of Contents3 Distributed Storage Planning in Smart Distribution Grids . . . 543.1 Linearized Power Flow Equations and the Loss Function . . . . . . 553.2 Formulation of the Economic Gains of the Distributed Storage Sys-tems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.2.1 The Arbitrage Gain . . . . . . . . . . . . . . . . . . . . . . . 623.2.2 The Expected Reduction in Active Power Loss . . . . . . . . 623.2.3 The Reduction in Expected Price of Renewable Energy Cur-tailed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.2.4 The Improvement in the System Resilience . . . . . . . . . . 703.2.5 The Economic Gain of the System Upgrade Deferral . . . . . 713.3 The Optimal Distributed Storage Planning Problem . . . . . . . . . 733.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.4.1 The Setting of the Simulations . . . . . . . . . . . . . . . . . 773.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834 Volt-VAR Optimization in Active Smart Grids . . . . . . . . . . . 844.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.1.1 DistFlow Equations . . . . . . . . . . . . . . . . . . . . . . . 864.1.2 Markov Chain Model of Wind Power Generation . . . . . . . 884.2 Formulation of the Volt-VAR Optimization Problem . . . . . . . . . 914.2.1 The Objective Function of the VVO Problem . . . . . . . . 924.2.2 Power Flow Equations, Distributed Storage Systems, Capac-itors, and ULTCs . . . . . . . . . . . . . . . . . . . . . . . . 934.2.3 Feeder Reconfiguration . . . . . . . . . . . . . . . . . . . . . 964.2.4 Convexification of the VVO Problem . . . . . . . . . . . . . 994.3 Case Studies and Numerical Results . . . . . . . . . . . . . . . . . . 102viTable of Contents4.3.1 The Setting of the Simulations . . . . . . . . . . . . . . . . . 1024.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125 Conclusions and Future Research . . . . . . . . . . . . . . . . . . . . 114Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117viiList of Tables2.1 Worst-case Total Estimation Variance for VMM Placement in the33-node Active Distribution System . . . . . . . . . . . . . . . . . . 402.2 Worst-case Total Estimation Variance for PMU Placement in the 33-node Active Distribution System . . . . . . . . . . . . . . . . . . . . 402.3 Worst-case Total Estimation Variance for VMM Placement in the70-node Active Distribution System . . . . . . . . . . . . . . . . . . 442.4 Worst-case Total Estimation Variance for PMU Placement in the 70-node Active Distribution System . . . . . . . . . . . . . . . . . . . . 442.5 Worst-case Total Estimation Variance for VMM Placement in the119-node Active Distribution System . . . . . . . . . . . . . . . . . . 482.6 Worst-case Total Estimation Variance for PMU Placement in the 119-node Active Distribution System . . . . . . . . . . . . . . . . . . . . 484.1 Performance of the Optimal VVO Solution Under Different Settings 108viiiList of Figures1.1 Schematic of an active smart grid with smart meters, DGs, and DSS. 42.1 Schematic of a reconfigurable distribution system with smart meters,VMMs, and PMUs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Optimal location of five VMMs in the 33-node test system using Al-gorithm 1 (yellow boxes), Algorithm 2 (blue boxes), and GA (greenboxes). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.3 Optimal location of five PMUs in the 33-node test system using Al-gorithm 1 (yellow boxes), Algorithm 2 (blue boxes), and GA (greenboxes). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.4 Optimal location of 9 VMMs in the 70-node test system using Al-gorithm 1 (yellow boxes), Algorithm 2 (blue boxes), and GA (greenboxes). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.5 Optimal location of 9 PMUs in the 70-node test system using Al-gorithm 1 (yellow boxes), Algorithm 2 (blue boxes), and GA (greenboxes). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.6 Optimal location of 9 VMMs in the 119-node test system using Al-gorithm 1 (yellow boxes), Algorithm 2 (blue boxes), and GA (greenboxes). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49ixList of Figures2.7 Optimal location of 9 PMUs in the 119-node test system using Al-gorithm 1 (yellow boxes), Algorithm 2 (blue boxes), and GA (greenboxes). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.1 Magnitudes and angles of the nodal voltages in the test system ob-tained using linearized power flow equations and ACPF for the peakhour of a typical day with light loading. . . . . . . . . . . . . . . . . 603.2 Magnitudes and angles of the nodal voltages in the test system ob-tained using linearized power flow equations and ACPF for the peakhour of a typical day with high loading. . . . . . . . . . . . . . . . . 613.3 The modified test system with wind turbines and solar cells and thestorage unit optimally located on node 29. . . . . . . . . . . . . . . . 803.4 The optimal charging and discharging strategy of the installed storageunit for three different segments of the year. . . . . . . . . . . . . . . 814.1 Schematic of an active smart grid with distributed wind turbines andvarious VVO equipment. . . . . . . . . . . . . . . . . . . . . . . . . . 844.2 The 33-node active smart grid with a wind turbine, DSS, and VVOequipment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.3 Voltage profile of the system on a typical day at a peak and an off-peakhour. Newton’s AC power flow versus linearized extended DistFlowequations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.4 The reduction in active power losses (%) after optimal VVO for var-ious test cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.5 The reduction in the peak load (%) after optimal VVO for varioustest cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110xList of Figures4.6 The increase in the minimum voltage of the system (p.u.) after opti-mal VVO for various test cases. . . . . . . . . . . . . . . . . . . . . . 1114.7 The decrease in the maximum voltage of the system (p.u.) afteroptimal VVO for various test cases. . . . . . . . . . . . . . . . . . . . 1114.8 The decrease in the voltage spread of the system (p.u.) after optimalVVO for various test cases. . . . . . . . . . . . . . . . . . . . . . . . 1124.9 Voltage profile of the system under different VVO test cases. Thevoltage profile is of a typical day at peak hour. . . . . . . . . . . . . 113xiList of AbbreviationsAAE Average Angle Error.ACPF AC Power Flow.AME Average Magnitude Error.AMI Advanced Metering Infrastructure.CAIDI Customer Average InterruptionDuration Index.CRLB Crame´r-Rao Lower Bound.DG Distributed Generators.DOD Depth of Discharge.DSS Distributed Storage Systems.FIM Fisher Information Matrix.GA Genetic Algorithm.LLN Law of Large Numbers.MAE Maximum Angle Error.MC Markov Chain.MLE Maximum Likelihood Estimate.MME Maximum Magnitude Error.MV Medium Voltage.NLE Normalized Loss Error.PMU Phasor Measurement Units.SAIFI System Average InterruptionFrequency Index.SGO Smart Grid Operator.SLEM Second Largest Eigenvalue Mod-ulus.ToU Time of Use.TVE Total Vector Error.ULTC Under Load Tap Changers.VMM Voltage Magnitude Meters.VVO Volt-VAR Optimization.WLS Weighted Least Squares.xiiAcknowledgmentsFirst and foremost, my sincerest gratitudes go to my advisor, Prof. Vikram Krishna-murthy, for his great academic guidance and invaluable intellectual support duringthis research. His deep knowledge and remarkable insights profoundly improved thequality of this thesis. I also want to thank Prof. Krishnamurthy specifically forthe enormous encouragement and enthusiasm that he brought to this research. Hisamazing concentration and incredible dedication to his job has definitely become anunforgettable lesson for my life.Also, I would like to express my deep gratitude to my co-advisor, Prof. Jose´ RMart´ı for his huge support during preparation of this thesis. Prof. Mart´ı spent anenormous amount of time on discussions about this research and provided strategicinsights, novel ideas, and invaluable feedbacks about the power engineering side ofthis thesis. Nonetheless, engineering is not the only area in which I learned a lotfrom Prof. Mart´ı. He symbolizes a true man of science who teaches his studentshow to think critically about all aspects of life.I feel deeply honored for having had the opportunity of working with Prof.Krishnamurthy and Prof. Mart´ı without whom this research could have never beendone.xiiiDedicationTo my parents.xivChapter 1Introduction1.1 Active Smart Grids: An OverviewThe next generation power grids are envisioned to operate in an active manner.They can benefit from high penetration of renewable energy sources distributedthroughout the grid. Parts of the future distribution systems are even envisioned towork as microgrids where during the islanded mode the demand power should becompletely supplied by local Distributed Generators (DG) [1]. These DGs are oftenhighly intermittent and can cause dramatic changes in the demand/generation ofthe nodes. They can also cause bidirectional flow of power in the feeder and evenfrom distribution level to the sub-transmission system [2].In addition, the topological configuration of active smart grids can be dynami-cally optimized using remotely controllable tie line switches. In fact, use of remotelycontrollable switches enables the SGO to reconfigure the distribution system evenin daily or hourly intervals [3–7]. In practice, feeder reconfiguration can be done fora variety of goals including load balancing [8], reliability improvement [9], servicerestoration [10], voltage profile improvement and power loss minimization [11], [12].In systems with high penetration of non-dispatchable sources, the reconfigurationroutine can also be employed to mitigate the effect of intermittent renewable gen-eration. Traditionally, the topological configurations of the distribution grid havebeen radial to reduce the short circuit current under fault conditions and to facili-11.1. Active Smart Grids: An Overviewtate protection of the system [13]. However, in the future systems the topologicalconfigurations may no longer be limited to radial and can possibly include weaklymeshed ones [14, 15].The active nature of future smart grids calls for an online monitoring of the sys-tem for state estimation and situational awareness. State estimation in present dis-tribution systems mostly relies on substation measurements and pseudo-measurements(historical data). However, in future grids new data from Voltage Magnitude Meters(VMM), smart meters and Phasor Measurement Units (PMU) can be incorporatedin the state estimation process [16–18]. The data reported by these meters willalso increase the visibility of the secondary network for monitoring power losses andelectricity thefts. Note that the reporting rate of the smart meters is expected to beevery 15 to 60 minutes in the future smart grid. The residential meters of some util-ities currently transmit their measurements even a few times a day due to concernsabout privacy and exposure to frequency radiation. Hence, the smart metering datamay not be adequate for a real-time, highly accurate monitoring of the future distri-bution systems. Therefore, VMMs and PMUs can play an important role for stateestimation as they provide highly accurate synchronized information about differ-ent parts of the system [19]. The PMUs can be specifically useful in monitoring thedistribution system for islanding/reconnection [2], bidirectional flow of power [20],and power quality [21, 22]. Moreover, they provide capabilities for fault detectionin radial and weakly meshed distribution systems [23]. Furthermore, synchronizedmeasurements are a major candidate for parallel monitoring of the energy distribu-tion and communication sub-systems of the smart grid [24]. It is expected that forsome critical applications in the future distribution grids, a time synchronizationof 1µs between multiple devices (possibly in different sub-systems) will be needed[24]. In a 60-Hz distribution system, this amount of time synchronization requires21.1. Active Smart Grids: An Overviewa maximum phase error less than 3.77× 10−4 rad [20].As much as the active smart grids can benefit from the renewable energy sources,the intermittent nature of those sources requires careful operation and control. Forexample, when the amount of non-dispatchable renewable energy generation is fore-cast to violate a system constraint, the Smart Grid Operator (SGO) may have tocurtail the excess power. In addition, when the amount of total non-dispatchablegeneration is higher than the total demand, the excess power has to flow to thetransmission system. If the amount of excess power is less than the allowable re-verse power limit of the substations, this will not cause any problem. However, if theamount of excess power is greater than the allowable reverse power limit, the SGOor the private DG owner will have to curtail some non-dispatchable sources. In con-trast, if the system is equipped with Distributed Storage Systems (DSS), the SGOwill have the option to store the excess generation in storage units [25]. Similarly,when one part of the smart grid is to work as a microgrid in an islanded mode, themismatch between the load and power generation of local DGs can be smoothed outusing DSS [26]. An islanded mode occurs when the microgrid is disconnected fromthe Medium Voltage (MV) grid due to faults or planned switching actions [27, 28].On the other hand, when the microgrid is connected back to the MV grid, the SGOcan exploit the installed DSS to improve some metric of the system performancesuch as voltage profile or power loss.Fig. 1.1 illustrates the schematic of an active smart grid with a distributedstorage unit. In this figure, the smart meters, distributed solar cells, distributedwind turbines, and DSS are symbolically shown on a radial distribution system.The smart grid shown in Fig. 1.1, is also topologically reconfigurable as representedby a schematic dotted branch.In virtue of DSS, the SGO will also be able to buy the excess energy from DG31.1. Active Smart Grids: An Overview Smart Grid Operator Figure 1.1: Schematic of an active smart grid with smart meters, DGs, and DSS.owners and sell it back to the network when the demand is higher. In marketswith dynamic energy pricing where the price of energy is proportional to the totaldemand, that naturally translates into an arbitrage gain for the SGO. Likewise,the DG owner will be able to gain some profit by selling to the SGO the excessenergy that would be spilled otherwise [25]. If the system undergoes a fault, theSGO will also have the degree of freedom to take advantage of these DSS duringsystem restoration [29] to minimize the energy-not-supplied. More importantly, useof DSS can shave the peak load which constantly increases in distribution systems[30]. Shaving the peak load of the system can in turn defer the inevitable upgradeof the feeders and substations which, depending on the size of the system, can resultin substantial financial gains [31, 32]. Other applications of DSS in distributionsystems include voltage control [33, 34], ancillary services, feeder load managementand congestion control, and power smoothing for solar arrays [35, 36].Storage units can also be deployed at the very low-voltage end of the power41.1. Active Smart Grids: An Overviewsystem. Recently, some authors have considered the use of DSS at the householdlevel [37]. Also, Tesla Motors recently introduced a 10 kWh home battery, calledPowerwall, for smart homes. The concept of community energy storage also deploys120/240-Volt storage units to protect the same group of houses that share a commonsecondary transformer [38].In addition to possible generation-demand mismatches in smart grids with highpenetration of renewable energy sources, the nodes of the system can also be subjectto sever voltage fluctuations. For example, when the instantaneous generation of therenewable sources is high but the load is low the nodes of the system may undergoover-voltage problems [39]. Another potential problem is the violation of feederampacities at times when the renewable generation is considerably high. Due tothe high variability of non-dispatchable sources, the Volt-VAR Optimization (VVO)problem will be a very important issue in active smart grids. The VVO problem, inthe classical sense, is the optimal operation of the capacitor banks and Under LoadTap Changers (ULTC). The objective of the VVO techniques has conventionallybeen to optimize some metric of the system performance such as active power lossor voltage profile. In practice, the stochasticity of renewable energy sources requiresthe VVO problem to optimize the expected performance of the system. Furthermore,in modern smart grids, the VVO technique should take advantage of the DSS andremotely controllable switches as additional degrees of freedom on top of classicalVVO equipment. The incorporation of DSS and reconfiguration routine into theVVO procedure can improve various performance metrics of the system includingthe voltage profile and power losses.51.2. Research Goals1.2 Research GoalsThis research is dedicated to the optimal planning and operation of active smartgrids with high penetration of renewable energy sources. Various aspects of activesmart grids including monitoring and state estimation, feeder reconfiguration, DSSplanning and operation, wind and solar power stochasticity, and VVO have beenstudied. The main goal of the research has been to come up with mathematicalformulations and algorithmic solutions that can be implemented efficiently for smartdistribution systems. In particular, three major problems have been considered inthis thesis which include the following:ˆ Robust meter placement in reconfigurable distribution systemsˆ Optimal DSS planning in smart distribution gridsˆ VVO in active smart grids considering feeder reconfiguration and DSS opera-tionFor each problem mentioned above, a mathematical formulation is presented in sucha way that the resulting optimization problem can be solved in an efficient way.The effectiveness of the proposed methodologies has been studied and validatedon different test systems using real data from smart meters and renewable energysources.1.3 Related WorksThis section provides a review of the existing literature related to the three majorproblems studied in this thesis.61.3. Related Works1.3.1 Robust Meter PlacementAs the first part of this thesis, the problem of meter placement for state estimationin reconfigurable distribution systems is considered. In what follows, a summary ofthe existing works related to the optimal meter placement problem in distributionsystems is provided.The meter placement problem for single configuration distribution systems hasbeen considered in [17, 18, 40]. In [41] and [42], the joint placement of PMUsand smart metering peripherals for reconfigurable distribution systems has beenpresented. Several authors have also considered the installation of PMUs in thedistribution system [43]. For example, a multiphase state estimator is proposedin [44] which relies on the synchronized voltage and current measurements of thefeeder. Also in [20] and [45], a linear distribution system state estimator is pre-sented which incorporates synchronized phasor measurements along with the smartmetering data. A branch-current-based distribution system state estimator is alsopresented in [46] which takes advantage of PMU measurements and handles radialand weakly meshed networks. In [47], the performance of various state estimators fordistribution system has been compared where data from optimally located PMUs isincorporated as real measurements. The application of synchronized meters for dis-tribution system monitoring is also suggested in [48] as the data from smart meterswill not be sufficient for full observability of the system. Use of one or few PMUs inthe distribution system is also an enabler of the multilevel state estimation paradigmproposed in [49]. In this paradigm, the state estimation of the system is done atdifferent levels (feeder level, substation level, transmission level, and regional level)and the results are integrated together in a synchronized fashion to provide a verylarge scale monitoring of the power system.71.3. Related Works1.3.2 Distributed Storage PlanningAs part of this research, the problem of DSS placement in distribution systemswith high penetration of wind and solar energy sources is considered. This sectionprovides a review of the existing literature on the optimal DSS planning problem.The optimal DSS planning has been considered in several recent works [39, 50–56]. In [25], a framework is presented which optimizes the capacity and power ratingof DSS to ensure that the renewable energy generated by DGs never spill. Nonethe-less, this work does not consider various other advantages that the DDS introduceto the system. In [57], the potential of DSS in the low-voltage distribution grid fordeferring upgrades needed to increase the solar penetration level is investigated. In[58], the optimal allocation of DSS in distribution systems using a multi-objectiveoptimization approach is considered. However, [57, 58] do not consider the roleof DSS in improving the resilience of the system. In [59], a methodology for DSSallocation in distribution systems is proposed which which aims at cost-effectiveimprovement of the system reliability. In, [60], the optimal planning of DSS usingthe point estimate method is considered. In [61], the problem of DSS allocation issolved using Bender’s decomposition and a scenario tree for wind power simulation.In [31, 32, 62], the optimal placement of DSS considering the benefits due to systemupgrade deferral is investigated. Nevertheless, none of the aforementioned papersincludes a complete and comprehensive consideration of the technical and financialaspects of distributed storage planning using mixed-integer convex formulation.1.3.3 Volt-VAR Optimization in Active Smart GridsThe third part of this research is focused on the VVO problem in active smart grids.Below, we review the recent literature on the VVO problem in distribution systems.Several works in the literature have considered the VVO problem for distribution81.4. Contributionssystems [63–67]. In [68], a VVO technique using oriented discrete coordinate descentmethod for minimization of power loss or the number of control actions is proposed.In [69], a VVO methodology is presented which minimizes the total reactive powersupplied by the DGs. In [70], a scenario-based multiobjective method for dailyVVO is presented which considers renewable energy sources. However, [68–70] donot consider feeder reconfiguration as a tool for VVO. Voltage control and VARoptimization methodologies are also presented in [71] and [72] that consider feederreconfiguration in addition to classical VVO devices. However, these papers donot consider the DSS and the stochasticity of wind power generation in their VVOmethodologies. In [73], the integration of energy storage systems into the voltagecontrol mechanisms of distribution systems is proposed. Nonetheless, none of theaforementioned works provides a mixed-integer convex program formulation of theVVO problem that takes care of the operation of all the available equipment andtreats the stochasticity of wind power generation in a statistically rigorous way.1.4 ContributionsThe main theme of the thesis is to propose comprehensive mathematical formula-tions and algorithmic solutions for some important planning and operational prob-lems in active smart grids. The concepts of submodularity and convexity have beenemployed in mathematical formulations such that the resulting problems can besolved efficiently. The mathematical formulations presented in this thesis are com-prehensive in that various relevant aspects of the problem have been taken intoaccount. The stochasticity of the renewable power generation and loads has alsobeen addressed in a statistically rigorous way.In what follows, the three major contributions of the thesis are briefly introduced.91.4. Contributions1.4.1 Robust Meter PlacementThe first part of the present thesis is focused on the formulation and solving theoptimal meter placement problem in reconfigurable distribution systems. In thefollowing, we summarize our contributions in this part of the thesis.In this thesis, the near-optimal placement of a limited number of VMMs andPMUs for state estimation in reconfigurable distribution systems is considered. Thetrace (sum of diagonal entires) of the inverse of the Fisher Information Matrix(FIM) is proposed as the estimation accuracy criterion. Under some conditions,the reduction in the trace of the inverse of the FIM is a submodular function [74]which lends itself for a greedy algorithm for a near-optimal meter placement in asingle configuration distribution system. For a given number of meters, the greedyalgorithm also provides a (1−1/e) approximation guarantee for the meter placementproblem [74, 75] in single configuration distribution systems. The meter placementproblem in reconfigurable distribution systems, however, should take into accountthe topological reconfigurations of the system [41, 42]. Therefore, the problemshould be formulated in a robust way so as to optimize the worst case estimationaccuracy among all the possible configurations of the system. To that end, a robustsubmodular optimization algorithm, called submodular saturation algorithm [76],is proposed in this thesis which outperforms the greedy algorithm and the GeneticAlgorithm (GA) in most cases and provides competitive results in other cases.1.4.2 Distributed Storage PlanningThe second contribution of the present thesis is a comprehensive formulation of theoptimal DSS placement problem in smart grids. This section provides a summaryof the contributions on the optimal DSS planning problem.This thesis presents a comprehensive methodology for optimal DSS planning in101.4. Contributionssmart distribution systems. The optimal DSS placement problem is formulated asa mixed-integer quadratic program to be solved using branch-and-bound methods.Various financial gains due to price arbitrage, reduction in the system losses, reduc-tion in the renewable energy curtailed, system resilience enhancement, and systemupgrade deferral have been considered in the presented methodology. The stochas-ticity of the loads and renewable energy sources are accounted for by evaluatingthe expected discounted gains using the Law of Large Numbers (LLN). Simulationresults on a radial distribution test system and with real data from smart metersand renewable energy sources are presented and discussed. The present thesis is thefirst work that provides a mixed-integer convex formulation that comprehensivelyaccounts for all the technical and financial aspects of distributed storage planningas well as the stochasticity of load and renewable energy sources.1.4.3 Volt-VAR Optimization in Active Smart GridsThe third contribution of this research is a comprehensive formulation of the VVOproblem for smart distribution grids. In what follows, we review our contributionsin this part of the thesis.This thesis provides a comprehensive formulation of the VVO problem in activesmart grids. The formulation provided in the thesis is comprehensive in that itjointly considers wind turbines, DSS, capacitor banks, ULTCs, and feeder reconfig-uration. The VVO problem presented is formulated as a mixed-integer, quadraticprogram which can be solved efficiently using commercial softwares. The formula-tion provided for the VVO problem tries to minimize the expected power loss ofthe system. In order to address the stochasticity of the wind power generation inevaluating the expected power loss of the system, a first-order Markov Chain (MC)model [77] is employed. Simulation results on a 33-node, 12.66 kV active smart grid111.5. Thesis Structureusing real data from wind turbines and smart meters are also presented and dis-cussed. Various test cases are considered in the simulations to compare the impactof different VVO equipment on the system loss and voltage profile.1.5 Thesis StructureIn this section, the structure of this thesis is presented.Chapter 2 is devoted to the problem of robust meter placement in single-configurationand reconfigurable smart grids. This chapter takes the reduction in the trace of theinverse of the FIM as the metric of the state estimation accuracy and provides aformulation of the meter placement based on the concept of submodularity and di-minishing returns. Numerical results of robust meter placement on three differentactive distribution systems are presented and discussed in this chapter.Chapter 3 is the dedicated to the problem of optimal DSS planning in smartdistribution grids. This chapter formulates various economic gains of the DSS asa function of the amount and location of storage units in the system. Based onthe real data from smart meters and renewable energy sources, the methodologypresented in this chapter evaluates and optimizes the total expected discountedgain due to installation of DSS. In order to come up with a mixed-integer convexprogramming1 formulation of the optimal DSS planning problem, the chapter utilizesthe linearized power flow equations in rectangular coordinates [78–80]. Numericalresults for optimal DSS placement and sizing on a test system are also provided inthis chapter.Chapter 4 is devoted to a comprehensive formulation of the VVO problem inactive smart grids. This chapter considers various VVO equipment and provides1With slight abuse of terminology, the term mixed-integer convex programming is used in thisthesis to refer to the problems that are non-convex merely due to the integrality of some variables.Therefore, these problems become convex if the integrality constraint is relaxed.121.5. Thesis Structurea mixed-integer convex formulation of the problem to be solved using branch-and-bound methods. To that end, the chapter uses the linearized DistFlow equationsfor radial distribution systems [8, 81] and extends them to accommodate the DGsand bi-directional flow of power. Also, an MC model is utilized in this chapter tomodel the stochasticity of the wind power generation. The chapter also presentssimulation results for VVO on an active smart grid and compares them for varioustest cases.Finally, chapter 5 concludes the thesis and provides some insights about thepossible future lines of research.It should be pointed out that the formulations provided throughout the thesisassume that the systems are balanced. However, in practice, the distribution systemsare often unbalanced. For an unbalanced distribution system, the formulationspresented in this thesis should be modified to include a full 3-phase power flowsolution. Nonetheless, the rest of the formulations remain the same.13Chapter 2Robust Meter Placement inReconfigurable Smart GridsThe active nature of future smart grids requires an online state estimation procedurefor system monitoring and situational awareness. The state estimation procedurecan be viewed as a building block of virtually all operational tasks in distributionsystems. That is because optimization of the system against any objective functionwould require up-to-date knowledge about the current state of the system. Stateestimation in current distribution systems mostly relies on substation measurementsand historical data. In future smart grids, however, the measurements from VMMs,PMUs, and smart meters will also be used for state estimation. Although the report-ing rate of the smart metering data may be around 15 to 60 minutes in the futuresmart grids, the current reporting rate is often much less (e.g. twice a day). This isbecause of concerns about customer privacy, exposure to frequency radiation, andcommunications bandwidth. This concept is schematically illustrated in Fig. 2.1.As a result, the smart metering data is currently insufficient for an accurate moni-toring of the smart grid. Therefore, it would make sense to install a limited numberof VMMs and/or PMUs in the distribution system to increase the grid visibility. Assuch, optimal meter placement in the system plays an important role in finding thebest trade-off between the upgrade costs and the state estimation accuracy. Thischapter presents a methodology for robust meter placement in reconfigurable smart14Chapter 2. Robust Meter Placement in Reconfigurable Smart Grids Smart Grid Operator Distribution System State Estimation VMM/PMU Higher Reporting Rate Lower Reporting Rate Figure 2.1: Schematic of a reconfigurable distribution system with smart meters,VMMs, and PMUs.grids [82].It should be pointed out that in distribution systems, the voltage angles withrespect to the slack node are typically small. However, the possibility of installa-tion of PMUs in distribution systems is included in this chapter for completeness.Even though the differences in the bus angles in distribution systems is small, anew generation of PMUs is under development that can measure these differences.Therefore, it is worthwhile to explore how much more accuracy in the state estima-tion is achievable if PMUs were to be placed in the system.This chapter is organized as follows. Sec. 2.1 describes the state estimationof the distribution systems using non-linear measurements. An expression for thecorresponding FIM is also provided in this section. Sec. 2.2 reviews the linearizedpower flow equations and the corresponding formulations for the meter placement152.1. State Estimation Using Non-Linear Measurements and the FIMproblem. Sec. 2.3 reviews the greedy algorithm for meter placement in a singleconfiguration distribution system. Sec. 2.4 considers the problem of meter place-ment in reconfigurable distribution systems. The submodular saturation algorithmis presented in this section. Sec. 2.5 presents the numerical results and discussions.Finally, Sec. 2.6 concludes this chapter.2.1 State Estimation Using Non-Linear Measurementsand the FIMThis section describes the state estimation of distribution systems using nonlinearmeasurements including substation measurements and smart metering data. Anexpression for the corresponding FIM is also provided.Consider a distribution system with N nodes, each having a voltage phasorvn = |vn|ejθn , n = 1, 2, . . . , N . Let v = [|v1|, |v2|, . . . , |vN |, θ1, θ2, . . . , θN ]′ be thestate of the system comprising the magnitudes and phases of all the voltages in thesystem. Here, [·]′ denotes transposition. The conventional substation measurementsand the measurements provided by smart meters are related to v through a nonlinearmodel as followszi = hi(v) + ηi, i = 1, 2, . . . ,M (2.1)where ηi ∼ N (0,Ri) is the Gaussian noise in the ith measurement andM is the totalnumber of non-linear measurements. The Maximum Likelihood Estimate (MLE) ofthe voltages, denoted by vˆ, can be found through the famous Weighted Least Squares(WLS) method as followsvˆ = arg minvM∑i=1[zi − hi(v)]′R−1i [zi − hi(v)]. (2.2)162.1. State Estimation Using Non-Linear Measurements and the FIMFor state estimation of the system one has to solve (2.2) using an iterative Newton-based algorithm.Let Z be the vector of all available data from substation measurements andsmart meters. The FIM matrix for the model in (2.1) is defined asJ(v) = Ep(Z|v){[∇v ln p(Z|v)]′[∇v ln p(Z|v)]}, (2.3)where p(Z|v) is the conditional probability density function of Z given v. Note thatin the literature of the power system state estimation, the FIM evaluated at v = vˆis usually referred to as the gain matrix. One can expand the FIM as follows (see[83] for this derivation)J(v) =M∑i=1[∇vhi(v)]′R−1i [∇vhi(v)]. (2.4)Eq. (2.4) shows that the FIM is the sum of M symmetric positive semi-definitematrices. We further assume that the system is fully observable such that J isindeed a positive definite, and hence, invertible matrix.The Crame´r-Rao Lower Bound (CRLB) establishes a bound on the conditionalcovariance matrix of the estimates based on the FIM. More precisely, C(vˆ|v)−J−1is guaranteed to be a positive semi-definite matrix. As a result, the diagonal entriesof J−1 establish a lower bound on the conditional estimation variances. To verifythis, let ur be a unit vector whose elements are zero except for the rth elementwhich is equal to one. It follows directly from CRLB thatu′r[C(vˆ|v)]ur ≥ u′r[J(v)]−1ur. (2.5)Because (2.5) holds for all r, the total conditional estimation variance is lower172.2. Meter Placement Using Linearized Power Flow Equationsbounded by trace of the inverse of the FIM. Therefore, one can design the me-ter places for minimization of the trace of J−1. However, the FIM depends on theactual state of the system, especially when the variance of the measurement noisesare proportional to the actual measured values. Therefore, a robust meter place-ment method should consider the worst case estimation variance among all possiblestates of the system. If the variances of the measurement noises are proportionalto the measured values, it will be shown in the following that the worst estimationvariance approximately corresponds to the peak loads. Hence, we design the meterplaces for peak loads.2.2 Meter Placement Using Linearized Power FlowEquationsAlthough the FIM of the nonlinear model (2.1) depends on the actual state of thesystem, the dependency of the functionsHi(v) = ∇vhi(v), (2.6)on v is very minimal, as mentioned in [84]. That is, the change in Hi(v) is verysmall as we start from the initial flat voltage point and iterate through the Newton’smethod. One way of interpreting this phenomenon is by considering the linearizedpower flow equations in rectangular coordinates.2.2.1 Linearized Power Flow Equations In RectangularCoordinatesLet the voltage of the nth node be represented in rectangular coordinates as vn =1 + v˜n = 1 + en + jfn. Without loss of generality, assume the first node is the slack182.2. Meter Placement Using Linearized Power Flow Equationsnode with a given voltage of 1 p.u.. Let v˜ = [v˜2, v˜3, . . . , v˜N ]′ denote the vector ofdeviations from the flat voltage profile at the remaining nodes of the system. Also,let e = [e2, e3, . . . , eN ]′ and f = [f2, f3, . . . , fN ]′ represent the real and imaginaryparts of v˜. These vectors can be approximated by linearized power flow equationsin rectangular coordinates provided that their elements are small [79]. For realdistribution systems it is a known fact that the elements of f are very small (i.e.,the voltage angles with respect to the slack node are very small.) The operationalconstraints of distribution systems also require that the elements of e remain small(e.g., within ±0.06 p.u.) at peak hours. At off-peak hours, the elements of e and fare even smaller which make the linearized power flow equations more accurate.Let YN×N = G + jB be the bus admittance matrix of the system where G andB are the bus conductance and bus susceptance matrices, respectively. Since G isa symmetric matrix, by separating its first row and column, we can partition it asfollows:G =g11 g′1g1 G˜ (2.7)where G˜(N−1)×(N−1) is a submatrix of the bus conductance matrix correspondingto the non-slack nodes and (·)′ denotes transposition. The same partitioning canalso be applied to the matrices Y and B to obtain the symmetric submatrices Y˜and B˜, respectively. Using these matrices and assuming that no shunt capacitor isinstalled in the system, the linearized power flow in rectangular coordinates can bewritten as [79]: pq−pDGqDG = A−1ef (2.8)where p = [p2, p3, . . . , pN ]′ and q = [q2, q3, . . . , qN ]′ are the vector of active and192.2. Meter Placement Using Linearized Power Flow Equationsreactive power consumptions of the nodes, respectively. Similarly,pDG = [pDG,2, pDG,3, . . . , pDG,N ]′,qDG = [qDG,2, qDG,3, . . . , qDG,N ]′are the vector of active and reactive power generation by local DGs, respectively.Moreover, A is defined as:A = G˜ −B˜−B˜ −G˜−1. (2.9)Rearranging the matrix form of the linearized power flow equations, (3.1) readsas:(p− jq)− (pDG − jqDG) = (G˜g − B˜f) + j(B˜g + G˜f)= (G˜ + jB˜)(g + jf), (2.10)or, equivalently,s∗ − s∗DG = Y˜v˜, (2.11)where s = p + jq is the vector of complex power consumption of the nodes, sDG =pDG+jqDG is the vector of complex power injections by local DGs, and (·)∗ denotescomplex conjugation. Eq. (2.11) is the complex form of the linearized equationsderived in [78] with appropriate modifications to account for DGs. In [78], theauthors have used this complex form to theoretically bound the approximation errorof the linearized power flow equations in rectangular coordinates.202.2. Meter Placement Using Linearized Power Flow Equations2.2.2 The Linearized Measurement ModelFrom (2.8) one can see that the gradients of the power consumptions with respectto g and f are constant (independent of the actual state of the system.) In addition,because the elements of f are very small, the magnitude of the voltages measuredby VMMs are quite close to the real part of the voltages. Therefore, the magnitudeof the voltage vn is approximately a linear function of gn whose gradient is equal toone.Even though the gradient of the nonlinear measurements is approximately in-dependent of the actual state of the system, the variance of the measurement noisecould depend on the actual state of the system. This will happen if the measure-ment error is a constant percentile of the measured values, as widely assumed in theliterature [17, 18, 20, 41, 47]. To see how this can impact the FIM, assume that themeasurement noises of different measured values are independent. Therefore, thecovariance matrices Ri are diagonal. Also, let Hi(v) be approximately independentof v. The FIM can now be expanded asJ =∑i∑r1σ2i,rhi,rh′i,r, (2.12)where σ2i,r is the rth diagonal element of Ri and hi,r is the rth column of Hi. It isseen from (2.12) that the FIM is a weighted sum of a series of Hermitian positivedefinite matrices. When σ2i,r increases, the single non-zero eigenvalue of the matrix1σ2i,rhi,rh′i,r decreases and all the zero eigenvalues remain the same. It follows thenfrom Weyl’s inequality that all the eigenvalues of J decrease as well. As a result, allthe eigenvalues of J−1 increase. Consequently, the total estimation variance whichis equal to the sum of the eigenvalues of J−1 increases as well. This shows thatthe maximum value of the total estimation variance corresponds to the maximum212.2. Meter Placement Using Linearized Power Flow Equationsmeasurement noises. If the variance of the measurement noises is proportional to themeasured values, then the maximum total estimation variance corresponds to thepeak loads. That is because in the peak hours the consumption powers can be severaltimes bigger than those of off-peak hours whereas the reduction in the voltages islimited. Our simulations also verify that hi(v) is approximately independent of vand the maximum estimation variance corresponds to peak hours. Finally, note thatthe same conclusions can be made if Ri are not diagonal. In that case, one has toexploit the eigenvalue decomposition of Ri. For the sake of brevity we omit thedetails of this case.Mathematically, designing the meter places for peak loads is equivalent to lin-earizing the measurement model given in (2.1) around vpeak as followszi = hi(vpeak) +Hi(v − vpeak) + ηi, (2.13)whereHi = ∇vhi(vpeak), (2.14)and vpeak is the vector of nodal voltages at peak hours. To realize this equivalence,define z˜i as follows:z˜i = zi − hi(vpeak) +Hivpeak. (2.15)Then the linearized model can be rewritten as:z˜i = Hiv + ηi. (2.16)It is readily seen that the FIM corresponding to the measurement model in (2.16)is equal to the FIM in (2.4) evaluated at v = vpeak, i.e., J(vpeak). In fact, in thislinearized model the conditional covariance matrix is independent of v and achieves222.3. Meter Placement in Single Configuration Distribution Systemsthe CRLB. That is,Clin(vˆ) = [J(vpeak)]−1, (2.17)where Clin(vˆ) is the unconditional covariance matrix of the linearized model.2.3 Meter Placement in Single ConfigurationDistribution SystemsIn this section the problem of optimal meter placement for minimization of thetotal estimation variance of nodal voltages in a single configuration distributionsystem is formulated. This formulation is identical to the one presented in [74]for optimal placement in transmission networks. The greedy algorithm [74, 75] isthen reviewed as a near-optimal meter placement scheme. Although the greedyalgorithm is presented here for meter placement in single configuration distributionsystems, it can also be used as a suboptimal algorithm for meter placement in activedistribution systems. Throughout the rest of the chapter a meter refers to a VMMor a PMU.2.3.1 Formulation of the Meter Placement ProblemA PMU measures the magnitude and the phase of the voltage of a node with highaccuracy [2, 19, 41]. The measurement model of the data provided by a PMUinstalled at the bus j ∈ {1, 2, . . . , N} is as follows:yj = Pjv + qj , (2.18)232.3. Meter Placement in Single Configuration Distribution SystemswherePj =e′j 00 e′j , (2.19)ej is a unit vector with a one as its jth element, and qj ∼ N (0,Qj) is the measure-ment noise.Similarly, a VMM measures the magnitude of the voltage of a node. Therefore,one should consider the same measurement model as in (2.18) for a VMM placed innode j with the following regression matrix:Pj =e′j 00 0 . (2.20)Let A be the set of nodes equipped with a meter (PMU or VMM). The MLE ofthe node voltages given substation measurements and the measurements providedby VMMs, smart meters and PMUs is found as follows:vˆ = arg minv{M∑i=1[zi − hi(v)]′R−1i [zi − hi(v)] +∑j∈A[yj − Pjv]′Q−1j [yj − Pjv]}.(2.21)For distribution system state estimation the MLE of the voltages should becomputed according to (2.21). However, the meter placement will be done so as tooptimize the trace of the inverse of the FIM at v = vpeak. As explained in Sec. 2.1,this is equivalent to linearizing the non-linear measurement models as in (2.16). Forthe linearized model, the MLE of the voltages is given by the following equationvˆlin = J−1lin[M∑i=1H ′iR−1i z˜i +∑j∈AP ′jQ−1j yj], (2.22)242.3. Meter Placement in Single Configuration Distribution Systemswhere Hi and z˜i are as defined in (2.14) and (2.15), respectively, andJlin =M∑i=1H ′iR−1i Hi +∑j∈AP ′jQ−1j Pj (2.23)is the FIM corresponding to the linearized model.The accuracy of the PMUs and VMMs is considerably higher than the historicaldata (pseudo-measurements) and smart metering data with relatively low reportingrates. Therefore, as in [74], we can assume that the MLE of the voltages of the meter-equipped nodes is equal to the actual measurements reported by the meter of thosenodes. That is, if a node is equipped with a VMM, the estimate of the magnitudeof the voltage of that node can be considered to be equal to the measured value.Similarly, if a node is equipped with a PMU, the magnitude and the phase of thevoltage of that node can be considered to be equal to the measured values. Denoteby Y the vector of all magnitude and phase measurements reported by the placedmeters. Also, denote by Ac the complement of the set A with respect to the set{1, 2, . . . , N}. LetHi be decomposed as [Hi,A|Hi,Ac ]. In the case of VMM placement,Hi,A corresponds to the magnitudes of the voltages of the VMM-equipped nodes andHi,Ac corresponds to the remaining magnitudes and phases. In the case of PMUplacement, Hi,A and Hi,Ac correspond to the columns of Hi associated with themagnitudes and phases of the PMU and non-PMU nodes, respectively. With thesenotations and for the linearized model in (2.16), the MLE of the voltages at thenon-meter nodes can be computed as:vˆlin[Ac] =[M∑i=1H ′i,AcR−1i Hi,Ac]−1×[M∑i=1H ′i,AcR−1i(z˜i − Hi,AY)]. (2.24)It is shown in [74] that for Qi = σ2I, the estimates obtained from this approach are252.3. Meter Placement in Single Configuration Distribution Systemsequal to those given in (2.22) as σ2 goes to zero.The FIM corresponding to the estimates in (2.24), which now coincides with theunconditional covariance matrix C(vˆlin[Ac]), is given byJlin[Ac] =M∑i=1H ′i,AcR−1i Hi,Ac . (2.25)According to (2.25), the total variance reduction due to installing meters at nodesj ∈ A is given byF (A) = tr([M∑i=1H ′iR−1i Hi]−1)− tr([M∑i=1H ′i,AcR−1i Hi,Ac]−1), (2.26)where tr(·) denotes the trace of the matrix. Therefore, the meter placement problemfor minimization of the total estimation variance (maximization of the reduction intotal estimation variance) can be formulated asA∗ = arg max|A|≤KF (A), (2.27)where K is the number of available meters. It should be pointed out that the aboveapproach is only adopted for formulation of the meter placement problem and theactual task of state estimation will still be done according to (2.21).Problem (2.27) is an NP-hard, combinatorial optimization problem. The com-plexity of finding the optimal solution to this problem is exponential in the numberof nodes. Since the distribution systems usually have a large number of nodes, onehas to resort to low complexity algorithms for finding suboptimal solutions to thisproblem. In the next subsection, a simple greedy algorithm for solving this problemis presented.262.3. Meter Placement in Single Configuration Distribution Systems2.3.2 Submodularity and the Greedy AlgorithmAs stated above, the meter placement problem (2.27) is an NP-hard problem. How-ever, due to the submodularity of the set function F (A), a simple greedy algorithmcan provide a near-optimal solution to this problem.Formally, a set function F : 2{1,2,...,N} → R is defined to be submodular if forany two sets A,B ⊆ 2{1,2,...,N} the following inequality holds:F (A ∪ B) + F (A ∩ B) ≤ F (A) + F (B). (2.28)Also, F is called monotonic if for any two sets A,B ⊆ 2{1,2,...,N} such that A ⊆ B wehave F (A) ≤ F (B). A set function F (A) is called antitonic, if −F (A) is monotonic.By expanding the FIM as in (2.4) it can be shown that the trace of J−1 is anantitonic function of the set of available measurements and, hence, the reduction inthe total estimation variance is monotonic. To see this, first note that all eigenvaluesof the FIM are greater than zero because it is positive-definite. Now, let JM andJM+1 be the FIM when there exist M and M + 1 measurement vectors in thesystem, respectively. Here, the FIM may include terms corresponding to both linearand nonlinear measurements. Let H(v) = [∇vhM+1(v)]. From (2.4) we can writeJM+1 = JM +H′(v)R−1M+1H(v), (2.29)where tr(·) denotes the trace of the matrix. It follows from the structure of thesecond term in the right-hand side of (2.29) that it is a positive-definite matrix.Therefore, based on Weyl’s inequality for the sum of two positive-definite Hermitianmatrices, the eigenvalues of the JM+1 are greater than the corresponding eigenvaluesof JM . As a result, the eigenvalues of J−1M+1 are smaller than the corresponding272.3. Meter Placement in Single Configuration Distribution Systemseigenvalues of J−1M . Therefore, we havetr(J−1M+1)≤ tr(J−1M). (2.30)Now consider the case where the estimate of the voltages measured by VMMs orPMUs are considered to be equal to the actual measurements. Let J be the FIMwhere no measurements from VMMs or PMUs are available. Also, let J˜ be theFIM corresponding to estimation of the remaining voltages when some voltages(magnitudes and/or angles) are measured. It then follows from Cauchy’s interlacingtheorem that the eigenvalues of J˜ are greater than the corresponding eigenvalues ofJ (note that the number of eigenvalues of J˜ is less than that of J). Therefore, wecan conclude thattr(J˜−1)≤ tr(J−1). (2.31)For a monotonic, submodular function satisfying F (∅) = 0, a classical resultby Nemhauser et. al. [75] shows that the greedy algorithm shown in Algorithm 1achieves a (1−1/e) approximation to the problem (2.27). That is, ifA∗ is the optimalsolution to the problem (2.27) and AG is the solution returned by Algorithm 1, thenF (AG) ≥ (1− 1/e)F (A∗). (2.32)Algorithm 1 GreedySpecify the number of meters KInitialize A = ∅while |A| < K doa∗ = arg maxa∈Ac{F (A ∪ {a})− F (A)}A ← A∪ {a∗}end whilereturn AFrom (2.26), it is obvious based on the structure of the total variance reduction282.4. Meter Placement in Reconfigurable Distribution Systemsfunction, F (A), that F (∅) = 0. Furthermore, if the columns of the matrixH =H1H2...HM(2.33)are nearly orthogonal, F (A) is a submodular function (see [74]). If the submodu-larity of F (A) holds, then the greedy algorithm results in a (1−1/e) approximationto the meter placement problem (2.27) [74]. Moreover, there exists no polynomialtime algorithm for a better approximation guarantee [85].2.4 Meter Placement in Reconfigurable DistributionSystemsIn this section the problem of meter placement for minimization of total estimationvariance of the magnitudes and phases of the voltages in a reconfigurable distributionsystem is considered. Unlike the meter placement problem for single configurationdistribution systems, the meter placement problem for active distribution systemsdoes not admit any approximation guarantee. Therefore, a robust algorithm formeter placement in reconfigurable distribution systems is proposed which outper-forms the greedy algorithm and GA in most cases and provides competitive resultsin other cases.2.4.1 Formulation of the Robust Meter Placement ProblemIn an active distribution system, the configuration of the system may change dueto opening/closing some tie line switches. Therefore, the meter placement problem292.4. Meter Placement in Reconfigurable Distribution Systemsshould be formulated in a robust fashion to take these changes into account. Inthe following, we formulate the robust meter placement problem for reconfigurabledistribution systems.The model for the ith measurement in the lth configuration of the system isgiven byzli = hli(v) + ηli. (2.34)Denote by vlpeak the vector of nodal voltages corresponding to peak load of the lthconfiguration of the system. LetH li = ∇vhli(vlpeak) (2.35)be the gradient of hli(v) evaluated at v = vlpeak. Decompose Hli as [Hli,A|H li,Ac ],where H li,A and Hli,Ac correspond to the columns of Hli associated with the magni-tudes and/or phases of the meter-equipped and non-meter nodes, respectively. Theminimization of the worst case total estimation variance is equivalent to maximizingthe reduction in total estimation variances compared to the worst initial configu-ration. With the above-mentioned notations, the reduction in the total estimationvariance of the lth configuration of the system with respect to the worst initialconfiguration is given by the following expression:Fl(A) = max0≤l≤L{tr([M∑i=1[H li ]′R−1i [Hli ]]−1)}−tr([M∑i=1[H li,Ac ]′R−1i [Hli,Ac ]]−1),(2.36)where tr(·) denotes the trace of the matrix. We further assume that the columns of302.4. Meter Placement in Reconfigurable Distribution SystemsHl =H l1H l2...H lM(2.37)are nearly orthogonal such that Fl(A) is a submodular function [74].Finally, the robust meter placement problem for minimization of the total esti-mation variance (maximization of the reduction in total estimation variance) in areconfigurable distribution system can be formulated as follows:A∗ = arg max|A|≤Kmin0≤l≤LFl(A), (2.38)where K is the number of available meters and L is the total number of possibleconfigurations of the system. Observe that by defining the functions in a specificway as in (2.36), the optimization problem in (2.38) is equivalent to minimizing theworst case total estimation variance of the system.Remark 2.1 An alternative way of formulating the meter placement problem ina reconfigurable distribution system is to assign probabilities to the configurationsand optimize the expected reduction in the total estimation variance. Let λl be theprobability that the system will have the lth configuration, where∑Ll=1 λl = 1. Then,the expected reduction in total estimation variance of the system is given by:E{F (A)} =L∑l=1λlFl(A), (2.39)where Fl(A), defined in (2.36), is the reduction in the total estimation varianceof the lth configuration of the system when meters are placed in the set A. Sincesubmodular functions are closed under convex combinations, it follows from the sub-312.4. Meter Placement in Reconfigurable Distribution Systemsmodularity of the functions Fl(A) that the expected reduction in the total estimationvariance is also a submodular function in A. Therefore, the (1− 1/e) performanceguarantee will be provided by the greedy algorithm. However, in this chapter weconsider the robust meter placement problem where the objective is to optimize theworst configuration.2.4.2 The Submodular Saturation AlgorithmIn this part, the submodular saturation algorithm for robust submodular optimiza-tion problems is presented. In [76], this algorithm has been successfully used forrobust sensor placement and observation selection over various temperature andprecipitation data sets. In this chapter, this algorithm is employed for robust meterplacement in reconfigurable distribution systems.The minimum of a set of submodular functions is not submodular in general.Therefore, the application of the greedy algorithm for the robust meter placementproblem (2.38) does not provide the (1 − 1/e) approximation guarantee. More im-portantly, it is shown in [76] that problem (2.38) does not admit any approximationguarantee. Therefore, we consider the following relaxed problem:A∗ = arg max|A|≤αKmin0≤l≤LFl(A), (2.40)where α ≥ 1 is a fixed parameter. The objective in (2.40) is to maximize the reduc-tion in the total estimation variance with respect to the worst initial configuration,where the number of available meters is increased with a factor of α. It is obviousthat for α = 1, (2.40) reduces to (2.38). Problem (2.40) can be reformulated in the322.4. Meter Placement in Reconfigurable Distribution Systemsfollowing equivalent formmax t (2.41)s.t. min0≤l≤LFl(A) ≥ t|A| ≤ αKNow suppose there exists an algorithm that for any t solves the following relatedproblemmin |A| (2.42)s.t. min0≤l≤LFl(A) ≥ t.Problem (2.42) seeks the minimum number of meters to be installed in the systemsuch that the reduction in the total estimation variances is greater than a thresholdt for all the configurations. If the solution to this problem satisfies the constraint|A| ≤ αK, then t is a feasible solution to the problem (2.41). On the other hand,if the the solution to the problem (2.42) has more elements than αK, then t is nota feasible solution to the problem (2.41). Our task is then to find the largest t forwhich problem (2.41) is feasible. It is evident that such a t is indeed the solution tothe problem. Now consider the following two extreme values for t:tmax = min0≤l≤LFl({1, 2, . . . , N})tmin = 0 (2.43)It is obvious that the solution to the problem (2.42) with t = tmin (which is A∗ = ∅)is a feasible solution to the problem (2.41). If the solution to the problem (2.42)332.4. Meter Placement in Reconfigurable Distribution Systemswith t = tmax is also a feasible solution to the problem (2.41), then the optimalsolution of the problem (2.41) is found. Otherwise, we need to do a binary searchbetween tmin and tmax to find the largest feasible t. This is the main idea of thesubmodular saturation algorithm [76].According to the above discussion, for solving (2.40) and (2.41), it remains tofind an algorithm to solve (2.42). The main difficulty with (2.42) is that it containsa non-submodular constraint. However, this constraint can be reformulated to bein a submodular form. The trick is to define a set of truncated functions as followsF ′l (A) = min{Fl(A), t}. (2.44)Note that if Fl(A) is monotonic and submodular, F ′l (A) will be monotonic andsubmodular, too. Now define the following average truncated functionF¯ ′(A) = 1LL∑l=1F ′l (A). (2.45)Since submodular functions are closed under convex combinations, F¯ ′(A) is sub-modular. Using above definitions, problem (2.42) can be written asmin |A| (2.46)s.t. F¯ ′(A) ≥ t.One can easily verify that problems (2.42) and (2.46) are equivalent. Problem (2.46),if feasible, is a submodular set covering problem as it is equivalent to the following342.4. Meter Placement in Reconfigurable Distribution Systemsproblem:min |A| (2.47)s.t. F¯ ′(A) = F¯ ′({1, 2, . . . , N}).Submodular set covering problems are NP-hard in general. However, a near-optimalsolution can be found by a greedy algorithm which starts with the empty set anditeratively adds elements with the best increment until F¯ ′(A) = F¯ ′({1, 2, . . . , N}).If the functions Fl(A), l = 1, 2, . . . , L, are integral-valued, a famous result due toWolsey shows that the solution obtained by greedy algorithm has a logarithmicperformance guarantee [86]. In practice, the greedy algorithm usually performsvery well even if the functions are not integral-valued.Having the greedy algorithm for solving problem (2.42), or its equivalent form(2.47), we are now ready to solve problem (2.40). The pseudo-code provided in thenext page presents the submodular saturation algorithm for solving this problem.If α is greater than a certain lower bound and the initial estimation accuracy isthe same for all the configurations, Algorithm 2 provides approximation guaranteesfor the relaxed problem (2.40), see [76]. If the initial estimation accuracy is notequal for all configurations, the performance guarantee provided in [76] requires aslight modification in the definition of the submodular functions. It is also shownin [76] that it is very unlikely that any other algorithm provides the same perfor-mance guarantee with a less restricting condition on α. Note, however, that theperformance guarantee provided by Algorithm 2 requires the functions Fl(A) to beintegral-valued. In our application for meter placement in active distribution sys-tems, the total reduction in variances is not an integral-valued function. However,one can round the functions with an arbitrary number of high order bits and thenrun Algorithm 2 for these approximate functions. Although the approximate func-352.4. Meter Placement in Reconfigurable Distribution SystemsAlgorithm 2 Submodular SaturationInitialize:tmin = 0tmax = min0≤l≤L Fl({1, 2, . . . , N})A = ∅Specify:A small non-negative tolerance δThe number of meters KThe number of configurations Lwhile tmax − tmin ≥ δ dot← tmax+tmin2Define F¯ ′(A) = 1L∑Ll=1 min{Fl(A), t}Let Aˆ = ∅while |Aˆ| < αK + 1 doa∗ = arg maxa∈Aˆc{F¯ ′(Aˆ ∪ {a})− F¯ ′(Aˆ)}Aˆ ← Aˆ ∪ {a∗}end whileif |Aˆ| ≤ αK thenA ← Aˆtmin ← telsetmax ← tend ifend whilereturn A362.4. Meter Placement in Reconfigurable Distribution Systemstions do not necessarily remain submodular in this case, the corresponding error inthe theoretical guarantees can be bounded as explained in [76] and [87]. In addition,observe that Algorithm 1 can also be used for meter placement in an active distribu-tion system. In this case, Algorithm 1 considers the minimum of all the submodularfunctions as a submodular function and greedily finds the meter places. In orderto fairly compare Algorithm 2 with Algorithm 1 for such a case, we need to runAlgorithm 2 for α = 1. Even though the performance guarantee of the Algorithm 2does not hold in this case, numerical results show that in most cases it outperformsAlgorithm 1 even for α = 1 and without rounding.Remark 2.2 In practice, the measurements provided by VMMs and PMUs mightinclude bad data. In such cases, the estimation error will be higher than what isconsidered in the meter placement problem, simply because the measurement erroris higher. However, the state-of-the-art state estimators usually perform a hypothesistest on the measurements for identifying and removing bad data. In addition, themetering network might face sensor failures due to various hardware or softwarecauses. From a mathematical point of view, these effects are equivalent and can becaptured in the robust meter placement problem as follows:A∗ = arg max|A|≤Kmin0≤l≤Lmin|B|≤KBFl(A\B), (2.48)where B is the set of bad data (or failed sensors) and KB is the maximum numberof bad data. Even thought Algorithm 2 can be directly applied to this problem, thecorresponding performance guarantee might become loose. Nonetheless, it is possibleto modify the definition of F¯ ′ to come up with a tighter performance guarantee,especially for small number of meters. We do not consider this case here and referthe interested reader to [76] for a full treatment of this approach.372.5. Numerical Results and Discussions2.5 Numerical Results and DiscussionsIn this section numerical results are provided that show the effectiveness of the sub-modular saturation algorithm (Algorithm 2) for meter placement in active distribu-tion systems. In all the simulations, Algorithm 1 considers the worst case objectivefunction among all configurations as a submodular function and greedily finds themeter places. Also, Algorithm 2 is employed with α = 1 and without rounding thesubmodular functions. The voltage magnitude and the active and reactive powerflows are assumed to be always available in the substation. In the case of PMUplacement, one PMU is always assumed in the substation as the reference node.The VMMs are assumed to measure the magnitudes of the voltages with a max-imum error of 1%. Also, the PMUs are assumed to measure the phasor of thevoltages with a maximum Total Vector Error (TVE) of 1%. Moreover, the activeand reactive consumption powers at the nodes are found by aggregating the mea-surements of the smart meters. A maximum error of 10% for such measurements isconsidered which can model the low reporting rate of the smart meters comparedwith the requirements of the real-time applications. Note that in future smart gridsthe state estimation may be done in real-time but the reporting rate of the smartmeters can be hourly. In some present distribution systems, the reporting rate ofthe smart meters is every few hours. The 10% error in the smart metering data iscompliant with the maximum signal dynamics considered in [88] for non-stationaryconditions. In real applications, one will have to consider the actual reporting rateof the smart meters to approximately adjust the measurement error of the smartmeters. The measurement noises have been considered to be white and Gaussianwith a standard deviation equal to one third of the maximum measurement error.The results of the proposed algorithms are compared with that of the GA forthree different test systems. In addition, the computational costs of the three algo-382.5. Numerical Results and Discussionsrithms are compared in terms of the number of times that the objective function isevaluated for each configuration of the system. For GA, the number of populationsand the number of generations are considered to be 200 and 500, respectively.In all the following test cases, each system has one primary substation whichfeeds several secondary transformers via feeders. Accordingly, the substation mea-surements refer to the measurements performed at the primary substation of thesystem.2.5.1 Meter Placement in a 33-node, 12.66 kV Active DistributionSystemIn this section a modified version of the 33-node, 12.66 kV distribution systempresented in [89] is considered. For this system, five weekly meshed configurationsare created by first opening all the tie lines and then closing one of the tie lines at atime. To make sure that these configurations satisfy the operational limits on nodalvoltages the maximum loads of the test system are reduced by 30%. Note that thismodification has to be done to the system as any practical configuration of a realdistribution system should satisfy these operational constraints.For the 33-node test system, the installation of up to 5 meters is considered.Table 2.1 lists the total estimation variances for VMM placement in the 33-nodetest system using Algorithm 1, Algorithm 2, and GA. The table suggests that Al-gorithm 2 outperforms Algorithm 1 in all cases. Moreover, in only one case the GAhas been able to find a better solution than Algorithm 2. Also, Table 2.2 shows thetotal estimation variance for PMU placement in the 33-node test system. The Tableshows that in all cases Algorithm 2 outperforms Algorithm 1 and GA.392.5. Numerical Results and DiscussionsTable 2.1: Worst-case Total Estimation Variance for VMM Placement in the 33-nodeActive Distribution SystemKAlgorithm 1(×10−6)Algorithm 2(×10−6)GA(×10−6)1 3.36 3.36 3.652 2.15 2.15 2.253 1.17 1.14 1.464 0.93 0.90 0.765 0.69 0.65 0.65Table 2.2: Worst-case Total Estimation Variance for PMU Placement in the 33-nodeActive Distribution SystemKAlgorithm 1(×10−7)Algorithm 2(×10−7)GA(×10−7)1 7.39 7.39 10.132 2.38 2.38 2.413 0.97 0.97 1.574 0.36 0.30 0.385 0.21 0.16 0.21Fig. 2.2 depicts the optimal location of five VMMs in the 33-node test systemusing Algorithm 1, Algorithm 2, and GA. Similarly, Fig. 2.3 shows the optimallocation of five PMUs in the test system using Algorithm 1, Algorithm 2, and GA.Note that Fig. 2.2 and Fig. 2.3 merely show the layout of the 33-node test systemand not its possible configurations.402.5. Numerical Results and Discussions 29 28 27 26 1 2 3 4 5 6 7 8 9 10 14 15 16 13 12 11 17 18 23 25 24 30 31 32 33 19 20 21 22 V3 V2 V2 V3 V4 V4 V5 V5 V1 V1 V1 V3 V2 V4 V5 Figure 2.2: Optimal location of five VMMs in the 33-node test system using Algo-rithm 1 (yellow boxes), Algorithm 2 (blue boxes), and GA (green boxes).412.5. Numerical Results and Discussions 29 28 27 26 1 2 3 4 5 6 7 8 9 10 14 15 16 13 12 11 17 18 23 25 24 30 31 32 33 19 20 21 22 P4 P1 P1 P1 P2 P2 P5 P3 P3 P3 P4 P4 P2 P5 P5 Figure 2.3: Optimal location of five PMUs in the 33-node test system using Algo-rithm 1 (yellow boxes), Algorithm 2 (blue boxes), and GA (green boxes).422.5. Numerical Results and Discussions2.5.2 Meter Placement in a 70-node, 11 kV Active DistributionSystemIn this section the robust meter placement problem for a modified version of anactive 70-node, 11 kV distribution system [90] is considered. For this test system,10 weekly meshed configurations are considered. These configurations are createdas follows. First, all the tie branches are opened except branches (9, 50), (21, 27),and (22, 67). The reason these three tie lines are kept closed is that, based on oursimulations, they are crucial for keeping the minimum voltage of the system withinoperational constraints. Then, two pseudo-random tie switches are closed at a time.These two tie switches are selected from a large set of tie switches such that theyprovide a better minimum voltage in the system. Note that in real applications onewould expect that those configurations be preferable as they meet the operationalconstraints and provide a better voltage profile. Finally, the maximum loads of thesystem are reduced by 20% such that all configurations of the system satisfy theoperational constraints on nodal voltages.For the 70-node test system, the robust placement of up to 9 meters is studied.Table 2.4 compares the performance of Algorithm 1, Algorithm 2, and GA for ro-bust VMM placement in the system. The table shows that Algorithm 2 provides abetter placement compared to Algorithm 1 and GA for all cases. Also, Table 2.5demonstrates the results for PMU placement in the test system. The table suggeststhat Algorithm 2 results in the best placement for K ≤ 7. For K = 8, 9, however,GA provides a better solution than Algorithm 1 and Algorithm 2.432.5. Numerical Results and DiscussionsTable 2.3: Worst-case Total Estimation Variance for VMM Placement in the 70-nodeActive Distribution SystemKAlgorithm 1(×10−6)Algorithm 2(×10−6)GA(×10−6)1 4.76 4.76 4.782 3.45 3.45 3.493 3.13 3.13 3.284 2.92 2.89 3.105 2.83 2.80 3.006 2.77 2.74 2.907 2.72 2.64 2.838 2.62 2.60 2.759 2.45 2.44 2.62Table 2.4: Worst-case Total Estimation Variance for PMU Placement in the 70-nodeActive Distribution SystemKAlgorithm 1(×10−6)Algorithm 2(×10−6)GA(×10−6)1 3.53 3.53 3.582 1.29 1.29 1.423 0.87 0.87 0.994 0.61 0.61 0.715 0.46 0.41 0.556 0.40 0.22 0.517 0.37 0.14 0.218 0.13 0.11 0.089 0.08 0.08 0.06Fig. 2.4 depicts the optimal location of 9 VMMs in the 70-node test system usingAlgorithm 1, Algorithm 2, and GA. Similarly, Fig. 2.5 shows the optimal locationof 9 PMUs in the test system using Algorithm 1, Algorithm 2, and GA. Note thatFig. 2.4 and Fig. 2.5 merely show the layout of the 70-node test system and not itspossible configurations.442.5.NumericalResultsandDiscussions 1 2 3 4 5 6 7 89 10 11 12 13 14 15 30 31 32 33 34 35 36 37 3847 48 49 50 39 40 41 42 43 44 45 46 23 24 25 26 27 28 29 68 69 16 17 18 19 20 21 2251 52 53 54 55 56 61 62 63 64 67 66 65 57 58 59 60 V1 V1 V1 V4 V4 V4 V2 V2 V2 V3 V3 V3 V6 V6 V6 V5 V5 V5 V7 V7 V7 V8 V8 V8 V9 V9 V9 Figure 2.4: Optimal location of 9 VMMs in the 70-node test system using Algorithm 1 (yellow boxes), Algorithm 2 (blueboxes), and GA (green boxes).452.5.NumericalResultsandDiscussions 1 2 3 4 5 6 7 89 10 11 12 13 14 15 30 31 32 33 34 35 36 37 3839 40 41 42 43 44 45 46 23 24 25 26 27 28 29 68 69 16 17 18 19 20 21 2251 52 53 54 55 56 61 62 63 64 67 66 65 57 58 59 60 P1 P1 P1 P2 P2 P3 P3 P3 P4 P4 P4 P2 P5 P5 P7 P7 P7 P8 P8 P6 P6 47 48 49 50 P9 P9 P5 P6 P8 P8 Figure 2.5: Optimal location of 9 PMUs in the 70-node test system using Algorithm 1 (yellow boxes), Algorithm 2 (blueboxes), and GA (green boxes).462.5. Numerical Results and Discussions2.5.3 Meter Placement in an 119-node, 11 kV Active DistributionSystemIn this section the problem of optimal meter placement for state estimation in amodified version of the 119-node, 11 kV distribution system presented in [91] isconsidered. For this system, first all the tie lines except for the two tie lines (110, 118)and (75, 88) are opened. Then, 30 weekly meshed configurations are created byclosing three pseudo-random tie lines at a time. These three pseudo-random tielines are obtained from a large set of random lines and by choosing those tie lineswhich result in a better minimum voltage along the feeder. Finally, the loads of thetest system are reduced by 20% to make sure that all the configurations satisfy theoperational constraints.For the 119-node test system, the optimal placement of up to 9 meters has beenstudied. The results of the VMM placement using Algorithm 1, Algorithm 2, andGA are presented in Table 2.6. The table suggests that for five cases Algorithm 1outperforms Algorithm 2, for two cases Algorithm 2 outperforms Algorithm 1, andfor two cases the results are identical. Moreover, both algorithms outperform GAfor all values of K. Also, Table 2.7 compares the performance of Algorithm 1,Algorithm 2, and GA for PMU placement in the test system. It is observed fromthe table that for all values of K Algorithm 2 outperforms Algorithm 1 and GA.472.5. Numerical Results and DiscussionsTable 2.5: Worst-case Total Estimation Variance for VMM Placement in the 119-node Active Distribution SystemKAlgorithm 1(×10−6)Algorithm 2(×10−6)GA(×10−6)1 13.44 13.44 13.532 10.50 10.50 11.763 8.83 8.92 9.804 7.78 7.85 9.485 6.73 6.84 7.796 5.90 5.63 6.427 5.00 4.91 5.848 4.19 4.24 5.119 3.82 3.87 4.76Table 2.6: Worst-case Total Estimation Variance for PMU Placement in the 119-node Active Distribution SystemKAlgorithm 1(×10−6)Algorithm 2(×10−6)GA(×10−6)1 11.42 11.42 11.502 6.30 6.30 6.323 3.84 3.76 3.924 2.74 2.65 3.895 1.86 1.79 3.876 1.39 1.17 1.447 0.93 0.75 0.808 0.50 0.41 0.749 0.35 0.33 0.69Fig. 2.6 illustrates the optimal location of 9 VMMs in the 119-node test systemusing Algorithm 1, Algorithm 2, and GA. Similarly, Fig. 2.7 shows the optimallocation of 9 PMUs in the test system using Algorithm 1, Algorithm 2, and GA.Note that Fig. 2.6 and Fig. 2.7 merely show the layout of the 119-node test systemand not its possible configurations.482.5.NumericalResultsandDiscussions   10 11 12 13 14 15 16 17 1 2 4 5 6 7 8 9 28 29 30 31 32 33 34 35 38 39 40 41 42 43 44 45 46 100 101 102 103 104 105 106 107 108 109 110 111 114 115 116 117 118 119 112 96 97 98 99 89 90 91 92 93 94 95 86 87 88 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 18 19 20 21 22 23 24 25 26 27 78 79 80 81 82 83 84 85 36 37 55 56 57 58 59 60 61 62 3 V5 V1 V1 V2 V2 V6 V6 V9 V9 V9 V7 V7 V8 V8 48 49 50 51 47 52 53 54 V4 V4 V3 V3 V5 V5 V2 V3 V4 V6 V8 V7 V1 Figure 2.6: Optimal location of 9 VMMs in the 119-node test system using Algorithm 1 (yellow boxes), Algorithm 2 (blueboxes), and GA (green boxes).492.5.NumericalResultsandDiscussions   10 11 12 13 14 15 16 17 1 2 4 5 6 7 8 9 28 29 30 31 32 33 34 35 38 39 40 41 42 43 44 45 46 100 101 102 103 104 105 106 107 108 109 110 111 114 115 116 117 118 119 112 96 97 98 99 89 90 91 92 93 94 95 86 87 88 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 18 19 20 21 22 23 24 25 26 27 78 79 80 81 82 83 84 85 36 37 55 56 57 58 59 60 61 62 3 P8 P1 P1 P2 P2 P6 P6 P9 P9 P9 P7 P7 48 49 50 51 47 52 53 54 P4 P4 P3 P5 P5 P3 P3 P4 P6 P5 P7 P1 P8 P8 P2 Figure 2.7: Optimal location of 9 PMUs in the 119-node test system using Algorithm 1 (yellow boxes), Algorithm 2 (blueboxes), and GA (green boxes).502.5. Numerical Results and Discussions2.5.4 Computational CostThe number of objective function evaluations in Algorithm 1 is (slightly less than)KN per configuration. On the other hand, in the simulations conducted on all threetest systems, Algorithm 2 always converged in less than 11 iterations (in the caseof 33-node test system the algorithm often converged in 8 iterations). Therefore,the total number of function evaluations in Algorithm 2 is no more than 11KN perconfiguration. The interested reader is referred to [76] for a general upper bound onthe number of function evaluations of Algorithm 2.The number of function evaluations by GA is the number of populations timesthe number of generations which is equal to 200× 500 = 100000 in our simulations.Therefore, Algorithm 1 always has the minimum computational cost. Moreover,based on the parameters used in our simulations and the size of the test systems, thenumber of function evaluations of Algorithm 2 is always less than 11×10×119200×500 = 13%of that of the GA.To give an idea about the computational time of the proposed algorithms, theCPU time of the simulations are reported here. The simulations are performed onMATLAB using a computer with a processor of 3.4 GH and 4 GB of RAM. TheCPU time for all the test systems were less than one minute for Algorithm 1, lessthan 4 minutes for Algorithm 2, and between 15 to 34 minutes for the GA.2.5.5 DiscussionAs shown in the numerical results over the three test systems, the submodularsaturation algorithm outperforms the greedy algorithm in most cases and providescompetitive results in other cases. These results are similar to the results reported in[76] where the submodular saturation algorithm outperforms the greedy algorithmfor robust observation selection over various data sets.512.6. ConclusionThe GA has several parameters to tune and its performance depends on theseparameters. The submodular saturation algorithm, on the other hand, does nothave any parameter to tune which makes it very favorable for an easy implemen-tation. In many cases, the submodular saturation algorithm was capable of findinga better solution than GA even with a much less number of function evaluations.Therefore, the submodular saturation algorithm serves as a powerful tool for themeter placement problem in active distribution systems.The meter placement problem in a reconfigurable distribution system is an NP-hard problem which does not admit any approximation guarantee [76]. Moreover,as shown in our numerical results, none of the algorithms beats others in all cases.Nonetheless, note that the meter placement problem is a design problem which hasto be solved only once. Therefore, for a given test system and a given number ofmeters to be installed, one can compare the results of the submodular saturationalgorithm with that of the greedy algorithm and GA to obtain the best placement.2.6 ConclusionIn this chapter, we studied the problem of robust meter placement for state esti-mation in active distribution systems. The trace of the inverse of the FIM waschosen as criterion for the estimation accuracy. A robust meter placement algo-rithm, called submodular saturation algorithm, was proposed to optimize the worstcase estimation accuracy among all possible configurations of the system. The re-sults of the meter placement problem on three active distribution systems showedthat the submodular saturation algorithm outperforms the greedy algorithm andthe GA in most cases and provides competitive solutions in other cases. The nu-merical results presented show that a high level of accuracy in the state estimationtask can be achieved with a much less number of PMUs than voltage magnitude522.6. Conclusionmeters indicating the possible application of PMUs for state estimation in futuresmart grids.53Chapter 3Distributed Storage Planning inSmart Distribution GridsDistributed Storage Systems (DSS) are, conceptually, the storage units that can beinstalled on any node of the system independent of the location of the distributedgenerators. The DSS can play important roles in improving the performance ofmodern smart grids. These roles include coping with the inherent intermittenceof non-dispatchable energy sources, enhancing the system resilience, reducing thepossibility of energy curtailment, peak load shaving, energy price arbitrage, andpower loss reduction. Toward achieving those goals in the most efficient way, it iscrucial to find the best capacity, power rating, and location of DSS in the system.This chapter proposes a methodology for optimal DSS planning in smart distributionsystems.This chapter is organized as follows. Sec. 3.1 provides some review about thelinearized power flow equations in distribution systems and a quadratic form for theloss function. Sec. 3.2 formulates the various financial gains due to installation ofDSS in distribution systems. Sec. 3.3 presents a mixed-integer convex formulationof the problem of distributed storage planning in smart distribution systems. Sec.3.4 provides the numerical results of the optimal DSS placement in a test system.Finally, Sec. 3.5 concludes the chapter.543.1. Linearized Power Flow Equations and the Loss Function3.1 Linearized Power Flow Equations and the LossFunctionRecall from Sec. 2.2.1 that the system voltages v = 1 + e + jf , can be found usinglinearized power flow equations in rectangular coordinates as:pq−pDGqDG = A−1ef (3.1)whereA = G˜ −B˜−B˜ −G˜−1, (3.2)and G˜ and B˜ are, respectively, sub-matrices of the bus conductance and susceptancematrices corresponding to the non-slack nodes.Using linearized power flow equations, the active power loss in the system canalso be approximated as a quadratic function of power injections. To see this, letus first define the net demand vector d as:d =pq−pDGqDG . (3.3)Based on (3.3), the linearized equations (3.1) can be written in the following compactform: ef = Ad. (3.4)Using (3.4), one can show that the total active power loss in the system can bewritten as a quadratic function of the net demand vector. Particularly, if L(d) isthe total active power loss of the system as a function of the net demand vector,553.1. Linearized Power Flow Equations and the Loss Functionthen we have:L(d) =12d′Md, (3.5)whereM = 2A′G˜ 00 G˜A, (3.6)is a 2(N − 1) × 2(N − 1) real matrix that depends merely on the bus conductanceand bus susceptance matrices of the system. Note that since G˜ is symmetric, M issymmetric too.Eq. (3.5) can be established using the approach presented in [92]. For easeof notation define the augmented vectors eˆ = [0 e′]′, fˆ = [0 f ′]′, and pˆ =[pslack, p2 − pDG,2, p3 − pDG,3, . . . , pN − pDG,N ]. Also, let iˆ be the vector of currentinjections to the system. If the voltage of the slack node is 1 p.u., the active powerloss in the system can be written as:L = 1′pˆ= <{(1 + eˆ + j fˆ)′ˆi∗}= <{(1 + eˆ + j fˆ)′[Y(1 + eˆ + j fˆ)]∗}= <{(eˆ + j fˆ)′Y∗(eˆ− j fˆ)} (3.7)= <{(eˆ + j fˆ)′(G− jB)(eˆ− j fˆ)}= eˆ′Geˆ + fˆ ′Gfˆ= e′G˜e + f ′G˜f , (3.8)where (3.7) follows from the fact that Y∗1 = 0 and 1′Y∗ = 0. This is because theshunt admittances are assumed to be zero. Note also that (3.8) follows directly fromthe definition of vectors eˆ and fˆ along with the partitioning of G as in (2.7).563.1. Linearized Power Flow Equations and the Loss FunctionRemark 3.1 The shunt capacitors are assumed to be zero in the linearized powerflow equations (3.1). That is because the reactive power injected by shunt capacitorscan be modeled as negative components in the reactive sub-vector of the net demandvector d. As a result, in derivation of the quadratic form of the loss function in(3.5), we considered Y∗1 = 0 and 1′Y∗ = 0.Eq. (3.8) can be written in matrix form as:L(e; f) =ef′ G˜ 00 G˜ef . (3.9)Eq. (3.5) then follows by substituting the the linearized power flow equations (3.4)into (3.9).Using (3.5) and (3.6), it can be shown that the loss function L(d) is convex ind. Since ∇2L(d) = M, the convexity of L(d) can be established by demonstratingthat M is positive semi-definite. To that end, first note that the system admittancematrix is positive semi-definite, that is G  0. In [92], this has been shown usingthe fact that the active power loss in the system is non-negative for all values ofthe nodal voltages. Next, note that for a positive semi-definite matrix, all theprincipal submatrices have to be positive semi-definite. Therefore, it follows fromthe positive semi-definiteness of G that G˜ is positive semi-definite, too. That is, forany x ∈ RN−1, we have x′G˜x ≥ 0. Similarly, for any x and y belonging to RN−1we have:xy′ G˜ 00 G˜xy = x′G˜x+ y′G˜y≥ 0. (3.10)573.1. Linearized Power Flow Equations and the Loss FunctionFor any Hermitian, positive-definite matrix there exists a decomposition into theproduct of a lower triangular matrix and its conjugate transpose. This decom-position is called Cholesky decomposition or Cholesky factorization. Since G˜ issymmetric and real, the following Cholesky decomposition exists:G˜ 00 G˜ = UU′, (3.11)where U is a real, lower triangular matrix with positive diagonal entries. Therefore,for any z ∈ RN−1, we have:z′Mz = 2z′ATUU′Az= 2(U′Az)′(U′Az)= 2‖U′Az‖22 ≥ 0, (3.12)which demonstrates that M is positive semi-definite.To illustrate the effectiveness of the linearized power flow equations, we considera modified version of the test system presented in [90]. For this system, inclinedsolar cells are considered on half of the nodes selected randomly. Moreover, two windturbines are installed on the nodes 21 and 50 with power ratings of 400 kW and 800kW, respectively. Real data of the loads, wind and solar generations are exploitedfor this illustration. Details about the test system and the real data used will beprovided in Sec. 3.4. All the numbers reported here are obtained by simulatingthe system for 535 days, 24 hours each, with a total number of simulations equalto 535 × 24 = 12840. For each simulation, the error of the linearized power flowequations is computed in comparison with the Newton’s AC Power Flow (ACPF)based on five different metrics. These metrics are Average Magnitude Error (AME)583.1. Linearized Power Flow Equations and the Loss Functionin p.u., Average Angle Error (AAE) in degrees, Maximum Magnitude Error (MME)in p.u., Maximum Angle Error (MAE) in degrees, and Normalized Loss Error (NLE).In particular, these metrics are defined as follows:AME =1NN∑n=1∣∣∣|vn,lin| − |vn,ACPF|∣∣∣AAE =1NN∑n=1∣∣∣∠vn,lin − ∠vn,ACPF∣∣∣MME = maxn∣∣∣|vn,lin| − |vn,ACPF|∣∣∣MAE = maxn∣∣∣∠vn,lin − ∠vn,ACPF∣∣∣NLE =∣∣∣Llin − LACPF∣∣∣LACPF, (3.13)where the subscript lin indicates the solution of the linearized power flow equations.For the test system under study and using the real data of loads, wind, and solargeneration, the AME and AAE indexes averaged over all simulations are 3.84×10−4p.u. and 1.04 × 10−3 degrees, respectively. Also, the MME and MAE indexesaveraged over all simulations turn out to be 1.45 × 10−3 p.u. and 9.31 × 10−3degrees, respectively. Moreover, the NLE averaged over all simulations turns out tobe 3.05%. These numbers show that the linearized power flow equations can providegood approximations in terms of nodal voltages and system power losses.Fig. 3.1 illustrates the magnitudes and angles of the system voltages obtainedfrom linearized equations and ACPF and by using real data of loads, wind, and solarpower. The voltages plotted in this figure correspond to the peak hour of a typicalday of light loading. As it can be seen from this figure, the voltages obtained fromlinearized power flow equations match that of the ACPF very well. Also, Fig. 3.2depicts the magnitude and angles of the system voltages obtained using linearized593.2. Formulation of the Economic Gains of the Distributed Storage Systems10 20 30 40 50 600.9811.021.04Magnitudes (p.u.)  ACPFLPF10 20 30 40 50 60-0.500.511.52Node NumberAngles (deg.)  ACPFLPFFigure 3.1: Magnitudes and angles of the nodal voltages in the test system obtainedusing linearized power flow equations and ACPF for the peak hour of a typical daywith light loading.power flow equations and ACPF for peak hour of a typical day of high loading.Although the linearized power flow equations introduce a higher error in this case,they still provide a good approximation compared with the ACPF.3.2 Formulation of the Economic Gains of theDistributed Storage SystemsThis section formulates the various economic gains of the DSS for smart distributiongrids. Due to the stochasticity of the load, wind and solar generations, the expectedeconomic gain of the SGO due to installation of storage units is considered. Similarlyto [58] and [93], it is assumed that the storage units are charged and discharged oncea day. Therefore, for each day there is one charging cycle corresponding to off-peakhours and one discharging cycle corresponding to peak hours.In the proposed methodology for optimal DSS planning, the planning horizon603.2. Formulation of the Economic Gains of the Distributed Storage Systems10 20 30 40 50 600.920.940.960.981Magnitudes (p.u.)  ACPFLPF10 20 30 40 50 60-0.500.511.52Node NumberAngles (deg.)  ACPFLPFFigure 3.2: Magnitudes and angles of the nodal voltages in the test system obtainedusing linearized power flow equations and ACPF for the peak hour of a typical daywith high loading.consists of Y years and each year is divided into S segments. For hour h of segment s,let pDSSs,h be the vector of charging/discharging powers by the storage units installedin the system. Even though pDSSs,h is a 2(N − 1)× 1 vector, throughout this chapterit is assumed that the DSS are operated in a way that they only absorb and injectactive power. Therefore, it is implicitly assumed that the last N − 1 elements ofpDSSs,h are zero for all s and h. If a node is not equipped with a storage unit, theformulation trivially requires the active powers to be zero at all time. In general,the charging/discharging strategy of each storage unit can be optimized not only foreach hour and each segment, but also for each year. However, we restrict ourselves tothe case where the optimized strategy for each hour and segment remains the samefor all years. That is because even though the load and renewable generations growover years, the capacity of the DSS remains fixed for the whole planning horizon.Nonetheless, the proposed methodology can simply accommodate the case wherethe DSS strategy is optimized for each year, in addition to optimizing for each hour613.2. Formulation of the Economic Gains of the Distributed Storage Systemsand each segment. We will denote the operation strategy of the DSS in the sthsegment of each year bypis =pDSSs,1pDSSs,2...pDSSs,24. (3.14)Also, the matrix of DSS operation strategies for the whole planning horizon will bedenoted by Π = [pi1,pi2, . . . ,piS].3.2.1 The Arbitrage GainTo evaluate the arbitrage gain of the SGO due to installation of DSS, a Time ofUse (ToU) pricing scheme is considered in this chapter. Let ηys,h be the price ofelectricity per kWh at hour h of the sth segment of the yth year of the planninghorizon. The average arbitrage gain in the yth year can be written as:Γarby (Π) =365SS∑s=1{ ∑h∈Hpηys,h1′pDSSs,h −∑h∈Hoηys,h1′pDSSs,h}, (3.15)where Hp is the set of candidate discharging hours (peak hours) andHo = {1, 2, . . . , 24} \ Hpis the set of candidate charging hours (off-peak hours). Observe from (3.15) thatthe average arbitrage gain of the SGO is linear in Π.3.2.2 The Expected Reduction in Active Power LossIn this section, the expected economic gain of the SGO due to reduction in the activepower loss is formulated. The expected daily power loss of the system in the sth623.2. Formulation of the Economic Gains of the Distributed Storage Systemssegment of the yth year as a funciton of the DSS operation strategy can be writtenas:E{24∑h=1Lys,h(pDSSs,h)}=∑h∈HoE{L(dys,h + pDSSs,h)}+∑h∈HpE{L(dys,h − pDSSs,h)},(3.16)where the L(·) function is defined in (3.5) and E{·} denotes the expectation operator.Here, dys,h is the vector of stochastic net demands at hour h of the sth segment ofthe yth year. Expanding (3.16) using (3.5) and considering the ToU pricing scheme,the expected daily cost due to the active power loss in the sth segment of the ythyear can be written as:E{24∑h=1ηys,hLys,h(pDSSs,h)}=1224∑h=1ηys,h[pDSSs,h]′M[pDSSs,h]+∑h∈Hoηys,h[d¯ys,h]′M pDSSs,h−∑h∈Hpηys,h[d¯ys,h]′M pDSSs,h +24∑h=1ηys,hcys,h, (3.17)where d¯ys,h = E{dys,h}, andcys,h = E{Lys,h (0)}=12E{[dys,h]′ M [dys,h]}, (3.18)is the expected hourly power loss in the sth segment of the yth year when no DSS isinstalled in the system.Leveraging the LLN, it is possible to use the real data of the loads and renewablegenerations to approximate d¯ys,h with the empirical mean which is an unbiased esti-mator. Let d0s,h(l) be the lth random element of the net demand vector d0s,h at thefirst year. Also, assume that at the planning time a total number of Ks real data633.2. Formulation of the Economic Gains of the Distributed Storage Systemspoints of the load and renewable generation is available for each hour of segment s.With slight abuse of notation, let d0s,h(l , k) be the kth measured data correspondingto the lth element of the net demand vector at hour h and segment s. Then, basedon LLN, d¯0s,h(l) can be approximated as:d¯0s,h(l) = E{d0s,h(l)} ≈1KsKs∑k=1d0s,h(l , k). (3.19)In a similarly way, one can approximate the expected hourly power loss in the systemwhen no DSS is installed. Note, however, that computation of cys,h is not necessaryfor the optimal DSS placement problem as will be explained soon. Nonetheless, therelevant details are provided here for completeness. Using real data of loads andrenewable generation, the expected hourly power loss at the planning year can beapproximated as:c0s,h =12E{[d0s,h]′ M [d0s,h]}=122(N−1)∑l=12(N−1)∑r=1ml ,rE{d0s,h(l)d0s,h(r)}≈ 12Ks2(N−1)∑l=12(N−1)∑r=1Ks∑k=1ml ,rd0s,h(l , k)d0s,h(r , k) (3.20)where ml ,r is the (l , r) element of M.Computation of the above-mentioned expectations for the whole planning hori-zon requires the statistics of the vector of stochastic net demands in future years.This can be computed based on its current statistics as well as the anticipatedgrowth rate of the load and renewable generations. Let the net demand data pointsat the planning time be decomposed as d0s,h(l , k) = d0,loads,h (l , k) − d0,DGs,h (l , k), cor-responding to the load and DGs. Assume a fixed annual growth rate of γload and643.2. Formulation of the Economic Gains of the Distributed Storage SystemsγDG for the the load and renewable generation, respectively. Then d¯ys,h and cys,h fory ≥ 1 can be computed by (3.19) and (3.20), respectively, with d0s,h(l , k) replacedby dys,h(l , k) = [γload]yd0,loads,h (l , k)− [γDG]yd0,DGs,h (l , k).Finally, the expected economic gain of SGO due to the reduction in the activepower loss in the yth year of the planning horizon is given by:Γlossy (Π) =365SS∑s=124∑h=1ηys,hcys,h − E{24∑h=1ηys,hLys,h(pDSSs,h)}. (3.21)Observe from (3.21) and (3.17) that the terms corresponding to cys,h cancel out inΓlossy . Therefore, Γlossy contains only quadratic and linear terms of Π. This is indeedexpected as Γlossy (0) should be equal to zero. In other words, when no DSS is installedin the system, there will be no economic gain. Also note that the Hessian of Γlossy (Π)with respect to Π is equal to −1S ×diag{ηy1,1M, ηy2,1M, . . . , ηyS,1M, . . . , ηyS,24M}, whichis negative semi-definite, because M  0 and ηys,h > 0,∀h,∀s, ∀y. Therefore, Γlossy (Π)is concave in Π.3.2.3 The Reduction in Expected Price of Renewable EnergyCurtailedDue to the intermittent nature of renewable energy sources, there may be timesthat the total generation of the renewable sources exceeds the total demand of thesystem. In such cases, the excess generation can flow to the sub-transmission systemas long as the system constraints are satisfied. However, at times when the systemconstraints are about to be violated due to excess distributed generation, the SGOmay have to curtail the renewable generation. Such a case, particularly, happenswhen high amounts of non-dispatchable generation causes over-voltage problems inthe system. In contrast, if the system is equipped with DSS, they can be optimally653.2. Formulation of the Economic Gains of the Distributed Storage Systemsoperated so as to improve the voltage profile and reduce the chance of over-voltageproblems. As a result, in addition to the gains that the SGO obtains due to pricearbitrage and reduction in the power loss, it may also benefit from reduction inthe renewable energy spillage [25]. It is, however, difficult to explicitly formulatethe expected economic value of the spilled energy as a function of the amount ofDSS installed and their operation strategy. Instead, an indirect approach is takenin this chapter. Here, we regularize the objective function of the DSS placementwith a virtual term associated with the improvement in the voltage profile. Withthis virtual benefit included in the objective function, the DSS placement routinewill try to avoid the violation of voltage constraints. Therefore, the resulting DSSplacement and strategy is expected to lower the amount of spilled energy. Oncethe storage units are optimally placed in the system and their operation strategy isoptimized, the expected economic value of the spilled energy can be approximatedthrough Monte Carlo simulations. To that end, a power flow method is performedfor each hour of the available dataset to see if any of the constraints are violatedin the system with and without DSS. If so, an optimal curtailment strategy will beused to find the minimum amount of renewable energy that needs to be curtailed,as will be explained later in this section. Finally, the expected gain in the price ofenergy curtailed will be approximated based on LLN. The average economic gainbecause of lower energy curtailment due to installation of DSS in year y will bedenoted by Γcurty (Π).To formulate the regularized objective function, first note that in distributionsystems the imaginary part of the voltages are very small (the voltage angles arevery small). Therefore, one can approximate the magnitude of the nodal voltagesby their real parts. As a result, the total expected deviation in the real parts ofthe nodal voltages, with respect to the flat voltage profile, can serve as an index663.2. Formulation of the Economic Gains of the Distributed Storage Systemsfor evaluation of the voltage profile. Now consider the following partitioning for thematrix A defined in (2.9):A =AeAf . (3.22)where Ae and Af are (N − 1) × 2(N − 1) matrices corresponding to the real andimaginary parts, respectively. Let eys,h(pDSSs,h ) be the vector of deviations in the realparts of the nodal voltages at hour h, segment s, and year y. Similarly to theway that the expected reduction in the power loss was computed in Sec. 3.2.2,the average improvement in the voltage profile during the planning horizon due toinstallation of DSS can be formulated as:Γvol(Π) =365SY∑s,yE{24∑h=1∥∥eys,h(0)∥∥2 − ∥∥eys,h(pDSSs,h )∥∥2}=−1SY∑s,y24∑h=1[pDSSs,h]′A′eAe[pDSSs,h]+ 2∑h∈Ho[d¯ys,h]′ A′eAe[pDSSs,h]− 2∑h∈Hp[d¯ys,h]′A′eAe[pDSSs,h]. (3.23)In (3.23), d¯ys,h can be approximated using LLN as in (3.19). Observe that the Hessianof Γvol(Π) with respect to Π is equal to −2SY ×diag{A′eAe,A′eAe, . . . ,A′eAe} whichis negative semi-definite, because A′eAe  0 by structure. Therefore, Γvol(Π) isconcave in the operation strategy of the DSS.Once the optimal DSS planning strategy is obtained for the system, the expectedprice of renewable energy curtailed will be computed for the system with and withoutDSS. Without loss of generality, assume that the wind turbines are candidates ofenergy curtailment as they are more likely to be utility-owned. Nonetheless, theformulation presented in this part can simply accommodate the curtailment of solargeneration as well. Using linearized power flow equations, the optimal curtailment673.2. Formulation of the Economic Gains of the Distributed Storage Systemsstrategy seeking the minimum power curtailment to satisfy the system constraints,can be formulated as:min 1′pcurt (3.24)subject to.0 ≤ pcurt ≤ pwind (3.25)vmin ≤ Ae(d + pcurt) ≤ vmax (3.26)|Il (pcurt)|2 ≤ |Imaxl |2, l = 1, 2, . . . , L, (3.27)where pcurt is the vector of curtailed powers and pwind is the vector of power gen-erations by wind turbines. Also, vmin and vmax are the vector of minimum andmaximum allowable voltage limits, respectively. Here, d is the vector of forecasttotal demand of the nodes for the next hour. In (3.27), |Il | is the magnitude of thecurrent flowing in branch l , Imaxl is the ampacity of branch l , and L is the totalnumber of branches in the system. It is possible to leverage the linearized powerflow equations to approximate the magnitude of the currents as follows:|Il (pcurt)|2 = (d + pcurt)′Tl(d + pcurt), (3.28)where Tl is a positive semi-definite matrix. Therefore, (3.27) is indeed a convexconstraint in pcurt. Since the constraints (3.25) and (3.26) are also convex, problem(3.24) is convex. Note, however, that due to the inherent approximation of thelinearized power flow equations, the solution of (3.24) is a rough estimate of thecurtailed powers. To find an accurate solution, one can solve (3.24) for coarsetuning the curtailed powers and then do grid search along with ACPF to fine tunethe results.683.2. Formulation of the Economic Gains of the Distributed Storage SystemsTo derive the quadratic form of (3.28) for the magnitude of the currents, leti = ire + jiim be the vector of currents in the branches of the system. Also, let Ybrbe the branch admittance matrix of the system. Then the vector of branch currentscorresponding to the total demand of d + pcurt is given by:I = Ybr(1 + eˆ + j fˆ)= Ybr(eˆ + j fˆ) (3.29)= Y˜br(e + jf)= (G˜bre− B˜brf) + j(G˜brf + B˜bre)=[(G˜brAe − B˜brAf ) + j(G˜brAf + B˜brAe)](d + pcurt)= Tre(d + pcurt) + jTim(d + pcurt), (3.30)where again eˆ and fˆ are the augmented voltage deviation vectors and Tre and Timare defined as follows:Tre = G˜brAe − B˜brAf , (3.31)Tim = G˜brAf + B˜brAe. (3.32)Also, Y˜br = G˜br + jB˜br is the branch admittance matrix with the first columnremoved. Moreover, (3.29) stems from the fact that Ybr1 = 0. Let t′re,l and t′im,lbe the l th row of Tre and Tim, respectively. Then, the magnitude of the currentsin the branches of the system as a function of the curtailment vector pcurt will be693.2. Formulation of the Economic Gains of the Distributed Storage Systemsgiven by:|Il (pcurt)|2 =(t′re,l (d + pcurt))2+(t′im,l (d + pcurt))2= (d + pcurt)′tre,lt′re,l (d + pcurt) + (d + pcurt)′tim,lt′im,l (d + pcurt)= (d + pcurt)′Tl(d + pcurt), (3.33)whereTl = tre,lt′re,l + tim,lt′im,l . (3.34)Note that the matrices tre,lt′re,l and tim,lt′im,l are positive semi-definite by structure.Therefore, Tl is positive semi-definite, too.3.2.4 The Improvement in the System ResilienceIn this section the economic value of the DSS in terms of improving the systemresilience is formulated.Let Ho(h) ∈ Ho be the set of off-peak hours from the beginning of the off-peakperiod up to the hour h. Similarly, let Hp(h) ∈ Hp be the set of peak hours from thebeginning of the peak period up to the hour h. Also, let 0 < βch < 1 and 0 < βdis < 1be the charging and discharging efficiencies of the DSS technology used, respectively.Hence, βrt = βchβdis < 1 is the round-trip efficiency of the storage units. Denotethe total energy that the DSS can supply to the grid in segment s and at the endof hour h by Es,h(Π). Since Es,h is equal to the total energy stored in the storageunits times βdis, one can write:Es,h=βrt∑h′∈Ho(h) 1′pDSSs,h′ , h ∈ Hoβrt∑h′∈Ho1′pDSSs,h′ −∑h′∈Hp(h)1′pDSSs,h′ , h ∈ Hp703.2. Formulation of the Economic Gains of the Distributed Storage SystemsUpon imposing appropriate charging/discharging constraints, as will be done in Sec.3.3, Es,h is guaranteed to be non-negative. In the event that the primary supply ofthe system is interrupted, the distribution system may be operated as an islandedmicrogrid [27, 94, 95]. In such a case, the local DGs as well as the energy storedin the DSS can supply the loads of the system for a limited time. The non-criticalloads of the system may need to be shed depending on the availability of the energyby local sources [96]. Therefore, Es,h can be viewed as the additional amount ofload that can be preserved each time the supply from the primary source has failed.In distribution systems, the System Average Interruption Frequency Index (SAIFI)measures the average number of power interruptions a customer experiences duringone year. Let µ¯y be the expected interruption cost per 1 kWh of energy in year y ofthe planning horizon. In practice, the interruption cost per unit kWh is a function ofthe interruption duration as well [97]. However, one can use the Customer AverageInterruption Duration Index (CAIDI) to find an approximate value for the expectedinterruption cost. Let ξs,h be the fraction of the interruptions that occur at hour hduring segment s of each year, see [97]. The average economic value of the DSS interms of enhancing the resilience of the system can be approximated as:Γresy (Π) = µ¯y × SAIFI×∑h∑sξs,hEs,h. (3.35)Note that Γresy (Π) is linear in Es,h and, hence, in Π.3.2.5 The Economic Gain of the System Upgrade DeferralDue to the constant load growth in distribution systems, the system will requirean upgrade in the feeder ampacities and substation capacity at some point in thefuture. However, if DSS are installed in the system, they can be employed to shavethe load at peak hours and thereby defer the required upgrade of the system [31, 32].713.2. Formulation of the Economic Gains of the Distributed Storage SystemsIn order to defer the upgrade of the system, the objective function of the DSSplacement can include a term to shave the peak load in the system. To that end,form the expected total active power drawn from the substation during peak hoursas:P peaks,h (Π) = E{[1′ 0′](dYs,h − pDSSs,h )}, ∀s, h ∈ Hp, (3.36)where 1 and 0 are of size (N − 1) × 1. The optimization routine will try to lowerthe maximum of P peakh (Π) over peak hours as will be explained in the next section.Note that the expected total load in (3.36) has been computed for the last yearof the planning horizon because the last year is supposed to have the maximumdemand.Once the DSS is optimally planned in the system, the economic gain of deferringthe system upgrade due to installation of DSS will be computed. As per commonpractice of distribution system planning, the system constraints have to be satisfiedfor all combinations of the load and distributed generation. To evaluate the latestpossible time for the system upgrade, the histograms of the available real dataare examined to find the maximum values of the load and minimum values of thedistributed generation based a predetermined confidence interval. These extremumsare then scaled based on the given annual growth for the load and distributedgeneration (see Sec. 3.2.2) to estimate the extremums of all the years of the planninghorizon. Then, for each year of the planning horizon a power flow is run according tothe the extreme loading and generation to find the required ampacity of the feedersand substations. Then, the upgrade requirements of the feeders and substation ofthe system are calculated for the system with and without DSS [31]. Finally, theaverage yearly difference in the system upgrade cost for the yth year, denoted byΓupy (Π), is computed.723.3. The Optimal Distributed Storage Planning Problem3.3 The Optimal Distributed Storage PlanningProblemThis section aims at formulating the optimal DSS planning problem in smart dis-tribution grids. To that end, the regularized expected discounted gain of the SGOdue to installation of DSS is defined as:Γ′(Π) = ω1Γvol(Π)− ω2∆ +Y∑y=1λy{Γarby (Π) + Γlossy (Π) + Γresy (Π)}, (3.37)where ω1, ω2 > 0 are two regularization factors and ∆ is an auxiliary variable usedfor shaving the peak load as will be explained shortly (see (3.43)). Also, 0 < λ < 1is the discount factor. If the interest rate is int and a uniform inflation rate of inf isassumed for the energy price and the interruption cost, then λ = (1 + inf)/(1 + int).The regularization parameters ω1 and ω2 will be tuned based on simulations to yieldthe best trade-off between the investment cost and the economic gain of the DSS.Note that based on the formulations presented in Sec. 3.2, Γ′(Π) is concave in Π.In order to formulate the investment cost of the DSS, let b = [b2, b3, . . . , bN ]′ bethe capacity of the installed storage units in kWh. Also, let κ1 $/kWh be the unitcost of the DSS technology used and κ2 $/kW be the associated power electronicsand O&M costs. Then, the initial investment cost of DSS in the system is equal toκ11′b + κ21′r, where r = [r2, r3, . . . , rn]′ is the vector of rated powers of the storageunits. Now, assume that the planning horizon is greater than the lifetime of theDSS with a factor of K, i.e., Y = K×YDSS, where YDSS is the DSS lifetime in years.Then, the storage units need to be replaced in the system K times. Therefore, the733.3. The Optimal Distributed Storage Planning Problemdiscounted investment cost of storage units will be given by:Ω(b, r) =(κ1K−1∑k=0λkYDSS)× 1′b + κ21′r. (3.38)Observe from (3.38) that Ω(b, r) is linear in b and r.The optimal DSS planning aiming at maximizing the economic gain minus theinvestment cost can now be cast as the following optimization problem:maxb,r,ΠΓ′(Π)− Ω(b, r) (3.39)subject to a series of constraints as follows.ˆ The power rating of the storage units. The absorbed and injected power bythe storage units is limited by the inverter’s power rating, that is:0 ≤ pDSSs,h ≤ r, ∀s, h. (3.40)ˆ The charging/discharging capacity of the storage units.Let γDOD be the Depth of Discharge (DOD) of the storage units. The pa-rameter γDOD along with βch and βdis determines how much energy is neededto charge a storage unit of a given capacity during off-peak hours. The firsttime that a storage unit of size bn is connected to node n, a total energy of(1−γDOD)bn kWh is absorbed which remains in the unit and never discharges.Next, an amount of up to γDODbn kWh is absorbed from the grid during thenext charging cycle. Therefore, for charging cycles (off-peak hours) one can743.3. The Optimal Distributed Storage Planning Problemwrite:βch∑h∈HopDSSs,h = γDODb, ∀s. (3.41)Similarly, at peak hours when storage units get discharged, the total energyabsorbed by the units during off-peak hours times the discharging efficiencywill be equal to the total energy injected to the grid. That is,∑h∈HppDSSs,h = βdisβch∑h∈HopDSSs,h= βrt∑h∈HopDSSs,h , ∀s. (3.42)Since the values of the charging and discharging powers are assumed to be non-negative in our formulation, the constraints (3.41) and (3.42) require that thecharging/discharging power of each node be zero if no storage unit is installedon the node. Also note that our formulation assumes that the storage unitsare charged during off-peak hours and discharged during peak hours. That is,the charging and discharging pattern of the storage units is only once a day.This assumption ensures a longer lifetime for the storage units and has beenmade extensively in the literature, see, e.g., [58, 93].Note that in practice the peak and off-peak hours may differ for differentsegments of the year. Similarly, the solar radiation pattern may change indifferent segments. In such cases, the formulations provided in this chaptercan be easily modified to optimize the charging and discharging strategies ofdifferent segments by considering separate peak and off-peak hours for eachsegment.ˆ The maximum peak load. The parameter ∆ in (3.37) serves to represent the753.3. The Optimal Distributed Storage Planning Problemmaximum peak load over peak hours. Hence, the following set of linear con-straints has to be added to the optimal DSS placement problem:P peaks,h (Π) ≤ ∆, ∀s, h ∈ Hp (3.43)where P peaks,h (Π) is defined in (3.36).ˆ The allowable capacity of DSS that can be installed on each node.The installed capacity of the DSS for each node has to be non-negative. Dueto possible physical limitations, the maximum capacity at each node may alsobe limited. These constraints can be written as follows:0 ≤ b ≤ bmax, (3.44)where bmax is the vector of maximum installable storage capacity in the sys-tem.Once the optimal DSS planning problem (3.39) is solved, the total discountedeconomic gain of the SGO will be computed as:Γ(Π) =Y∑y=1λy{Γarby (Π) + Γlossy (Π) + Γresy (Π) + Γcurty (Π) + Γupy (Π)}. (3.45)Note that the explicit formulation of Γcurty (Π) and Γupy (Π) in terms of Π is in-tractable and, hence, the regularized expected discounted gain Γ′(Π) is consideredfor optimal DSS planning instead of Γ(Π).763.4. Numerical Results3.4 Numerical ResultsThis section provides the numerical results of the optimal DSS planning methodologyon a modified version of the distribution test system presented in [90].3.4.1 The Setting of the SimulationsThe modified test system is radial (all tie line switches are open) with four feederswhere node 1 is considered as the slack node. For this system, inclined solar cellsare considered on 34 randomly selected nodes of the grid. The nodes equipped withsolar cells are 3, 4, 6, 8, 10, 13, 17, 19, 20, 22, 23, 30, 31, 32, 33, 35, 36, 37, 40, 44, 45,46, 49, 51, 53, 55, 57, 58, 61, 62, 63, 65, 67, 68. A total rated power of 1000 kW isconsidered for the solar cells in this test system. This total rated power is allocatedto the solar cell-equipped nodes proportionally to their secondary transformer powerrating. Based on the time series of the real data from smart meters and solar cellsfrom July 2009 to December 2010, the total energy provided by the solar cells iscalculated to be 7.9% of the total energy demand of the system. In addition, twowind turbines are installed on the nodes 21 and 50 with power ratings of 400 kWand 800 kW, respectively. Based on the time series of demands measured by smartmeters and the generation by wind turbines, the total energy provided by the windturbines turns out to be 29.7% of the total energy demand of the system from July2009 to December 2010.In our simulations, the Advanced Metering Infrastructure (AMI) data releasedby the Commission for Energy Regulation (CER) [98] is utilized to model the loads.The dataset provided by CER is from the Electricity Customer Behaviour Trailstudy and has been collected from 5000 smart meters in Ireland from July 14, 2009 toDecember 31, 2010. This dataset was received by the author from Irish Social ScienceData Archive (ISSDA) [99]. The reactive power demands are modeled by assuming773.4. Numerical Resultsa constant power factor for the nodes. A growth of 5% [50] in the demands of thenodes is considered over the planning horizon. The time series of wind generationand solar generation are obtained from [100] and [101], respectively.The type of DSS used is Lead-Acid (LA) due to its lower installation and re-placement costs [102]. The DSS are assumed to come in 100 kW-100 kWh unitswith charging and discharging efficiency of βch = βdis = 0.85. The DOD of thestorage units is assumed to be γDOD = 0.75. The DSS costs are 305 $/kWh forstorage units and replacements, 175 $/kW for the inverter, and 15 $/kW for annualmaintenance. The planning horizon is assumed to be twice as the lifetime of theDSS and each year is divided into 12 segments. The maximum installable storageunit on each node is considered to be 500 kWh.The price of energy is considered to be 8.29 ¢/kWh for off-peak hours and 12.43¢/kWh for peak hours. The interest rate and the inflation rates are assumed 5%and 1%, respectively. Also, µ is considered 15 $/kWh based on the survey reportedin [97] and SAIFI is 1.5. These values are typical values but in practice may bedifferent for different cases.The system upgrade includes upgrading the distribution system feeders and sub-station. In addition, similarly to [32], it is assumed that a 20 km transmission linebetween the distribution substation and the HV/MV primary substation needs tobe upgraded too. An average length of 150 meters per branch is assumed for thedistribution feeders in accordance with the IEEE distribution test systems [103].The upgrade costs of lines, feeders and substations are obtained from [104].The simulations are conducted using MATPOWER [105] and the optimizationproblems are solved using the package CVX: Software for Disciplined Convex Pro-gramming, version 2.1 [106] bundled with Mosek, version 7.1.783.4. Numerical Results3.4.2 ResultsThe optimal DSS planning methodology on the system under study results in theinstallation of a storage unit of size 200 kWh with an inverter power rating of 100kW. The optimal location of this unit is node 29 which is on feeder 2. Fig. 3.3illustrates the test system under study with the optimal location of the storageunit. The locations of the two wind turbines are also shown in Fig. 3.3. In thisfigure, the nodes equipped with solar cells are indicated with shadow.793.4.NumericalResults 1 2 3 4 5 6 7 89 10 11 12 13 14 15 30 31 32 33 34 35 36 37 3847 48 49 50 39 40 41 42 43 44 45 46 23 24 25 26 27 28 29 68 69 16 17 18 19 20 21 2251 52 53 54 55 56 61 62 63 64 67 66 65 57 58 59 60 Figure 3.3: The modified test system with wind turbines and solar cells and the storage unit optimally located on node29.803.4. Numerical Results5 10 15 20020406080100120140HourStored Energy (kWh)  Segment 3Segment 6Segment 9Figure 3.4: The optimal charging and discharging strategy of the installed storageunit for three different segments of the year.Fig. 3.4 depicts the optimal charging and discharging strategy of the storageunit for three different segments of the year. Note that the optimal charging anddischarging pattern in Fig. 3.4 shows the energy stored in the unit in addition tothe amount that is always there due to DoD.The total investment cost for this storage unit will be $141, 310 which includesthe following:ˆ Initial investment cost: $78, 500ˆ Replacement cost: $43, 006ˆ Total discounted maintenance cost: $19, 804With the optimal DSS placement in the system, the SGO is expected to obtain thefollowing gains during the entire planning horizon:ˆ Expected discounted arbitrage gain: $6, 135ˆ Expected discounted gain due to reduction in the system losses: $1, 946813.4. Numerical Resultsˆ Expected discounted economic value of system resilience enhancement: $25, 038ˆ Expected discounted gain due to reduction in the renewable energy curtail-ment: $235ˆ Total economic gain due to deferring the upgrade of the system: $193, 540The gain in the deferral of the system upgrade comes from a gain of $31, 290 infeeder 2 upgrade deferral, a gain of $133, 500 in the transmission line upgrade de-ferral, and a gain of $28, 750 in the substation upgrade deferral. Based on thesenumbers, the total discounted gain that the SGO obtains due to installation of DSSwill be $226, 894. Given that the total investment and maintenance cost of DSSinstallation is $141, 310, we conclude that energy storage in this system results in atotal discounted saving of $85, 584.It is worth mentioning that the proposed DSS placement routine found a singlestorage unit to be installed in this test system. The reason why the optimal solutionin this system consists of just one store unit is because of the integrality of thecapacity and power rating of the units. If the integrality constraint is relaxed inthe problem formulation, the optimal solution will be different. Nonetheless, forpractical applications, the storage units are expected to come in predefined sizesand the integrality constraint should be preserved.We finish this chapter by emphasizing that a detailed and case-specific analysisshould be carried out to evaluate the actual economic gain of DSS in any distributionsystem. The test system studied in this chapter represents a case in which DSSinstallation is an economically justified investment for the SGO. However, dependingon how close the system is to requiring an upgrade, the situation may change,possibly rendering the DSS installation uneconomical. Nonetheless, the trade-offbetween investment costs and the return the SGO obtains is expected to change in823.5. Conclusionthe future in favor of DSS. That is because as more companies enter the bluishnessand energy storage efficiency improves, the cost of storing energy per kWh reduces.3.5 ConclusionThis chapter presented a methodology for optimal planning of DSS in smart distri-bution grids. The problem of optimizing the expected economic gain of the systemoperator was formulated as a mixed-integer convex program. In particular, theoptimal planning problem incorporated the arbitrage gain, the reduction in theactive power loss, the reduction in the non-dispatchable energy curtailment, theimprovement in the system resilience, and the financial gain due to deferring thesystem upgrade to future years. Numerical results using real data of smart metersand renewable energy sources on a typical distribution system was presented whichdemonstrated the effectiveness of using DSS in future smart grids.83Chapter 4Volt-VAR Optimization inActive Smart GridsDue to the high variability of non-dispatchable energy sources in active smart grids,the Volt-VAR Optimization (VVO) problem will be a very important operationaltask of the SGO. Fig. 4.1 illustrates the schematic of an active smart grid with var-ious equipment that can be used for VVO, including DSS, capacitor banks, ULTCs,and feeder reconfiguration.This chapter presents a comprehensive formulation of the VVO problem in activesmart grids considering wind turbines, DSS, capacitor banks, ULTCs, and feeder Smart Grid Operator Figure 4.1: Schematic of an active smart grid with distributed wind turbines andvarious VVO equipment.844.1. Preliminariesreconfiguration [107]. The VVO problem is formulated as a mixed-integer, quadraticprogram to be solved using branch-and-bound methods. In order to formulate theexpected power loss of the system, the stochasticity of the wind power generation isaddressed using a first-order Markov Chain (MC) model [77]. Simulation results ona 33-node, 12.66 kV reconfigurable smart grid using real data from wind turbinesand smart meters are presented and discussed. Various test cases are considered inthe simulations to compare the impact of different VVO equipment on the systemloss and voltage profile.The load model considered in this chapter is a constant load model. Althoughthe VVO problem is related to the voltage-dependence of the loads, in many casesthis information is not available to the system operator. As an approximation,the use of a constant load model enables us to formulate the problem of VVO asa mixed-integer, quadratic program which can be solved efficiently using existingsoftwares.This chapter is organized as follows. Sec. 4.1 provides some preliminaries aboutthe power flow equations in radial distribution systems as well as an MC modelfor wind power generation. Sec. 4.2 presents the formulation of the VVO problemconsidering DSS, capacitor banks, ULTCs, and network reconfiguration. Sec. 4.3provides the numerical results on a reconfigurable test system and by using real datafrom smart meters and wind turbines. Finally, Sec. 4.4 concludes the chapter.4.1 PreliminariesThis section reviews the DistFlow equations of radial distribution systems [8, 81]and extends them to accommodate the DGs and bi-directional flow of power. Also,an MC model for wind power generation based on [77] is reviewed.Consider a radial distribution system with N nodes and let vn = |vn|ejθn , n =854.1. Preliminaries1, 2, . . . , N be the voltage phasor of the nth node, where j =√−1. Let sn = pn+jqndenote the complex power demand of a generic node n. Similarly, let sGn = pGn + jqGnbe the complex power generation of node n. For any radial configuration of thesystem, let L be the set of all lines which connect two nodes of the system together.Note that L uniquely specifies the configuration of the system and, therefore, wedenote the configuration of the system by L. With some abuse of notation, we alsouse (n,m) ∈ L to specify that node n is connected to node m in configuration L. Inaddition, let NLn be the set of all nodes which are connected to node n via a line inL. Also, let Ssub be the set of substations in the system.Throughout the chapter, it is assumed that the renewable sources provide onlyactive power. Likewise, it is assumed that the DSS only store and supply activepower, and reactive power demands will be compensated by capacitors as needed.Nevertheless, the formulations can be easily modified to account for the possibilityof reactive power injection by DSS. In such a case, an appropriate joint constraint onthe active and reactive power injection capability of the DSS should be considered.Such a constraint is non-linear in general but can be easily replaced by a set ofapproximated linear constrains [58].4.1.1 DistFlow EquationsSuppose that node n is connected to node m in a radial configuration via line l ∈ Lwith impedance zl = rl + jxl. For any instant of time, let snm = pnm + jqnm be thecomplex power that flows from node n towards node m. Also, let smn = pmn+ jqmn864.1. Preliminariesdenote the complex power that flows from node m towards node n. Then we have:snm + smn = zl|snm|2|vn|2= zl|smn|2|vm|2 (4.1)vm = vn − zl s∗nmv∗n. (4.2)Separating the active and reactive parts in (4.1) and the voltage magnitudes in (4.2),yields:pnm + pmn = rlp2nm + q2nm|vn|2= rlp2mn + q2mn|vm|2 (4.3)qnm + qmn = xlp2nm + q2nm|vn|2= xlp2mn + q2mn|vm|2 (4.4)|vm|2 = |vn|2 − 2(rlpnm + xlqnm) + (r2l + x2l )p2nm + q2nm|vn|2 . (4.5)Equations (4.3)-(4.5) together with the following generation-demand equationspGn − pn =∑m∈NLnpnm (4.6)qGn − qn =∑m∈NLnqnm (4.7)874.1. Preliminariesare called the DistFlow equations and provide a full AC power flow model for aradial distribution system. Note that these equations are slightly different from theDistFlow equations in [11] and [8] in that they support bi-directional flow of powerwhich is required for radial systems with distributed generations. Accordingly, theactive and reactive powers that flow from one node to another can take both positiveand negative values.4.1.2 Markov Chain Model of Wind Power GenerationThe wind power generation can be modeled as a first-order homogeneous MC asproposed in [77]. To that end, the range of the power generation by the wind turbineis divided into S intervals, each denoted by a state level xi, i = 1, 2, . . . , S. Here, xiis the mean power generated by the wind turbine when the power generation is inthe ith interval. Next, a transition probability 0 ≤ θij ≤ 1 is assigned to the eventthat the wind power generation goes from state i at hour h to the state j at hourh + 1. These probabilities form the transition probability matrix T = [θij ]. Thematrix T is a stochastic matrix satisfying T1 = 1, where 1 denotes a vector withall elements equal to one.Let pi(h) = [pi1(h), pi2(h), . . . , piS(h)]′ be the state probability vector at hour h.Then, based on the Chapman-Kolmogorov equation, the probability of being in eachstate at hour h + τ will be pi(h + τ) = (T′)τ pi(h). In practice, the realization ofthe wind power generation is observed at the beginning of the optimization process.Therefore, all the elements of pi(0) but one are equal to zero. The initial stateprobability vector pi(0) is used to obtain subsequent state probability vectors.The entries of the matrix T can be estimated from real data using a Maximum-Likelihood (ML) approach. Let W = (W1,W2, . . . ,WK) be the wind MC and w =(w1, w2, . . . , wK) be the chain of K wind generation samples in the available dataset.884.1. PreliminariesIt follows from the Markovianity of W that:Pr.(W = w) = Pr.(W1 = w1)×K∏k=2Pr.(Wk = wk,Wk−1 = wk−1) (4.8)= Pr.(W1 = w1)×S∏i=1S∏j=1θnijij , (4.9)where Pr.() denotes the probability and nij is the number of times that the MC goesfrom state i to state j in the available dataset. Based on (4.9), the log-likelihoodfunction can be written as:log Pr.(W = w) = log Pr.(W1 = w1) +S∑i=1S∑j=1nij log θij . (4.10)Now since we have:∑jθij = 1, i = 1, 2, . . . , S, (4.11)the Lagrangian can be formed as:L(T) = log Pr.(W1 = w1) +S∑i=1S∑j=1nij log θij −∑iλi1−∑jθij , (4.12)where λi is the ith Lagrange multiplier. Taking the derivative with respect to θij foreach i and setting it equal to zero yields:ni1θi1=ni2θi2= · · · = niSθiS= λi, i = 1, 2, . . . , S. (4.13)It follows from (4.13) that θij is proportional to nij for each i, which together with894.1. Preliminaries(4.11), gives the following expression for the ML estimate:θˆij =nij∑j nij. (4.14)Remark 4.1 It is to be noted that for all practical purposes for a day-ahead VVOscheme, one needs to update the look-ahead state probability vectors ever day. Thatis because the convergence of the state probability vector of the wind MC is veryslow. To demonstrate this in a mathematically rigorous way, some background onMCs is briefly provided here. An MC is called irreducible if every state i can reachevery other state j in a finite amount of time. Intuitively, we expect the wind powerMC to be irreducible as starting from any generation state can lead to any otherstate in a finite amount of time. For an irreducible MC defined on a finite statespace, there exits a stationary state probability vector pi∞ which is the normalizedright eigenvector of T′. In other words, pi∞ is the unique solution of the followingfixed-point equation:pi = T′pi, (4.15)1′pi = 1. (4.16)An MC is called regular if there exists a fixed amount of time within which everystate can reach every other state. Again, intuitively the wind power MC is expected tobe regular. In fact, the regularity of the wind MC yields its irreducibility. A regularMC forgets its initial condition geometrically fast in the Second Largest EigenvalueModulus (SLEM) of T. The SLEM is also upper-bounded by the Dobrushin coeffi-cient which is defined as:r =12maxi,jS∑l=1|θil − θjl|. (4.17)904.2. Formulation of the Volt-VAR Optimization ProblemBased on our dataset of the wind power generation, the SLEM of T turns out to be0.9766 which is very close to 1. Also, r turns out to be equal to 1 which renders theDobrushin upper-bound trivial. Therefore, for small time horizons which are typicalfor operational optimizations of smart grids, the non-asymptotic state probabilityvectors need to be computed and used. The stationary probability vector, however,can be used for long horizons that are typical of planning problems.Unlike wind power generation, the loads of the system usually have daily patternsthat can be forecast with acceptable accuracy based on season, day of the week, timeof the day, and weather conditions. Therefore, similarly to [50, 63, 108–110], typicalload patterns obtained using smart meter measurements are employed for day-aheadVVO in this research. In particular, the load pattern of each node is simply obtainedby computing the average load of the node over a history of a prescribed length, e.g.,one month. Once the day-ahead schedule of the system equipments are obtainedbased on typical load patterns, finer adjustments can be made in real time based onthe actual measurements of the loads [108, 111].4.2 Formulation of the Volt-VAR OptimizationProblemThis section formulates the VVO problem in active smart grids considering windgeneration, DSS, ULTCs, capacitors, and feeder reconfiguration. Note that theformulation presented here provides a mathematical tool for VVO in smart grids.However, one should be aware that, in practice, the continuous use of tap changersand shunt capacitors increases their maintenance cost.914.2. Formulation of the Volt-VAR Optimization Problem4.2.1 The Objective Function of the VVO ProblemIn this chapter, the objective function of the VVO problem is considered to be theexpected active power loss in the system. Based on the DistFlow equations, thetotal active power loss in the system at hour h is given by:Loss(h) =∑(n,m)∈Lrlp2nm(h) + q2nm(h)|vn(h)|2 . (4.18)Approximating the nodal voltages by 1 p.u. in the right-hand side of (4.3), the totalactive power loss in the system at hour h will be given by the following quadraticform:Loss(h) ≈∑(n,m)∈Lrl(p2nm(h) + q2nm(h)). (4.19)The active power loss in the system at each hour is a random variable due to thestochasticity of the wind power generation. Therefore, the VVO problem should tryto minimize the expected active power loss in the system as:minE{1HH∑h=1Loss(h)}, (4.20)where E{·} is the expectation operator. Also, H is the horizon of the VVO problemwhich, in the case of a day-ahead approach, equals 24. One can employ the MCmodel of the wind power generation to compute the expectation in (4.20) as follows:E{1HH∑h=1Loss(h)}=1HS∑i=1H∑h=1pii(h) Loss(h; i), (4.21)where Loss(h; i) is the total power loss at hour h assuming that the wind powergeneration is in the ith state. It is seen from (4.19) and (4.21) that the objectivefunction of the VVO problem is quadratic and convex.924.2. Formulation of the Volt-VAR Optimization Problem4.2.2 Power Flow Equations, Distributed Storage Systems,Capacitors, and ULTCsThe VVO problem needs to be solved subject to a series of constraints. Below, theconstraints corresponding to power flow equations as well as the operation of DSS,capacitors, and ULTCs are formulated.A full set of AC power flow equations can be described by (4.3)-(4.5) alongwith the generation-demand equations. To include the charging and dischargingstrategies of the storage units as well as the reactive power injection of capacitors,the generation-demand equations (4.6)-(4.7) have to be modified as follows:pGn (h)− pn(h)− pDSSn (h) =∑m∈NLnpnm(h), h ∈ Ho, ∀n, (4.22)pGn (h)− pn(h) + pDSSn (h) =∑m∈NLnpnm(h), h ∈ Hp, ∀n, (4.23)qGn (h)− qn(h) =∑m∈NLnqnm(h), h ∈ Ho ∪Hp, ∀n. (4.24)In this formulation, pDSSn (h) is the amount of average power that the storage unitinstalled on node n stores from or supplies to the grid at hour h. Also, qGn (h) is theamount of reactive power generation of the capacitor installed on node n at hour h.Moreover, Hp and Ho denote the set of peak and off-peak hours, respectively.The constraints corresponding to the rated power limit of the storage unitsduring charging and discharging periods can be written as:0N ≤ pDSS(h) ≤ pDSS,max, ∀h, (4.25)934.2. Formulation of the Volt-VAR Optimization Problemwhere 0N is an all-zero vector of length N . Also,pDSS(h) = [pDSS1 (h), pDSS2 (h), . . . , pDSSN (h)]′is the vector of power storage and injections of the storage units at hour h, andpDSS,max = [pDSS,max1 , pDSS,max2 , . . . , pDSS,maxN ]′is the vector of power ratings of the storage units. If a node n is not equipped witha storage unit, then its power rating is set to zero, that is:pDSS,maxn = 0, n /∈ SDSS, (4.26)where SDSS is the set of nodes equipped with DSS.The constraints corresponding to the capacity of the storage units can be for-mulated as:∑h∈HopDSS(h) =γDODβchb, (4.27)∑h∈HppDSS(h) = βdisγDODb, (4.28)where b = [b1, b2, . . . , bn]′ in kWh is the vector of DSS capacities installed in thesystem. If a node n is not equipped with DSS, then bn = 0. Also, γDOD is theDOD of the storage units and 0 < βch < 1 and 0 < βdis < 1 are the chargingand discharging efficiencies of the DSS technology, respectively. Hence, βchβdis isthe round-trip efficiency of the storage units. The derivation of (4.27) and (4.28) isprovided in Sec. 3.3.The shunt capacitors can inject reactive power into the feeder. This reactive944.2. Formulation of the Volt-VAR Optimization Problempower is modeled as:qGn (h) = cn(h)qcapn , n ∈ Scap, (4.29)cn(h) ∈ {0, 1, 2, . . . , cmaxn }, ∀h. (4.30)where qcapn is the reactive power injection by a unit module of the capacitor bankinstalled on node n and cn(h) is the number of modules connected at hour h. Also,Scap is the set of nodes equipped with a shunt capacitor and cmaxn is the maximumnumber of modules available in the capacitor bank installed on node n.A ULTC that is installed on the branch connecting node n to node m withbranch impedance of zl can be modeled as follows [72, 112]. An ideal transformer ismodeled between node n and an auxiliary node m′, in series with the impedance zlbetween node m′ and node m. Therefore, the ULTC operation can be formulatedas:|vm|2 = |vm′ |2 − 2(rlpm′m+xlqm′m) + (r2l + x2l )p2m′m + q2m′m|vm′ |2 . (4.31)|vm′(h)| = (1 + γδ)|vn(h)|, (4.32)where γ is the ULTC tap position and δ is the tap step size. For instance, if thetransformer ratio is between 1− a and 1 + a with step size δ, thenγ ∈{γi = i∣∣i = −aδ,−aδ+ 1, . . . ,aδ− 1, aδ}. (4.33)For the sake of simplicity, no explicit index has been used for the ULTC tap positionand step size to indicate the corresponding node numbers.Finally, the operational constraints on the nodal voltages and feeder ampacities954.2. Formulation of the Volt-VAR Optimization Problemshould be formulated. The nodal voltages should be bounded as:vmin ≤ |v(h)| ≤ vmax, ∀h, (4.34)where v(h) = [|v1(h)|, |v2(h)|, . . . , |vN (h)|]′ is the vector of nodal voltage magnitudesat hour h. Also,vmin = [vmin1 , vmin2 , . . . , vminN ]′,vmax = [vmax1 , vmax2 , . . . , vmaxN ]′,are the vector of minimum and maximum allowable voltage magnitudes, respectively.The limits on feeder ampacities can be written as:p2nm(h) + q2nm(h) ≤ |smaxnm |2, (m,n) ∈ L, ∀h (4.35)p2mn(h) + q2mn(h) ≤ |smaxnm |2, (n,m) ∈ L, ∀h (4.36)where |smaxnm | is the maximum apparent power that can flow in the line which connectsnode n to node m.4.2.3 Feeder ReconfigurationModern smart grids are equipped with remotely-controllable tie-line switches fortopological reconfiguration. This section formulates the role of feeder reconfigurationon the VVO problem. It is assumed that the reconfiguration routine is performedonce in 24 hours, i.e., in a day-ahead fashion.To formulate the constraints corresponding to feeder reconfiguration, the ex-tended formulation of [11] is exploited which accommodates the existence of DGsand bi-directional flow of power. Let L∞ be the layout of the network, i.e. the con-964.2. Formulation of the Volt-VAR Optimization Problemfiguration of the system when all the tie-line switches are closed. Denote by |L∞|the cardinality of L∞. Define |L∞| binary variables ynm ∈ {0, 1} associated witheach line between any two nodes in the final radial configuration as:ynm =1, if node n is connected to node m0, otherwise(4.37)and stack them in y. Using these notations, the constraints corresponding to thereconfiguration routine can be formulated as follows. For any n ∈ {1, 2, . . . , N},(n,m) ∈ L∞, the generation-demand equations (4.22)-(4.24) in a reconfigurabledistribution system should be augmented with the following constraints:−Dynm ≤ pnm(h) ≤ Dynm, ∀h (4.38)−Dynm ≤ qnm(h) ≤ Dynm, ∀h (4.39)−Dynm ≤ pmn(h) ≤ Dynm, ∀h (4.40)−Dynm ≤ qmn(h) ≤ Dynm, ∀h (4.41)where D  1 is a large disjunctive constant.The radiality constraint can be imposed by requiring that the resulting config-uration should be loop-free and connected. As explained in [13], for distributionsystems with DGs this can be achieved by introducing a set of fictitious loads onthe nodes that can be fed only from the substation. Let kn be the fictitious load onnode n. Also, let knm be the fictitious power that flows from node n towards nodem. Then, the following set of linear constraints are equivalent to the radiality of the974.2. Formulation of the Volt-VAR Optimization Problemsystem.1′y = N − 1, (4.42)kn =∑m∈NL∞nknm, ∀n, (n,m) ∈ L∞, (4.43)−Dynm ≤ knm ≤ Dynm, (4.44)where 1 is an all-one vector. Note that it is required to impose the radiality con-straint for only one hour. Therefore, km and knm do not depend on h.In addition, it is possible to limit the number of switching actions during thereconfiguration procedure to prevent excessive costs. Let y¯ denote the status ofthe switches in the current configuration of the system. Also, let α be an auxiliaryvector. Then, the following set of linear constraints limits the number of switchingactions to ρ:y − y¯ ≤ α, (4.45)y¯ − y ≤ α, (4.46)1′α ≤ ρ, (4.47)α ≤ 1. (4.48)Note that for the system to remain connected, opening one switch requires closinganother switch. Therefore, ρ should be an even number in (4.47).Finally, if a branch of the system is not equipped with remotely controllableswitches, its connection is forced to remain unchanged. That is,ynm = y¯nm, (n,m) /∈ Stie (4.49)984.2. Formulation of the Volt-VAR Optimization Problemwhere Stie is the set of branches equipped with a remotely controllable switch.4.2.4 Convexification of the VVO ProblemThe VVO problem derived above is non-convex due to some non-linearities in thepower flow equations, the multiplication of variables in the right-hand side of (4.32),and the integrality constraints (4.30) and (4.33). However, it is possible to makeapproximations to the power flow equations [11] to come up with a mixed-integerconvex optimization problem. To that end, we will need to consider the square of thevoltage magnitudes as independent optimization variables. Also, the multiplicationof variables in (4.32) can be replaced by alternative convex constraints as will bediscussed next.Although mixed-integer convex optimization problems are NP-hard, the relaxedversion of those problems is convex and, hence, they can often be solved in a rea-sonable time. In practice, mixed-integer convex programs are solved using a com-bination of a convex optimization technique and an exhaustive search algorithm,such as branch-and-bound methods. Two main characteristics that makes solvingmixed-integer convex programs particularly easier are the following. First, whenperforming the exhaustive search over integer variables along a tree, some branchesof the tree can be shown (through solving a relaxed problem) not to include theoptimal solution. Therefore, there is no need to follow those branches. Second, if anoptimal solution is found before searching all combinations of the integer variables,it may be possible to prove that this solution is optimal through solving a relaxedproblem. This is in contrast with non-convex integer programs, where even if theoptimal solution is obtained before searching all the combinations, there is no wayto prove that this solution is actually optimal.To convexify the VVO problem, first, constraint (4.32) is replaced by a series994.2. Formulation of the Volt-VAR Optimization Problemof linear constraints using the approach presented in [112]. Since the square of thevoltage magnitudes are considered as the optimization variables, (4.32) should bewritten as:|vm′(h)|2 = (1 + γδ)2|vn(h)|2, (4.50)and(1 + γδ)2 = 1 + 2δγ + δ2γ2. (4.51)The integer variable γ can be expanded using a series of binary variables κi ∈ {0, 1}as:γ =aδ∑i=−aδκiγi, (4.52)aδ∑i=−aδκi = 1. (4.53)Similarly,γ2 =aδ∑i=−aδκiγ2i . (4.54)Therefore, one can rewrite (4.50) as:|vm′(h)|2 = |vn(h)|2 + 2δ∑iγi|v˜n,i(h)|2 + δ2∑iγ2i |v˜n,i(h)|2, (4.55)where|v˜n,i(h)|2 = κi|vn(h)|2, i = −aδ, . . . ,aδ. (4.56)Constraint (4.55) is linear in |vm(h)|2, |vn(h)|2, and the 2aδ + 1 variables |v˜n,i(h)|2.Constraints (4.56) which are multiplications of a binary variable and a boundedcontinuous variable can each be replaced by the following linear constraints [108,1004.2. Formulation of the Volt-VAR Optimization Problem112]:|vn(h)|2 − (1− κi)(vmax2 )2 ≤ |v˜n,i(h)|2, (4.57)|v˜n,i(h)|2 ≤ |vn(h)|2 − (1− κi)(vmin2 )2, (4.58)κi(vmin2 )2 ≤ |v˜n,i(h)|2 ≤ κi(vmax2 )2, (4.59)If κi is zero, (4.59) forces |v˜n,i(h)|2 to be zero and (4.57) and (4.58) become redundant(same as voltage magnitude constraints (4.34)). If κi is one, (4.57) and (4.58) force|v˜n,i(h)|2 to be equal to |vn(h)|2 and (4.59) becomes redundant.To come up with a mixed-integer convex approximation of the VVO problem,it remains to convexify the power flow equations (4.3)-(4.5). Note that rl(p2nm +q2nm)/|vn|2 and xl(p2nm + q2nm)/|vn|2 are, respectively, the active and reactive powerloss on the line l which connects node n to node m. Therefore, by ignoring thepower loss on the lines, the power flow equations (4.3)-(4.4) assume the followinglinear form:pnm(h) + pmn(h) = 0, ∀h, (n,m) ∈ L (4.60)qnm(h) + qmn(h) = 0, ∀h, (n,m) ∈ L. (4.61)Moreover, since rl and xl have small values in real distribution systems, one can dropthe third term in the right-hand side of (4.5). That is, (4.5) can be approximatedwith the following constraint:|vm(h)|2 = |vn(h)|2 − 2(rlpnm(h) + xlqnm(h)), ∀h, (n,m) ∈ L. (4.62)Notice that if |vn|2 is considered as an independent variable of the optimization then(4.62) is a linear constraint.1014.3. Case Studies and Numerical ResultsThe VVO problem (4.20) subject to (4.60)-(4.62), (4.22)-(4.30), (4.34)-(4.36),(4.38)-(4.49), (4.55), and (4.57)-(4.59) is a mixed-integer quadratic problem andoptimizes the joint operation of the DSS, shunt capacitors, ULTCs, and remotelycontrollable reconfiguration switches for power loss minimization.4.3 Case Studies and Numerical ResultsIn this section, the presented VVO methodology is tested on an active distributionsystem and several cases with different VVO equipment availability are compared.4.3.1 The Setting of the SimulationsThe VVO problem for loss minimization is considered for a 33-node, 12.66 kV [89]active distribution system depicted in Fig. 4.2. In Fig. 4.2, the branches equippedwith remotely controllable switches are shown with dotted lines. In the presentedsimulations, the AMI data released by the Commission for Energy Regulation (CER)[98] is utilized to model the loads. The author received this dataset from Irish SocialScience Data Archive (ISSDA) [99]. The reactive power demands are modelled byassuming a constant power factor for the nodes. The horizon of the VVO problemis considered to be 24 hours corresponding to a day-ahead scenario.A wind turbine of a rated power of 1000 kW is installed on node 14. The timeseries of wind generation is obtained from [100]. The rated capacity of the windturbine is divided into 10 states for MC modeling. Based on the real data of thewind turbine, the mean power generation in the states (i.e., the state levels) is 53.96kW, 147.16 kW, 246.9 kW, 347.47 kW, 446.96 kW, 549.96 kW, 648.19 kW, 745.16kW, 848.05 kW, 948.07 kW, respectively.Two storage units of capacities 200 kWh and 300 kWh are installed on the nodes14 and 15, respectively. The power rating of both units is assumed to be 100 kW.1024.3. Case Studies and Numerical Results 29 28 27 26 1 2 3 4 5 6 7 8 9 10 14 15 16 13 12 11 17 18 23 25 24 30 31 32 33 19 20 21 22 Figure 4.2: The 33-node active smart grid with a wind turbine, DSS, and VVOequipment.1034.3. Case Studies and Numerical ResultsThe specifications of the DSS are obtained from [113] and [50] and are as follows.The charging and discharging efficiency of the storage units is βch = βdis = 0.85.The DOD of the storage units is assumed to be γDOD = 0.75.Two shunt capacitors are installed on the nodes 11 and 25. The reactive powerrating of the the capacitors are assumed to be 400 kvar consisting of 4 modules of size100 kvar. A ULTC with the ratio in the range of 0.9 ∼ 1.1 and with tap step sizesof 0.01 is installed between nodes 6 and 26. The branches equipped with remotelycontrollable tie line switches are the normally-open branches (8, 21), (9, 15), (12, 22),(18, 33), (25, 29) as well as branches (6, 7), (10, 11), (14, 15), (29, 30). The maximumnumber of switching actions per day is considered to be ρ = 6 which is equivalentto opening 3 switches and closing 3 other switches. The minimum and maximumvoltage limits for all nodes is considered to be 0.94 p.u. and 1.06 p.u., respectively.The disjunctive parameter considered for the reconfiguration routine is equal to 100.The VVO technique presented in the chapter is conducted on the system understudy for Nday = 30 days and the average results are reported. Typical load patternsobtained by averaging over a history of one month are used for optimization. Also,the MC model of the wind power generation is used during the optimization. Oncethe optimized operation of the VVO equipment is obtained, the system is testedwith the actual loads and wind power generation of the next day to compute thenodal voltages and the active power losses. In addition to the average active powerloss of the system, four more metrics of quality are also reported. These metrics arethe average peak load of the system, the average minimum voltage in the feeder,the average maximum voltage in the feeder, and the average voltage spread in thefeeder. The average voltage spread in the feeder is computed to measure the voltage1044.3. Case Studies and Numerical Resultsdeviation in the system and is defined as:Voltage Spread =1NdayHNday∑i=1H∑h=1[vmax(h)− vmin(h)] , (4.63)wherevmax(h) = maxn|vn(h)|, (4.64)vmin(h) = minn|vn(h)|. (4.65)All simulations are done using MATPOWER [105] and the optimization prob-lems are solved using the package CVX: Software for Disciplined Convex Program-ming, version 2.1 [106] bundled with Mosek, version 7.1.4.3.2 ResultsAs explained in Sec. 4.1, a linearized version of DistFlow equations is employed inthis chapter which allows for bi-directional flow of power in a radial system. To firstexamine the accuracy of the linearized DistFlow equations, a study is conducted onthe system. Unlike the VVO optimization procedure, it is assumed in this studythat the actual load and wind power generation is available for solving the powerflow equations. This study in conducted in the default configuration of the systemwithout shunt capacitors and DSS. The position of the ULTC is fixed at a ratioof 1.05. The simulations are done for 30 days, 24 hour each, and the results arecompared with that of the Newton’s AC power flow method. The results of thisstudy show that the average difference in the voltage magnitudes between the lin-earized extended DistFlow equations and the Newton’s method is 3.35× 10−4 p.u..Moreover, the average maximum nodal error of the linearized extended DistFlowequations turns out to be 5.36× 10−4 p.u.. Also, the average error and the average1054.3. Case Studies and Numerical Results5 10 15 20 25 300.950.960.970.980.9911.011.021.031.04  Newton-peakLinear DistFlow-peakNewton-off peakLinear DistFlow-off peakFigure 4.3: Voltage profile of the system on a typical day at a peak and an off-peakhour. Newton’s AC power flow versus linearized extended DistFlow equations.maximum error under peak load are 5.39× 10−4 p.u. and 8.55× 10−4 p.u., respec-tively. In addition, the average error in the active power loss computed using thelinearized DistFlow equations turns out to be 1.56 kW or 3.3%. Fig. 4.3. depictsthe voltage profile of the system for a typical day and for a peak and an off-peakhour. The figure contrasts the voltage profile obtained by the linearized extendedDistFlow equations and that of the Newton’s method.Table 4.1 presents the performance of the optimal VVO solution obtained un-der various case studies. In particular, 9 cases, including a default test case, aresimulated. The default test case corresponds to the test system without any VVOequipment while the remaining 8 cases have different VVO equipment. Specifically,the following list of VVO equipment is considered for the test cases studied:ˆ Case 1: ULTCˆ Case 2: Reconfiguration switchesˆ Case 3: Capacitors1064.3. Case Studies and Numerical Resultsˆ Case 4: Capacitors and storage unitsˆ Case 5: ULTC and reconfiguration switchesˆ Case 6: ULTC, reconfiguration switches, and storage unitsˆ Case 7: ULTC, reconfiguration switches, and capacitorsˆ Case 8: ULTC, reconfiguration switches, capacitors, and storage unitsAlso included in Table 4.1 is the differences between the optimized system comparedwith the default test system in terms of the five metrics of quality. A negativesign in the row of comparisons indicates that optimizing the objective function (i.e.,expected active power losses) in the system worsens that particular metric of quality.1074.3.CaseStudiesandNumericalResultsTable 4.1: Performance of the Optimal VVO Solution Under Different SettingsLosses Peak Load Minimum Voltage Maximum Voltage Voltage SpreadValue Reduced by Value Reduced by Value Increased by Value Reduced by Value Reduced by(kW) (%) (kW) (%) (p.u.) (p.u.) (p.u.) (p.u.) (p.u.) (p.u.)Default 37.94 - 2.256 - 0.951 - 1 - 0.036 -Case 1 37.29 1.72 2.255 0.05 0.955 0.004 1.031 −0.031 0.047 −0.011Case 2 31.30 17.51 2.241 0.64 0.958 0.007 1.001 −0.001 0.030 0.006Case 3 30.47 19.69 2.241 0.64 0.953 0.002 1.002 −0.002 0.032 0.004Case 4 30.15 20.52 2.169 3.85 0.954 0.003 1.001 −0.001 0.031 0.005Case 5 30.66 19.19 2.240 0.69 0.967 0.016 1.033 −0.033 0.046 −0.010Case 6 30.55 19.47 2.174 3.61 0.969 0.018 1.027 −0.027 0.040 −0.004Case 7 26.97 28.91 2.233 1 0.975 0.024 1.033 −0.033 0.041 −0.005Case 8 25.69 32.27 2.165 4.02 0.966 0.015 1.007 −0.007 0.023 0.0131084.3. Case Studies and Numerical ResultsTable 4.1 suggests that optimal capacitor switching and feeder reconfigurationcan result in maximal improvements in the active power loss of the system. Theresults also show that optimal control of DSS marginally improves on the systemlosses as it decreases the peak load in the system. While feeder reconfigurationand optimal capacitor control can reduce the peak load of the system, the greatestreduction in the peak load comes from optimal DSS scheduling. In terms of minimumvoltage of the feeder, the table shows that feeder reconfiguration in conjunction withULTC can play a very important role. This impact on increasing the minimumvoltage of the feeder becomes more significant if shunt capacitors are also employedfor VVO in the system. Although the improvement in the minimum voltage reducesif all the equipment in the system are jointly controlled for power loss minimization,the average maximum voltage and the voltage spread improve significantly in return.Particularly, if reconfiguration switches, the ULTC, and shunt capacitors are jointlycontrolled in the system, the voltage spread in the feeder is 0.041 p.u.. If the storageunits are also employed for VVO, the voltage spread reduces to 0.013 p.u. whichis in agreement with a much better maximum voltage in this case. All in all, weconclude that the DSS can improve the peak load and the voltage profile of thesystem significantly.To visualize the results presented in Table 4.1, different performance metrics areillustrated in the following figures. Fig. 4.4 presents the reduction in the activepower losses in percentage points, Fig. 4.5 shows the reduction in the peak load inpercentage points, Fig. 4.6 illustrates the increase in the minimum voltage of thesystem in p.u., Fig. 4.7 depicts the decrease in the maximum voltage of the systemin p.u., and Fig. 4.8 shows the decrease in the maximum voltage of the system inp.u..1094.3. Case Studies and Numerical Results   051015202530351 2 3 4 5 6 7 8Case NumberReduction in Losses (%) Figure 4.4: The reduction in active power losses (%) after optimal VVO for varioustest cases.   00.511.522.533.544.51 2 3 4 5 6 7 8Case NumberReduction in Peak Load (%) Figure 4.5: The reduction in the peak load (%) after optimal VVO for various testcases.1104.3. Case Studies and Numerical Results   00.0050.010.0150.020.0250.031 2 3 4 5 6 7 8Case NumberMin. Voltage Increase (p.u.) Figure 4.6: The increase in the minimum voltage of the system (p.u.) after optimalVVO for various test cases.   -0.035-0.03-0.025-0.02-0.015-0.01-0.00501 2 3 4 5 6 7 8Case NumberMax. Voltage Decrease (p.u.) Figure 4.7: The decrease in the maximum voltage of the system (p.u.) after optimalVVO for various test cases.1114.4. Conclusion   -0.015-0.01-0.00500.0050.010.0151 2 3 4 5 6 7 8Case NumberVoltage Spread Decrease (p.u.) Figure 4.8: The decrease in the voltage spread of the system (p.u.) after optimalVVO for various test cases.Fig. 4.9 illustrates the voltage profile of the system under peak load of a typicalday. Two test cases have been removed from the figure to make other curves morevisible. Clearly, use of any VVO equipment in a day-ahead fashion has resultedin an improved voltage profile. Also, as expected, the smoothest voltage profilecorresponds to the case where all VVO equipment are jointly operated in an optimalfashion.4.4 ConclusionIn smart grids with high penetration of renewable energy sources, the voltages ofthe system are subject to sever fluctuations. This is mostly because of the highvariability of non-dispatchable wind generation. Therefore, the operational task ofVVO is of great importance in systems with considerable wind power penetration.In this chapter, the optimal VVO problem aiming at minimizing the expectedactive power loss was formulated and solved. The VVO problem was formulated1124.4. Conclusion5 10 15 20 25 300.950.960.970.980.9911.011.02  DefaultReconfigCapacitorCapacitor, DSSULTC, Reconfig., DSSULTC, Reconfig., CapacitorULTC, Reconfig., Capacitor, DSSFigure 4.9: Voltage profile of the system under different VVO test cases. The voltageprofile is of a typical day at peak hour.using mixed-integer convex programming to be solved using existing branch-and-bound methods. The methodology presented in the chapter is comprehensive in thatit jointly formulates the operation of shunt capacitors, storage units, reconfigurationswitches, and ULTCs. The stochasticity of wind power generation was also addressedusing a Markov chain model. Numerical results on an active smart grid and usingreal data from smart meters and wind turbines was presented and discussed. Theresults indicated the significance of shunt capacitors and feeder reconfiguration inactive power loss minimization and the importance of DSS in alleviating the peakload and improving the voltage profile of the system.113Chapter 5Conclusions and FutureResearchThis section provides the conclusions and some possible directions for future researchabout active smart grids.Smart grids are the next-generation distribution systems that that are morevisible due to advanced metering infrastructures and more controllable due to thetwo-way communication between the SGO and the demand side. The widespreaduse of renewable energy sources, energy storage systems, and remotely controllablereconfiguration switches will enable the future smart grids to be operated in a muchmore active fashion. The active nature of smart distribution systems will require amore accurate monitoring and a more sophisticated planning and operation.In the present thesis, various aspects of active smart grids was studied andmathematical formulations and algorithmic solutions was presented to be used bythe SGO. In particular, three major problems was considered which included robustmeter placement for state estimation and situational awareness in reconfigurablesmart grids, optimal planning of energy storage systems for smart distribution grids,and VVO for active smart grids. The main focus of the thesis was to mathematicallyformulate different planning and operational problems in such a way that can besolved efficiently for real smart grids. The methodologies presented in the thesiswere validated on sample test systems using real data of smart meters and renewable114Chapter 5. Conclusions and Future Researchenergy sources.It was shown in this thesis that the energy storage systems can have considerableimpacts on the planning and operation of future smart grids. In fact, the usage ofenergy storage systems can improve the voltage quality and alleviate the peak loadof the system. The peak load alleviation of the distribution systems can in turn allowthe system operator to defer the upgrade of the system which readily translates intonoticeable financial gains. The peak load of the system can also be further improvedby reducing the losses. The reduction in the system losses can be achieved througha VVO scheme which optimally operates the system equipment such as ULTCs,capacitor banks, and reconfiguration switches.The work presented in this thesis can be extended in various aspects. In whatfollows we propose a few possible directions for future research about active smartgrids.ˆ It was assumed in this thesis that the location of the DGs are already de-termined in the system. The joint placement of DGs and DSS in the systemcan be a topic of future research for systems in which the DGs are yet to belocalized.ˆ A major advantage of storing the energy within the grid is to improve theresilience of the system. To estimate the improvement in the system resiliencemore accurately, the criticality and the inter-dependence of the of the loadsshould be taken into account. To model the inter-dependence of the criticalloads in the system, the Infrastructure Inter-dependence Simulator (I2Sim)platform [114] can be employed. Therefore, another possible line of researchis to integrate the I2Sim platform into the optimal DSS planning procedure.Use of I2Sim also enables the SGO to operate the DSS in the system in sucha way that the reliability of critical infrastructure is enhanced.115Chapter 5. Conclusions and Future Researchˆ In future smart grids, the customers (especially the private owners of criticalloads) may be willing to pay more for an improved reliability level. A possibleline of research could be to investigate how the cost-benefit trade-off of DSSinstallation is affected in systems with such reliability-oriented customers. Anenabler of such a paradigm can be a customer specific energy pricing in futuresmart grids.ˆ A unique feature of smart grids is a two-way communication between the SGOand the grid. It would be interesting to investigate the impact of cyber attacksand communication errors on the real time operation of smart grids. 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