Load Capability for Smart Distribution Systems. A Thesis. Submitted to the Faculty. Drexel University. Nicholas Stephen Coleman

Transcription

1 Load Capability for Smart Distribution Systems A Thesis Submitted to the Faculty of Drexel University by Nicholas Stephen Coleman in partial fulfillment of the requirements for the degree of Master of Science in Electrical Engineering June 2013

2 Copyright 2013 Nicholas Stephen Coleman. All Rights Reserved.

3 ii Acknowledgements First, I would like to my advisor, Dr. Karen Miu, for her encouragement, guidance, and thoughtful input both as a teacher and as an advisor. I am extremely grateful to Dr. Miu for giving me the opportunity to conduct research and complete my thesis in the Center for Electric Power Engineering. This experience has been formative and invaluable. I give thanks to Dr. Chika Nwankpa and Dr. Harry G. Kwatny for serving on my thesis committee. I deeply appreciate the commitment of their time, input, and expertise. I would like to thank all of the members of the Center for Electric Power Engineering, who have each contributed to my success in some way throughout this process. I am especially thankful to Christian Schegan and Nicole Segal, who have consistently made themselves available when I needed some advice or extra brainpower. Finally, I am most grateful and thankful to Mom, Dad, Steve, Ryan, and Jessica. Their commitment, support, and encouragement made this thesis possible.

4 iii Table of Contents LIST OF TABLES... vi LIST OF FIGURES... vii ABSTRACT... ix 1. INTRODUCTION Motivation Previous Works Objectives MODELING Constraints Electrical Constraints Operating Constraints Load Variation Concepts Units Load Update Function Inputs and Outputs of the Implicit Temporal Formulation Selected Parameters for the General Load Capability Model Branch Current-Limited Load Variation Factor Power Factor-Limited Load Variation Factors Forecast-Limited Parameter of the Implicit Temporal Load Capability Model PROBLEM FORMULATIONS General Load Capability Model Implicit Temporal Load Capability Model...15

5 iv 4. SOLUTION ALGORITHMS Algorithm Inputs Algorithm I: Main Solution Algorithm Sub-Algorithm I: General Load Capability Estimator Sub-Algorithm II: Unscheduled Control Operation Capacitor Scoring Heuristics Scoring Heuristic 1: Branch Current Magnitude Constraint Violation Scoring Heuristic 2: Power Factor Constraint Violation Simultaneous Occurrence of Multiple Constrain Violatios SIMULATION RESULTS Test Circuits Load and Branch Models Load Categories Seasonal Loading Conditions Shunt Capacitor and Battery Storage Models Three-Phase Powerflow Solver Simulation Assumptions and Parameters General Load Capability Simulations Results of General Load Capability Simulations Discussion of General Load Capability Simulation Results Implicit Temporal Load Capability Simulations Daily Forecast Battery Deployment Schedules...33

6 v Implicit Temporal Load Capability Simulation Parameters Results of Implicit Temporal Load Capability Simulations Discussion of Implicit Temporal Load Capability Simulation Results CONCLUSION Contributions Future Work...44 BIBLIOGRAPHY...46 A. DERIVATION OF POWER FACTOR-LIMITED LOAD VARIATION FACTORS...47 A.1 Lagging Power Factor-Limited Load Variation Factor...47 A.2 Extension to the Leading Power Factor-Limited Load Variation Factor...49 A.3 Alternative Approach...50 B. DETAILED LOAD DATA...51 C. CIRCUIT 1 VOLTAGE PROFILE UNDER EXTREME LOADING CONDITIONS..57

7 vi List of Tables 5.1. Seasonal load levels for each of the loading conditions, recorded at midnight. Each load category row gives the lumped value of all the loads belonging to that category Bit numbers, bus numbers, and nominal sizes of batteries and capacitors in the simulated circuits. The batteries are modeled as constant real power. The capacitors are modeled as constant admittance with a nominal-lineline voltage of 12.47kV Single load variation vectors for the general simulations. Variation units defined in fraction of nominal load per hour General load capability simulation parameters General load capability simulation results Sequence of load variation vectors for the daily forecast. = 1 h,. Variation vector units defined in fraction of nominal load per hour Battery deployment schedules. All batteries are nominally off. The on and off values specify time-window numbers Implicit temporal load capability simulation parameters B.1. C.1. Detailed nominal load data for each simulated circuit and load condition pair. Load model key: S = constant complex power, I = constant current, Z = constant impedance. Load category key: 1 = office, 2 = public works, 3 = residential, 4 = industrial Circuit 1 voltage profile under ten-time-seasonal-peak midnight loading conditions....57

8 vii List of Figures 2.1. Example illustrating the use of the subscripting indices. vector directions are represented by the load slopes, which are constant for each time-window., limits each sub-window length to ensure that constraints are not violated. Both capacitor operations are unscheduled control actions Descriptions of conditions that parameterize the power factor-limited load variation factors. Note that ± indicates that both and should be tested to determine the conditions. For each phase, during any subwindow,, no more than one of the power factor-based estimators can possibly limit load variations Heuristic 1, used in the event of an overcurrent violation Heuristic 2, used in the event of a power factor violation One-line diagram of circuits. The full diagram represents Circuit 1. The outlined sub-circuit represents Circuit 2. Subscripts on the load symbols indicate the load model (S = constant complex power, I = constant current, Z = constant impedance), and the load category (1 = office, 2 = public works, 3 = residential, 4 = industrial) Graphical solution to simulation T1.1. The green, blue, and red lines show the apparent, real, and reactive substation power output over time, respectively. The cyan and magenta lines show the battery real power injections and the capacitor reactive power injections over time, respectively Graphical solution to simulation T1.2. The green, blue, and red lines show the apparent, real, and reactive substation power output over time, respectively. The cyan and magenta lines show the battery real power injections and the capacitor reactive power injections over time, respectively Graphical solution to simulation T2.1. The green, blue, and red lines show the apparent, real, and reactive substation power output over time, respectively. The cyan and magenta lines show the battery real power injections and the capacitor reactive power injections over time, respectively Graphical solution to simulation T2.2. The green, blue, and red lines show the apparent, real, and reactive substation power output over time, respectively. The cyan and magenta lines show the battery real power injections and the capacitor reactive power injections over time, respectively

9 5.6. Graphical solution to simulation T3.1. The green, blue, and red lines show the apparent, real, and reactive substation power output over time, respectively. The cyan and magenta lines show the battery real power injections and the capacitor reactive power injections over time, respectively Graphical solution to simulation T3.2. The green, blue, and red lines show the apparent, real, and reactive substation power output over time, respectively. The cyan and magenta lines show the battery real power injections and the capacitor reactive power injections over time, respectively viii A.1. Power triangle example...47

10 ix Abstract Load Capability for Smart Distribution Systems Nicholas Stephen Coleman The load capability problem has traditionally been associated with long-term system planning, but as distribution networks become increasingly strained, load capability is becoming a daily challenge for grid operators. In a smart grid environment, operators can measure system states remotely and in real-time, which would allow real-time load capability assessment if desired. Utilities could benefit from smarter measurement schemes, in which data collection frequencies adapt to the condition of the circuit. This thesis presents the implicit temporal load capability formulation, which, given circuit parameters and a load forecast, identifies times when the system would benefit from closer measurement monitoring. In addition to the new formulation, this thesis presents an updated version of general load capability, which includes new equations to compute load capability with respect to power factor limits, clarifies the units of the problem, and introduces time as a solution parameter. Corresponding solution algorithms are proposed and implemented in a MATLAB-based software package that solves both problems. Simulation results were obtained on 21- and 105-bus distribution networks. One set of simulations solved general load capability for the 105-bus system with respect to both positive and negative load variations. These simulations successfully demonstrate efficient computation of currentand power factor-limited load variation factors as a part of the updated general formulation. A second set of simulations solved implicit temporal load capability for both circuits with a given load forecast, and various loading conditions and control settings. These simulations efficiently identify critical times and lull periods in a load forecast, where utilities may benefit from modified data collection schemes.

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12 1 1. INTRODUCTION As we improve the infrastructure of our aging power grid, we must also improve the engineering methods used in the planning and operation of the grid. Engineers should strive to take advantage of increasingly available real-time measurement data when making operating decisions. Specifically, engineers can use this improved measurement data to evaluate network load capability in real-time at the distribution level. Load capability is a measure of how much loads can vary in a certain direction before a constraint violation occurs. This thesis presents two applications of distribution load capability: an updated version of the general load capability problem introduced in [1], [2], and a new formulation, titled implicit temporal load capability, which applies load capability to daily load forecasts. 1.1 Motivation The load capability problem has traditionally been associated with long-term system planning; transmission planners use load capability tools to identify bottlenecks to major load growth, and to determine when and where infrastructure upgrades will be necessary. However, as power demands continue to grow, distribution systems are operating closer to their capacities. Smart grid technologies provide sustainable solutions to this challenge by allowing engineers to operate the power grid more efficiently and intelligently; this thesis aims to embrace smart grid technologies as part of a load capability application for distribution systems. In a smart grid environment, operators can measure system parameters automatically and remotely. Using this wealth of real-time data, operators could form strong predictions of load movement, and in turn, assess load capability over a week or a day, rather than over a year or a decade. Since every type of collected data point comes with an associated cost, the problem now is to determine how frequent is frequent enough to ping deployed measurement devices for data. Presently,

13 2 regulatory bodies dictate how often transmission system operators must collect measurement data, but these data collection frequencies are maladaptive. For example, PJM s Manual 18B: Energy Efficiency Measurement & Verification [3] (2010) requires that any measurement, monitoring and data recording equipment that sample continuously and integrate values should collect data at a frequency of one hour or less. In order to take full advantage of smart grid tools, standards like data collection frequencies must become more sophisticated. This thesis identifies distribution networks, which have few or no standards with regard to measurement pinging frequency, as a candidate for implementation of an adaptive measurement schedule. The implicit temporal load capability formulation presented in this thesis can help distribution grid operators design more intelligent, individualized measurement schedules, which collect data more frequently when circuits are operating near their limits, and less frequently when constraint violation is unlikely. Additionally, since automation is an integral component of real-time interaction with the smart grid, this thesis strives to improve upon previous formulations by presenting both the general and the implicit temporal load capability algorithms in a manner that is directly translatable to computer programming logic. 1.2 Previous Works The first presentation of the general load capability model appeared in Sauer, et. al. [4] (1990), in which the authors measured load capability as the maximum scalar load multiplier for which a load flow solution existed. By considering only electrical constraints, [4] essentially found an upper bound on the true load capability of a given system. Both Miu & Chaing [1] (1997), [2] (2000) and Irissari et. al. [6] (1997) presented load capability estimates with respect to electrical and operational constraints. The authors of [6] studied the load capability of transmission networks of over 4,000 buses, and used a voltage stability constraint to parameterize the problem. The authors of [1], [2] applied load capability to distribution-

14 3 level networks, and selected bus voltage magnitude limits and branch current ratings as the constraining parameters. Several works have demonstrated the importance of load capability in short-term planning and evaluation tools. For example, Liu et. al. [6] (2009) use a very simple load capability estimator to conduct an efficient voltage stability assessment. The post-processing capacitor control algorithm presented in Miu & Wan [7] (2000) successively executes the load capability algorithm in [2] to identify potential constraint violations as a preliminary step towards identifying a capacitor turn-on sequence that would maintain system feasibility between two discrete load levels. The implicit temporal analysis presented in this thesis uses an adapted version of the algorithm presented in [2]. 1.3 Objectives The objectives of this thesis are to: introduce a model to compute load capability with respect to lagging and leading power factor limits. present an improved general load capability problem formulation that clarifies units and includes time as a problem parameter. design the implicit temporal load capability problem formulation. design and implement a solution algorithm for solving both the general and implicit load capability problems for a given load forecast on a radial circuit. simulate both the general and implicit load capability problems for given load forecasts on radial, three-phase, distribution test circuits.

15 4 2. MODELING Given a set of initial load sizes and a load forecast, the solution to the implicit temporal load capability problem includes a set of critical times when a network operator must take action in order to maintain system feasibility, and a corresponding set of possible control options. These critical times are defined to occur when the forecasted loads will result in an operating constraint violation. This chapter outlines the quantities that parameterize the electrical and operating constraints that bound the problem, and the components required to model both the general and implicit temporal load capability problems for a given circuit. 2.1 Constraints The physical laws dictating the behavior of all electric circuits give rise to electrical constraints. To ensure that any set of states associated with a particular circuit solution are physically realizable, solvers and simulators must guarantee that the electrical constraints are satisfied. In addition to the physical laws that give rise to electrical constraints, distribution networks are subject to operating constraints defined by equipment ratings, operator knowledge, and economic considerations. Operating constraints restrict the system states to the set of operationally acceptable system states. The intersection of the sets of physically realizable and operationally acceptable states is the set of feasible system states; the purpose of load capability problem is to ensure that system states remain within the set of feasible states Electrical Constraints For a circuit with three-phase buses, the electrical constraints consist of up to 6( 1) nonlinear, real-valued, simultaneous equations known as the power flow equations. For notational convenience, the power flow equations take on a compact representation:

16 , = 0 (2.1) 5 where =,,,,, R () are the unknown states, =,, R are the voltage phase angles across the phases at bus, =,, R are the voltage magnitudes across the phases at bus, and represents the susceptences and conductances associated with the active control settings. The 6( 1) power flow equations that define the electrical constraints are the real and reactive parts of Tellegen s Theorem applied at each phase of each of the 1 load buses. This work assumes that the slack bus is always denoted bus 1. The power flow equations are: cos + sin =, = 2,, (2.2) sin cos =, = 2,, (2.3) where R are the three-phase real power injections at bus, R are the three-phase reactive power injections at bus, R are the three-phase voltage magnitudes at bus, R are the three-phase voltage magnitudes at bus, = R are the voltage phase angle differences between nodes at buses,, R are the three-phase network conductances connecting buses,, and R are the three-phase network susceptences connecting buses, Operating Constraints Like the electrical constraints, the operating constraints of a system take on a compact form:

17 , 0 (2.4) 6 Grid operators may choose to define the constraints represented by (2.4) in different ways depending on the circuit properties and operational objectives. Common operating constraints include limits on bus voltage, branch current, and apparent power magnitudes:,, (2.5), (2.6), (2.7) for phases =,,, buses = 1,,, and feeders = 2,,. As measurement and control capabilities improve, operators become more able to interact with circuits at the distribution level. This gives planners and operators the freedom to impose additional operating constraints that further reduce cost and maintenance requirements. Specifically, this thesis includes lagging and leading power factor limits across each phase at the substation (bus 1) as a member of the operating constraints represented by (2.4):,, if is lagging if is leading (2.8) 2.2 Load Variation Concepts The purpose of studying load capability is to compute how much network loads can vary in a certain direction before a violation occurs. Each potential violation type is associated with a different measure of load capability, quantified with a scalar load variation factor, R. The direction of load variation associated with this factor is called a load variation vector, = R, where is the active time-window. Note that in the general load capability formulation, only one time-window exists per problem. Time-window subscripting is only necessary for the implicit

18 7 temporal formulation, but subscripts are included for generality. The vector contains separate real and reactive variation vectors, each with dimension equal to the number of load nodes, assuming a three-phase, bus system, with a substation at bus 1 (the slack bus) Units Previous works on load capability have defined and as unitless quantities; this thesis departs from this practice and assigns units of time to, and units of inverse time to in order to better accommodate the objectives of the work. The specific time unit depends on the available load forecast. For example, if a load forecast projects that loads will change at the same rate and that no scheduled control actions will occur over the day, the unit for is inverse days, and the unit for is days; = 1 then corresponds to an allowable duration of one day Load Update Function For a load variation along the vector, over the period [,,, + ], the load update function is:, + =, + + (2.9) where, at continuously valued time, C () are the three-phase complex power injections of the loads, R () are the three-phase real power injections of the loads, R are the three-phase reactive power injections of the loads, R () are three-phase variation vectors for the real part of the load, R () are three-phase variation vectors for the reactive part of the load,, is the time at the beginning of sub-window,,, is the time when the nominal loads are recorded, and the

19 8 operator represents the Hadamard product. Section explains the concept of a sub-window as part of a load forecast Inputs and Outputs of the Implicit Temporal Formulation The implicit temporal formulation requires a fully specified forecast as an algorithmic input. A forecast includes the following elements: =,,,, a sequence of given load variation vectors, where =, = 1,,, the given variation vector during time-window, =,,,, a sequence of vector durations associated with each, and =,,,, a sequence of scheduled control settings, where is a binary string indicating the scheduled status of each control device in the network at the beginning of time-window. A new time-window begins when a new variation vector is available, or when a control action is scheduled. Note that if a scheduled control action will violate an operating constraint, the implicit temporal load capability algorithm will repeatedly delay the control action by one time-window until the action is feasible. If the forecast ends before the action is feasible, or if a second scheduled action undoes the intended effect of the infeasible action, then the algorithm cancels the scheduled infeasible action entirely. Each time-window contains one or more sub-windows. Within a time-window, secondary sub-windows ( > 1) begin when load capability determines that time-window cannot be completed without unscheduled operator (or automated) intervention. The general load capability algorithm runs once during each sub-window and outputs an associated,, which is equal to the duration of sub-window,, and an associated,, which includes the realized control settings active during sub-window,, if any. For a fully specified circuit and forecast, the solution to implicit temporal load capability is modeled as:

20 Λ =,,,,,,,,,,, the sequence of load variation factors, where, is the =, variation factor in the time-window, = 1,,, and = 1,,, and,,,,,,,,,, the sequence of realized control actions, where, is the binary string representing the scheduled and unscheduled control settings active at during sub-window,. 1 is the number of sub-windows (and therefore the number of variation factors) contained within time-window. is not known until the time-window has ended; therefore, the dimensions of Λ and are determined as a part of the solution. Figure 2.1 presents a simple example to illustrate the relationship between the subscripting indices of the variation factors, variation vectors, and time-windows. 9 Temporal Load Profile Example 12 Operating Constraint Violation 10 Load (KVA) 8 6 Capacitor On Capacitor Off 4 Sub-Windows: Operating Constraint Violation λ 1,1 λ 2,1 λ 2,2 λ 3,1 λ 4,1 λ 5,1 λ 6,1 λ 6,2 2 Time Windows: T 1 T 2 T 3 T 4 T 5 T Time (h) Figure 2.1. Example illustrating the use of the subscripting indices. vector directions are represented by the load slopes, which are constant for each time-window., limits each sub-window length to ensure that constraints are not violated. Both capacitor operations are unscheduled control actions that maintain feasibility.

21 Selected Parameters for the General Load Capability Model As evidenced by [2], [4], [5], [6], and other works that present unique load capability formulations, there are no universally applied parameters of load capability other than the requirement that the load flow solution exists; however, as discussed in Chapter 1, most authors tailor additional parameters to their specific purposes. The general load capability model presented in this thesis parameterizes load capability by current magnitude limits, as adapted from [2], and introduces a power factor-based parameterization. The current- and power factor-based parameters are of immediate concern when dealing with increasingly strained and diverse circuits at the distribution level. This section introduces the individual load variation factors that parameterize the general load capability formulation,,, Branch Current-Limited Load Variation Factor The current-limited load variation factor,, adapted from [2], requires the greatest computational expense of any of the selected measures. The equation for is non-linear, and its solution must be computed iteratively. Chapter 4 presents the full computation algorithm. Across all phases and branches, the current-based estimator update function given in [2], with modified notation, is: Δ,, =, Δ, + Δ,,,, =,, ; = 2,, Δ = min min Δ,, (2.10) = Δ where

22 11 Z is the iteration number, Δ,, R is the current-limited load variation factor update quantity at bus, phase, at iteration, R is the margin between actual and rated branch current magnitudes feeding bus, phase, is the set of buses downstream of bus, Δ, C is the change in current injection at bus, phase due to scheduled capacitor control actions, Δ, C is the change in current injection at bus, phase due to scheduled cogenerator control actions,, =,,, +,,, C is the forecasted change in complex load injections per unit time at bus, phase,, C is the line-to-neutral voltage at bus, phase, and Δ R is the branch current-limited load variation factor update quantity over the entire network at iteration. Note that the per-time unit selected for, is consequently the time unit of Δ. Equation (2.10) includes a dependence on anticipated changes in current injections due to future capacitor and cogenerator control actions. These terms are correctors that improve the robustness of the general load capability formulation presented in [2], but the definition of a timewindow renders these terms unnecessary in the formulations presented in this thesis. Omitting these corrector terms and adding the necessary subscripts gives the equation for the current-limited update quantity for this work:

23 12 Δ,,, =,,,,,,, =,, ; = 2,, Δ, = min,, min Δ, (2.11), = Δ, Power Factor-Limited Load Variation Factors This section introduces and, the lagging and leading power factor-limited load variation factors, respectively. Given the loading condition during sub-window, the equations for the lagging power factor-limited load variation factors are:,,,,, =,,,,, Condition II, Condition I, III = min,, (2.12),,,,, =,,,, Condition III, Condition I, II = min,,, (2.13) where + superscripting indicates a lagging quantity, superscripting indicates a leading quantity, ±,, R are the power factor-limited load variation factors on phase,

24 ± = ± tan ± R ± are the maximum allowable substation power ratios, which corresponds to the minimum allowable lagging and leading power factors, ± = cos ( ± ), = [, ] is the power factor angle at the substation,,, R is the total substation real power output on phase at the beginning of window,,,, R is the total substation reactive power output on phase at the beginning of window,,, = =, R is the forecasted rate of change of substation real power output on phase during time-window,, R is the forecasted rate of change of substation reactive power output on phase during time-window, ±, R are the power factor-limited load variation factors over all substation phases, and the conditions that parameterize these equations are: Condition I: the substation power factor is tending towards an acceptable value Condition II: the substation power factor is tending towards an unacceptable lagging value Condition III: the substation power factor is tending towards an unacceptable leading value Figure 2.2 illustrates the procedure for determining the variation condition on each phase,. 13 No No No Yes Yes Yes Yes Yes Condition I Condition II Condition III No Yes Yes No Figure 2.2. Descriptions of conditions that parameterize the power factor-limited load variation factors. Note that ± indicates that both and should be tested to determine the conditions. For each phase, during any sub-window,, no more than one of the power factor-based estimators can possibly limit load variations.

25 (2.12), (2.13) are one-to-one and monotonic over the subsets of their respective domains that do not map to infinity. Full derivations of (2.12), (2.13) are available in Appendix A Forecast-Limited Parameter of the Implicit Temporal Load Capability Model The implicit temporal load capability includes all of the parameters defined in Section 2.3, and one additional parameter of load capability, the forecast-limited load variation factor,,. This parameter is limited by the expiration of the present time-window,, and is computed using:, =, (2.14) where R is the total duration of time-window. The summation in (2.14) represents the summed duration of all the sub-windows that have already occurred during time-window. The purpose of is to limit variation vector to the prescribed duration, regardless of whether or not a constraint violation will occur before the end of the window.

26 15 3. PROBLEM FORMULATIONS This chapter presents two problem formulations. The general formulation improves on previous work by allowing for non-uniform, 6( 1) dimensional load variations, and by introducing time as a solution parameter. The implicit temporal model is a new formulation that incorporates control actions and successively utilizes the general formulation over a given load forecast. 3.1 General Load Capability Model The purpose of the general model is to evaluate the maximum duration,, for which a circuit with constant control setting can withstand load changes in the direction of a given. The general model does not make use of time- or sub-windows; therefore, time- and sub-window subscripting is omitted. The general load capability problem is: max (3.1) s.t.,, = 0,, 0 where = min,, (3.2) and,,,,, indicate that the constraints must remain satisfied after load variations proportional to. 3.2 Implicit Temporal Load Capability Model While the general load capability problem focuses on a variation vector, the implicit temporal model assesses the capability of a circuit to withstand load changes over a time-varying load forecast that spans multiple time-windows. A sequence of load variation vectors, a corresponding sequence of

27 finite durations, and a planned control action schedule fully specify a forecast. The primary results of temporal load analysis are a sequence of load variation factors that identify when control actions are necessary, and a sequence of corresponding control settings that maintain circuit feasibility. Given a fully specified load forecast, which includes a load variation vector sequence,, time-window durations,, and a control action schedule,, the goal of the temporal load profile analysis is to find: s.t. Λ =,,,,,,,,,, (3.3) =,,,,,,,,, = 0,,, (3.4) 16,, 0,,, where, = min,,,,,,, (3.5) Please refer to Figure 2.1 for a graphical example of this solution. Note that the realized control output,, does not necessarily reflect all of the elements of the control schedule input,. In order to maintain network feasibility, the solution will omit any scheduled control action that will cause a constraint violation.

28 17 4. SOLUTION ALGORITHMS The solution algorithm for the implicit temporal load capability problem requires a set of input data and multiple iterative routines during each time-window. This chapter outlines the inputs to and the steps of each of these routines. Algorithm I is the main algorithm for the temporal load capability formulation. Sub-Algorithm I is the general load capability estimator, which computes the load variation factor during each sub-window. Sub-Algorithm II ranks and executes control actions; this is necessary only when a constraint violation occurs. 4.1 Algorithm Inputs The general and implicit temporal load capability solution algorithms each require a set of inputs. Both models require: specified topology and impedance data for a given radial distribution network. specified nominal load values. specified initial control settings. a stopping tolerance,, for computing. In addition, the general load capability solution algorithm (Sub-Algorithm I) requires: a single load variation vector,, which specifies the load variation vector in terms of fraction of nominal load per unit time for the real and reactive portions of each load in the network. The implicit temporal algorithm (Algorithm I) requires: a load variation forecast,, which consists of sequential load variation vectors,, = 1,,. a sequence of durations,, which correspond to each variation vector. a control schedule,, which consists of control settings, = 1,,, which are scheduled to take effect at the beginning of each time-window,.

29 Algorithm I: Main Solution Algorithm Step 1. Run three-phase power flow. Step 2. Set time-window = 1. Set sub-window = 1. Step 3. [Sub-Algorithm I] Compute, along variation vector for sub-window, (3.5) If, 0, the control actions required by have caused an infeasible circuit condition. Set, = 0. Move to the next sub-window: = + 1. Undo and return to Algorithm I, Step 3. Else, continue. Step 4. Step 5. Record the realized control setting for sub-window,,,. Update all of the loads (2.9). Run three-phase power flow If, =, move to the next time-window: = + 1, = 1. Else, [Sub-Algorithm II] attempt an unscheduled control action to maintain feasibility. Perform three-phase power flow. Move to the next sub-window: = + 1. Step 6. Check If >, stop. Return a full solution: Λ (3.3), (3.4). Else, return to Algorithm I, Step 3, and continue. 4.3 Sub-Algorithm I: General Load Capability Estimator Step 1. Compute, iteratively as follows: 1.1. Determine the spare capacity,, in each branch Compute Δ,,, (2.11) Update the loads (2.9). Run three-phase power flow If Δ, <, continue.

30 19 Else, return to Sub-Algorithm I, Step 1.1, and continue. Step 2. Determine the load variation condition using Figure 2.2. Step 3. Step 4. Step 5. Compute, (2.12). Compute, (2.13). If computing general load capability, select the load variation factor (3.2), stop and return a solution,. Else (if computing implicit temporal load capability), continue. Step 6. Step 7. Compute, (2.14). Select, (3.5). Return to Algorithm I, Step 3.1, and continue solving implicit temporal load capability. 4.4 Sub-Algorithm II: Unscheduled Control Operation Step 1. Select a control option as follows: 1.1. Assign a score, to all of the allowable control options using the heuristics given in Section Eliminate control options with a score Rank the remaining control options in order of descending (list the highest scoring control option first). Step 2. Let be the number of eligible control operations. Set = 1. Step 3. Step 4. Execute the control option on the list. Run three-phase power flow. Check for operating constraint violations If the control action has cleared all operating constraint violations, stop. Return to Algorithm I, Step 5.1. Else, undo the previous control action. Set = + 1 and continue.

31 If >, stop and exit the program. Return a partial solution: Λ =,,,,, U =,,,,. The constraint violation is unavoidable under the given load forecast. The forecast is infeasible. Else, return to Sub-Algorithm II, Step 3, and continue. 20 Sub-Algorithm II, Step assumes that all operating constraints are hard constraints. When a constraint violation is unavoidable, this step exits the program, reports a partial solution, and declares the given load forecast infeasible. If desired, one can easily modify Sub-Algorithm II to soften these constraints in an attempt to make it to the end of the load forecast. For example, let represent the maximum duration for which a constraint violation may exist. An operator may decide that in the event of a violation during sub-window,, that if,, then to continue the forecast and ignore the constraint violation until the beginning of the next time-window. This inherent flexibility is an advantage of implicit temporal formulation; previous load capability formulations, which do not include time as a problem parameter, do not provide operators with the ability to make such decisions. 4.5 Capacitor Selection Scoring Heuristics This section describes the scoring heuristics called in Sub-Algorithm II, Step 1. Given the type and location of a constraint violation, and the present switch settings of all of the allowable controllable options in the network, each heuristic assigns a score to each possible control option. This thesis only considers single capacitor operations as possible unscheduled control actions, although one could easily modify the algorithm to include a wider search space of control options or entirely different control type.

32 Scoring Heuristic 1: Branch Current Magnitude Constraint Violation Overcurrent violations can be relieved by turning on a shunt capacitor bank located downstream of the violation (capacitors at the violation bus are included in the set of downstream capacitors). Figure 4.1 illustrates the process for assigning a score to each capacitor bank in the network in the event of a branch current violation at branch, phase, at the end of sub-window,. Figure 4.1 uses the following notation for scoring capacitor bank, located at bus : N is the number of times that has previously operated during the present forecast,, R is the reactive power injection of, phase, at the end of sub-window,,, C is the voltage at bus, phase, at the end of sub-window,, and R is the score assigned to for the purposes of clearing the present violation. Is the bank at bus presently off? Yes Is bus downstream of violation bus Yes No No Figure 4.1. Heuristic 1, used in the event of an overcurrent violation. The process shown in Figure 4.1 assigns a score of zero to capacitors that cannot or are unlikely to clear the violation. The remaining capacitors are scored based on current relaxation at the

33 violation node, and penalized if the selected capacitor has already operated during the present forecast. For notational consistency, the scoring equation is numbered: 22, 1 = ( + 1),,,,, if C is eligible to clear the violation 0, else (4.1) Scoring Heuristic 2: Power Factor Constraint Violations A power factor constraint violation indicates that the substation power factor is unacceptable. Turning on any inactive capacitor bank in the network will improve a lagging substation power factor. Similarly, turning off any active capacitor bank in the network will improve a leading substation power factor. Therefore, for a lagging (leading) power factor violation at sub-window,, any capacitor bank that is inactive (active) at the end of sub-window, is eligible for operation. Figure 4.2 illustrates the process for assigning a score to each capacitor bank in the network in the event of a power factor violation at phase, at the end of sub-window,. Leading Leading or Lagging PF Violation? Lagging Is bank presently on or off? Off On Is bank presently on or off? On Off Figure 4.2. Heuristic 2, used in the event of a power factor violation.

34 For consistency, the scoring equation for capacitor to clear a power factor violation at the end of sub-window, is: 23 1 = + 1, + Δ,,, if C is eligible to clear the violation 0, else (4.2) where R is the apparent power rating of the substation, and Δ,, R is the change in three-phase reactive power injection by capacitor that would occur due to bank operation the end of sub-window,. Equation (4.2) assigns the highest scores to the capacitors whose operation would bring the substation power factor closest to unity. The quantity, + Δ,, gives the total reactive power at the substation after operation of. Subtraction from ensures that smaller, + Δ,, values map to higher scores Simultaneous Occurrence of Multiple Constraint Violations In the event that two or more constraint violations of different types occur simultaneously, the system operator must either design an additional scoring heuristic, or prioritize the existing heuristics and select the most restrictive one. In this thesis, the heuristic associated with branch current violations, Heuristic 1, has a higher priority than Heuristic 2. In the event of simultaneous currentand power factor-based violations, Heuristic 1 would be used to clear the current violation, and then, if the power factor violation still exists, Heuristic 2 would be used to clear the remaining power factor violation.

35 24 5. SIMULATION RESULTS This chapter presents and discusses the results obtained using the proposed approaches. A program written in MATLAB 7.11 executed the solution algorithms to obtain simulation results. The goal of the simulations was to demonstrate the functionality of the algorithms presented in Chapter 4 and the accompanying MATLAB program. 5.1 Test Circuits Simulations modeled the operation of two balanced, three-phase, radial distribution circuits. Circuit 1 has 105 buses. Circuit 2 is a 21-bus sub-circuit of Circuit 1. Both circuits included: a single 12.47kV substation (slack bus). constant current, constant complex power, and/or constant impedance load models. constant admittance models of switchable gang-operated three-phase shunt capacitors, which operated automatically to clear violations. constant real power models of a switchable gang-operated three-phase batteries, which operated on a user-defined schedule. Figure 5.1 shows a one-line diagram of Circuit 1 and highlights the portion of the circuit that is equivalent to Circuit 2. These circuits illustrate characteristics specific to power distribution system operation and control. The remainder of this section defines the load models, load categories, and loading conditions on the circuits.

36 SUBSTATION ABC 1100 ABC 1101 ABC 1102 ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC S3 S3 S3 S3 S3 S4 S3 S3 S3 S3 S3 S ABC ABC 1116 ABC 1113 ABC 1117 ABC ABC ABC 1200 ABC 1201 ABC S4 S4 S4 S1 S4 S4 S ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC I3 I3 I3 I3 I3 I4 I3 I3 I3 I3 I3 I ABC ABC 1216 ABC 1213 ABC 1217 ABC ABC ABC I4 I4 I4 I1 I4 I4 I4 Circuit ABC 1301 ABC 1302 ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC AC S3 S ABC ABC S3 S3 S3 S4 S3 S3 S3 S3 S3 S ABC 1313 ABC 1317 ABC ABC ABC S4 S4 S4 S1 S4 S4 S4 Legend 1400 ABC 1401 ABC 1402 ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC AC Source Z3 Z3 Z3 Z3 Z3 Z4 Z3 Z3 Z3 Z3 Z3 Z3 Bus No. Phases MC M = Model C = Category Bus Line Load Capacitor ABC ABC 1500 ABC 1501 ABC 1416 ABC 1413 ABC 1417 ABC ABC ABC Z4 Z4 Z4 Z1 Z4 Z4 Z4 Z ABC S ABC ABC ABC ABC ABC ABC ABC ABC ABC 1516 ABC 1513 ABC 1517 ABC ABC ABC ABC ABC ABC I3 S3 Z3 I4 I3 Z3 S3 S3 I3 S3 Battery Closed Switch Open Switch 1601 ABC S ABC I2 S4 Z4 S4 I1 Z4 S4 S ABC S1 Figure 5.1. One-line diagram of circuits. The full diagram represents Circuit 1. The outlined sub-circuit represents Circuit 2. Subscripts on the load symbols indicate the load model (S = constant complex power, I = constant current, Z = constant impedance), and the load category (1 = office, 2 = public works, 3 = residential, 4 = industrial).

37 Load and Branch Models All of the simulations used constant current (I), constant complex power (S), and/or constant impedance (Z) load models, as defined in [8]. The load subscripts in Figure 5.1 indicate the models of each load. Circuit 1 contains a mixed selection of load models, and Circuit 2 contains all constant complex power models. To model the branches, all simulations used standard distribution line models and zero-impedance switch models, also defined in [8] Load Categories Each load falls into one of four load categories: (1) office, (2) public works, (3) residential, or (4) industrial. The load subscripts in Figure 5.1 indicate the category of each load. Each of the four categories is associated with a different load forecast. For any given simulation, all of the loads assigned to a certain category share a common load variation forecast Seasonal Loading Conditions Seasonal population swings strongly influence the power demands on the circuits. Basic infrastructure needs and year-round residents contribute to the seasonal base load level of approximately 4.73MVA at midnight in Circuit 1. Seasonal occupants contribute to the much larger seasonal peak load level, which, in Circuit 1, is approximately 10.63MVA at midnight. In Circuit 2, the seasonal peak load level is approximately 0.93MVA at midnight; the seasonal base was not considered for Circuit 2. Table 5.1 summarizes the estimated midnight power demands under each of these conditions for each simulation. Note that the midnight power demands are used as arbitrary nominal values; they do not necessarily represent daily minimums, and are subject to variation throughout the day. More detailed load data is available in Appendix B.

38 Table 5.1. Seasonal load levels for each of the loading conditions, recorded at midnight. Each load category row gives the lumped value of all the loads belonging to that category. Load Category Circuit 1, Circuit 1, Circuit 2, Seasonal Base Seasonal Peak Seasonal Peak 1 Office P (kw) 2, , Q (kvar) 1, , Public Works P (kw) 1, , Q (kvar) Residential P (kw) Q (kvar) Industrial P (kw) , Q (kvar) , Total P (kw) 4, , Q (kvar) 2, , S (kva) 4, , PF lagging lagging lagging Shunt Capacitor and Battery Storage Models The battery storage and shunt capacitor models in the circuits are balanced, three-phase, gang-operated components. Constant real power battery models come directly from given nominal real power ratings. To emulate energy storage units used in a 2010 microgrid project [9] with similar circuit properties, bank sizes are specified in multiples of 300 kw. The program computes constant admittance capacitor models with a specified reactive power rating and a nominal line-to-line voltage of 12.47kV. In order for the program to interpret the statuses of the batteries and capacitors, and, are defined as binary strings. Each bit in a string represents one battery or capacitor bank; a 0 indicates that the bank is inactive, and a 1 indicates that the bank is actively injecting power into the network. Table 5.2 lists the bit numbers, bus locations, and nominal three-phase power ratings of the battery and capacitor banks in the circuits. Examples of and, are given after Table 5.2.

39 Table 5.2. Bit numbers, bus numbers, and nominal sizes of batteries and capacitors in the simulated circuits. The batteries are modeled as constant real power. The capacitors are modeled as constant admittance with a nominal-line-line voltage of 12.47kV. Bit Number Type Bus Number Nominal Size (Three-Phase) Unit Ckt. 1 Ckt. 2 Batteries Capacitors kw kw kw kw kw ,500 kw kvar kvar kvar kvar kvar kvar kvar kvar kvar kvar kvar ,200 kvar 28 As an example using the data in Table 5.2 for Circuit 1, = indicates that at the beginning of time-window, only the battery bank at bus 1213 is scheduled to be on. contains scheduled battery statuses only., = indicates that at the beginning of sub-window,, the battery bank at bus 1213, and the capacitor banks at buses 1219 and 1701 are actively injecting power into the network. The spaces in the binary strings are only included for clarity. 5.2 Three-Phase Powerflow Solver Each simulation required multiple calls to solve three-phase distribution powerflow. The program completed this task with a traditional Newton-Raphson solver for MATLAB [8].

40 Simulation Assumptions and Parameters The load capability studies operated under the assumptions that: all nominal load sizes were available before beginning the assessment. subject to the scoring heuristics, capacitors could be operated automatically and instantaneously in the event of a constraint violation. battery storage elements were capable of injecting a constant wattage into the network during their scheduled operation periods. the circuits would always reach power factor or current magnitude limits before reaching voltage magnitude limits. The fourth listed assumption takes advantage of the fact that bus voltage magnitudes profiles are very flat for the simulated circuits. This assumption would not hold for transmission level circuits, in which voltage magnitude limits are critical parameters of load capability [2], [4]. To demonstrate the voltage stability of the circuits, Appendix C presents a voltage profile of Circuit 1 under a ten-timespeak-seasonal loading condition with no reactive power support. The simulations in this chapter parameterize general load capability by branch current magnitude and substation power factor. All of the branches in the circuit are conducting lines or normally closed switches with current magnitude ratings of 395A and 400A, respectively. The minimum allowable power factor in both the leading and lagging directions is 0.8, which corresponds to ± = ± General Load Capability Simulations Four simulations solved the general load capability problem under various conditions using Sub-Algorithm I. Required inputs to Sub-Algorithm I include nominal load sizes, a single load variation vector, shunt element switch statuses, and energy storage element switch statuses in order to solve (3.1), (3.2). For these general load capability simulations, shunt capacitors and battery storage