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Optimal Inverter VAR Control in Distribution Systems with High PV Penetration
Masoud Farivar, Russell Neal, Christopher Clarke, Steven Low
TL;DR
High PV penetration can produce rapid voltage fluctuations and reverse power flow that slow switched devices may not adequately regulate. The paper formulates fast inverter volt/var control as a radial OPF, solves it through an exact convex relaxation, and evaluates its benefits on a lightly loaded SCE feeder with 5 MW of PV. The proposed control nearly eliminates voltage-limit violations and achieves energy savings above 1% in the reported setting, while accounting for inverter losses.
Problem
High PV penetration creates rapid voltage fluctuations and reverse power flow, motivating faster voltage/var regulation than conventional slow switched devices provide.
Method
The paper formulates inverter volt/var control as a radial OPF minimizing line losses and energy consumption under voltage constraints, with inverter losses included.
Results
1,214 hours per year outside the feasibility region under unity power factor were reduced to an almost completely eliminated undesirable mode, while total-cost energy savings exceeded 1%.
Takeaways & Limitations
Optimal inverter reactive-power injection can be nonmonotone with real-power output because topology, loading, voltage limits, line losses, and CVR effects interact.
Takeaways & Limitations
The reported evaluation uses one very lightly loaded SCE distribution feeder with a 5 MW PV system installed about 6 miles from the substation.
Abstract
from arXiv · showhide
The intent of the study detailed in this paper is to demonstrate the benefits of inverter var control on a fast timescale to mitigate rapid and large voltage fluctuations due to the high penetration of photovoltaic generation and the resulting reverse power flow. Our approach is to formulate the volt/var control as a radial optimal power flow (OPF) problem to minimize line losses and energy consumption, subject to constraints on voltage magnitudes. An efficient solution to the radial OPF problem is presented and used to study the structure of optimal inverter var injection and the net benefits, taking into account the additional cost of inverter losses when operating at non-unity power factor. This paper will illustrate how, depending on the circuit topology and its loading condition, the inverter's optimal reactive power injection is not necessarily monotone with respect to their real power output. The results are demonstrated on a distribution feeder on the Southern California Edison system that has a very light load and a 5 MW photovoltaic (PV) system installed away from the substation.
I. INTRODUCTION
High photovoltaic penetration creates rapid voltage fluctuations and reverse power flow that traditional slow volt/var devices may not adequately manage. The paper proposes fast inverter reactive-power control formulated as an optimal power-flow problem balancing voltage regulation, losses, and energy consumption.
- Motivation for inverter var control: Traditional capacitor banks, OLTCs, and voltage regulators switch only a few times daily, limiting their response to rapid renewable-generation fluctuations.The paper identifies faster monitoring and control as a major volt/var-control challenge.
- Motivation for inverter var control: Inverters can inject or absorb reactive power, but IEEE 1547 historically excluded them from active voltage/var regulation.The paper argues that higher renewable penetration makes this inverter capability increasingly relevant.
- Motivation for inverter var control: The proposed approach augments slow switched control with fast inverter control and optimizes line losses, CVR energy consumption, and inverter losses subject to voltage limits.The study also does not impose substation power-factor maintenance as a constraint.
- Motivation for inverter var control: High PV penetration can cause local voltage rise, adverse fluctuations, and reverse power flow on lightly loaded distribution feeders.The studied SCE feeder has a 5 MW generator near the circuit end and can experience peak reverse flow exceeding 3 MW.
- Motivation for inverter var control: Reactive power can provide substantial voltage regulation because voltage rise with kvar is typically 2 to 3 times its response to kW.An inverter sized around 110% of maximum kW output may retain 46% reactive-power capacity at full real-power output.
III. PROBLEM FORMULATION AND SOLUTION
The formulation models balanced radial feeder power flow with DistFlow equations, representing loads and PV real power as given quantities while treating inverter reactive generation as the control variable. The resulting variables support a time-varying inverter volt/var optimization.
- Problem formulation and solution: The model represents the distribution circuit as a radial graph whose buses are nodes and whose lines are links governed by balanced DistFlow equations.The equations recursively describe feeder power-flow quantities across each link.
- Problem formulation and solution: The model defines complex voltage at each bus and active, reactive, and current-flow quantities for each feeder link.These variables support recursive radial power-flow calculations.
- Problem formulation and solution: Load demand and PV real-power generation are given quantities, whereas inverter reactive-power generation is the control variable.Shunt capacitors provide additional reactive power and are reconfigured on a slower control timescale.
- Problem formulation and solution: Shunt-capacitor reactive generation is zero when no capacitor is present or when the capacitor is switched off.When active, its generation is represented using the capacitor rating at the node.
- Problem formulation and solution: Current magnitude on each link is determined from the feeder power-flow variables through the stated current relation.The current calculation is part of the DistFlow representation used by the optimization.
A. Constraints
The constraints maintain acceptable bus voltages and limit inverter reactive-power output according to each inverter’s available apparent-power capacity and real-power generation.
- Constraints: Voltage magnitudes at all buses are constrained to remain within the acceptable operating range.Voltage regulation is identified as the primary purpose of distribution-circuit volt/var control.
- Constraints: Inverter reactive-power magnitude is upper bounded by a quantity that depends on real power generated at the node.This captures the inverter’s apparent-power capability.
- Constraints: The reactive-power bound uses the inverter nameplate capacity, which is assumed known at each time.The capacity constraint links feasible reactive output to the inverter’s real-power operating point.
B. Objective Function
The objective function accounts for line losses, Conservation Voltage Reduction energy effects, and inverter losses to minimize overall power consumption.
- Objective Function: The optimization objective includes line losses, CVR energy consumption, and inverter losses.Including inverter losses captures the cost of operating away from unity power factor.
1) Line losses:
The formulation accounts for CVR-related energy consumption and inverter losses alongside line losses. Reactive-power operation can reduce network losses and consumption while increasing inverter real-power losses away from unity power factor.
- Line losses:: CVR savings are formulated as an energy-consumption minimization under an exponential load-consumption model.The exponent n_i ranges from 0 to 2, covering constant-power, constant-current, and constant-impedance loads.
- Line losses:: Inverter losses are modeled as a quadratic function of apparent power, including standby, voltage-dependent, and ohmic components.The apparent-power magnitude is represented by s = √(p^2 + q^2).
- Inverter losses:: Reactive-power control can reduce line losses and CVR-measured energy consumption, but non-unity power factor increases inverter real-power loss.This trade-off is explicitly included in the objective function.
C. Overall Problem
The overall fast-timescale inverter volt/var problem is formulated over network flows, inverter variables, squared voltage magnitudes, and squared current magnitudes. Its objective is convex, but the feasible set remains nonconvex.
- Overall Problem: The fast-timescale control problem is formulated over a state vector containing network flows, inverter generation and consumption, squared voltages, and squared currents.The variable set includes P, Q, p^g, p^c, q^g, q^c, |V|^2, and |I|^2.
- Overall Problem: The objective function is convex because the network and consumption terms are linear and the inverter-loss term combines a norm with a quadratic function.This convexity concerns the objective as a function of the state vector X.
- Overall Problem: The feasible set defined by the network constraints is nonconvex, making the original optimization problem difficult to solve directly.Thus, objective convexity alone does not make the overall problem convex.
D. Solution Method
The solution method relaxes the radial OPF constraints into a second-order cone program. The paper states that this relaxation is convex and exact, enabling efficient real-time inverter volt/var control.
- Solution Method: Substituting the variable transformation makes the constraints linear except for one nonlinear equality, which is the source of nonconvexity.The method relaxes this equality to obtain a tractable formulation.
- Solution Method: The relaxation represents the relevant constraints as second-order cones in active power, reactive power, squared voltage, and squared current variables.New variables are introduced for buses, lines, and inverter-related quantities.
- Solution Method: The relaxed SOCP is used to solve the fast-timescale inverter volt/var control problem.The optimization is performed over X := (P, Q, p^g, p^c, q^g, q^c, ν, ℓ, s, t).
- Solution Method: The formulation relaxes current-magnitude equalities to inequalities and permits over-satisfaction of active and reactive loads.The authors state that the relevant inequalities are tight at optimal solutions.
- Solution Method: Theorem 1 states that the relaxed volt/var problem is convex and exact, yielding valid optimal inverter reactive generation and voltage magnitudes.The result supports efficient real-time control for random, rapid, and large solar-generation fluctuations.
- Solution Method: The proof extends an earlier result despite added inverter-loss terms and second-order-cone constraints.The argument remains applicable because active and reactive generation variables are not involved in the cited proof structure.
IV. EVALUATION
The evaluation uses peak load data from an SCE distribution circuit and presents PCC voltage magnitude as a function of solar output. The figure targets voltage behavior under changing photovoltaic generation.
- IV. EVALUATION: The evaluation uses load data from an SCE distribution circuit, with the reported values representing peak load conditions.Historical operation is described separately as approximately 20% of peak daytime loading with a 0.9 power factor.
- IV. EVALUATION: Figure 3 plots voltage magnitude at the point of common coupling against solar output.The plotted PCC voltage is the voltage magnitude associated with the solar-generation connection point.
- IV. EVALUATION: The evaluation therefore examines how PCC voltage varies as photovoltaic output changes.The figure provides the direct voltage-versus-generation comparison.
A. Simulation setup
The study evaluates fast inverter var control on a lightly loaded SCE feeder with a remote 5 MW PV plant, using voltage limits and operating conditions to examine optimal reactive-power injection.
- The test system is a very lightly loaded rural feeder with less than 1 MW of load and a 5 MW PV plant nearly 6 miles from the substation.
- Substation voltage is fixed at 1 pu, while all other bus voltages are constrained between 0.97 pu and 1.03 pu.
- Without var control, PCC voltage fluctuations can exceed 5% as solar output varies from 0 to 5 MW.
- The optimal inverter var injection balances line-loss reduction, CVR energy consumption, voltage limits, and changes in injection as solar output and load vary.More capacitive injection supports low voltage or high solar transfer, whereas more inductive injection can limit high voltage or reduce CVR energy consumption.
- The circuit diagram and equipment ratings describe the SCE distribution system used for the simulation.
C. Results
The results show that optimal inverter var injection depends jointly on solar output and loading. Its direction can reverse across operating conditions because line-loss and CVR objectives dominate in different regions.
- At low load, optimal var injection initially increases with solar output to reduce line losses, then absorbs var above a threshold to maintain the upper voltage bound.
- At high load, optimal var injection decreases as solar output increases because the CVR term favors lower energy consumption.
- At low solar output, injection decreases as load rises until voltages approach their lower bounds, then increases to maintain those bounds.
- Figures 4 and 5 compare optimal inverter reactive power in kvar with PV output under low-load and high-load conditions.
- At high solar output, injection initially increases with load when line losses dominate the CVR term.
D. Benefits of optimal inverter var control
Optimal inverter var control expands feasible operation under high solar and low load while reducing total cost, including inverter real-power losses. Simulations show near-elimination of voltage-limit violations and energy savings above 1% when both controls are feasible.
- 1,214 hours per year outside the feasibility region under unity power factor control with 3% voltage-drop tolerance.These violations occur when total load is low and solar power is high.
- The proposed optimal inverter var control almost completely eliminates the undesirable operation mode where voltage magnitudes violate their specified limits.As voltage-drop tolerance decreases, unity power factor control becomes infeasible more often while optimal control remains feasible.
- Above 1% energy savings are achieved in total cost, including inverter real-power loss, when both unity power factor and optimal controls are feasible.The inverter loss model assumes a maximum efficiency of 97%; savings exclude periods when either control is infeasible.
- The study computes optimal inverter var injections and demonstrates improved voltage regulation and efficiency on an SCE feeder with a 5 MW PV system installed 6 miles from the substation.The simulated feeder has a very light load, and the analysis uses a convex relaxation of the radial OPF problem.