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Bi-Level Volt-VAR Optimization to Coordinate Smart Inverters with Voltage Control Devices

Rahul Ranjan Jha, Anamika Dubey, Chen-Ching Liu, Kevin P. Schneider

arXiv:1912.03596v1math.OC

TL;DR

Three-phase branch-flow models are nonconvex, and relaxed formulations can be difficult to make feasible or extend to voltage-dependent loads and legacy controls. This paper proposes a bi-level VVO framework coordinating legacy devices with smart inverters; results report reasonably accurate power-flow approximations, successful coordination for CVR, and computational efficiency for real-time operation.

  • Problem

    Three-phase branch-flow models are nonconvex, and relaxed formulations can be difficult to make feasible or extend to voltage-dependent loads and legacy control devices.

    Source: A. Three-Phase Power Flow using Branch… · p. 3

  • Method

    A bi-level VVO framework coordinates legacy devices and smart inverters, using linear approximate three-phase OPF in Level 1 and nonlinear OPF to adjust inverter settings for a feasible solution in Level 2.

    Source: §I · p. 2, B. Approximate Three-Phase Power Flow… · p. 3, §VI · p. 11

  • Results

    The results report reasonably accurate power-flow approximations, successful coordination of legacy and new devices for CVR benefits, and computational efficiency of both levels for real-time operation.

    Source: §VI · p. 11

  • Takeaways & Limitations

    The reported findings support coordinated CVR control on radially operated feeders.

    Source: §I · p. 2, §VI · p. 11

    The paper focuses on radially operated feeders.

    Source: §I · p. 2

Abstract

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Conservation voltage reduction(CVR) uses Volt-VAR optimization(VVO) methods to reduce customer power demand by controlling the feeders' voltage control devices. The objective of this paper is to present a VVO approach that controls the systems' legacy voltage control devices and coordinates their operation with smart inverter control. An optimal power flow (OPF) formulation is proposed by developing linear and nonlinear power flow approximations for a three-phase unbalanced electric power distribution system. A bi-level VVOapproach is proposed where Level-1 optimizes the control of legacy devices and smart inverters using a linear approximate three-phase power flow. In Level-2, the control parameters for smart inverters are adjusted to obtain an optimal and feasible solution by solving the approximate nonlinear OPF model. Level-1 is modeled as a Mixed Integer Linear Program(MILP) while level-2 as a Nonlinear Program(NLP) with a linear objective and quadratic constraints. The proposed approach is validated using 13-bus and 123-bus three-phase IEEE test feeders and a 329-bus three-phase PNNL taxonomy feeder. The results demonstrate the applicability of the framework in achieving the CVR objective. It is demonstrated that the proposed coordinated control approach help reduce feeders' power demand by reducing the bus voltages, the proposed approach maintains an average feeder voltage of 0.96 pu. A higher energy saving is reported during the minimum load conditions. The results and approximation steps are thoroughly validated using OpenDSS.

NOMENCLATURE

The nomenclature defines the distribution-system graph and phase-indexed sets, along with electrical quantities and control variables used in the optimization.

Sets and graph

  • The feeder is represented by a directed graph with sets for lines, regulator branches, buses, capacitor-bank nodes, smart-inverter DG nodes, and bus phases.The notation also defines phase pairs across connected buses.Source: NOMENCLATURE · p. 1

Electrical variables

  • Phase-indexed line currents, bus voltages, and complex three-phase line impedances describe feeder electrical quantities.The impedance notation includes elements for individual phase pairs on a branch.Source: NOMENCLATURE · p. 1

DG quantities

  • Smart-inverter DG notation includes projected per-phase active generation, available reactive power, and rated apparent-power capacity.The available reactive power is specified per phase and at the current time instance.Source: NOMENCLATURE · p. 1

Control variables

  • Capacitor banks have rated per-phase reactive-power generation, while regulator taps and capacitor-bank states use binary control variables.The capacitor-bank control variable is defined for nodes with capacitor banks.Source: NOMENCLATURE · p. 1

Power quantities

  • Complex bus demand and branch power flow are represented with corresponding active and reactive components.Demand is indexed by bus phase, and branch flow by the phase pair on that branch.Source: NOMENCLATURE · p. 1

I. INTRODUCTION · II. THREE-PHASE UNBALANCED ELECTRIC POWER DISTRIBUTION SYSTEM · A. Three-Phase Power Flow using Branch Flow Model

The paper develops a bi-level Volt-VAR optimization approach for conservation voltage reduction that coordinates legacy voltage-control devices and smart inverters in radial, three-phase unbalanced distribution systems. It combines linear and nonlinear power-flow approximations to model voltage-dependent loads and seek feasible operating points.

I. INTRODUCTION

  • CVR reduces customer demand by lowering service voltage; prior studies report 3.04% lower annual energy consumption if implemented on all U.S. distribution feeders.Traditional Volt-VAR control operates legacy devices while keeping service voltages within the recommended 0.95–1.05 pu ANSI limits; smart inverters can add local reactive-power control.Source: §I · pp. 1–2
  • The proposed approach coordinates legacy voltage-control devices and smart inverters for CVR in radial, three-phase unbalanced feeders, addressing gaps in coordination, load modeling, and scalability.Level 1 uses a linear OPF to set device controls; Level 2 uses a nonlinear OPF to adjust smart-inverter parameters and obtain a feasible power-flow solution.Source: §I · p. 2
  • The paper proposes voltage-dependent load and device models, including a CVR-based approximation of ZIP loads that fits both optimization levels without changing equation types.The modeled devices include capacitor banks, voltage regulators, and smart inverters.Source: §I · p. 2
  • The proposed three-phase power-flow approximations include a DistFlow-inspired linear model and a nonlinear model that approximates phase mutual coupling with quadratic equality constraints.The nonlinear formulation reduces the number of variables compared with standard three-phase power-flow formulations.Source: §I · p. 2
  • The bi-level framework decomposes a hard MINLP into a MILP and NLP, solving the 123-bus feeder within 4 minutes and the reduced 329-bus feeder within 9 minutes.The demonstrated feeders include 267 single-phase nodes for the 123-bus case and 860 for the 329-bus case.Source: §I · p. 2
  • The paper validates its power-flow approximations on IEEE 13-bus and 123-bus feeders and a 329-bus PNNL feeder, and validates the load model and OPF results against ZIP models and OpenDSS.The stated validation covers the approximation steps and the resulting OPF.Source: §I · p. 2

II. THREE-PHASE UNBALANCED ELECTRIC POWER DISTRIBUTION SYSTEM

  • Section II introduces branch-flow-based three-phase power-flow equations and proposes linear and equivalent quadratic approximations to reduce the original formulation.The model targets radial systems and aims to incorporate system components without greatly increasing inherent nonlinearity.Source: §II · pp. 2–3, A. Three-Phase Power Flow using Branch… · p. 3

A. Three-Phase Power Flow using Branch Flow Model

  • The three-phase branch-flow formulation has voltage-drop and power-balance equations plus a nonconvex Rank-1 constraint; relaxations can make feasible solutions difficult to obtain,.The passage also notes that extending this formulation to voltage-dependent loads and legacy controls is difficult,.Source: A. Three-Phase Power Flow using Branch… · p. 3

B. Approximate Three-Phase Power Flow Equations · 1) Assumption 1 - Approximating Phase Voltages:

The paper approximates three-phase power flow equations by decoupling branch power and voltage equations phase by phase, reducing the number of variables while retaining an unbalanced-system model. The phase-voltage approximation assumes nearly balanced magnitudes and 120° phase separation.

B. Approximate Three-Phase Power Flow Equations

  • Phase-decoupling approximates mutual coupling and characterizes three-phase power flow with fewer variables.The formulation addresses power-flow nonlinearity from both power-voltage-current relationships and mutual coupling; it approximates the latter.Source: B. Approximate Three-Phase Power Flow… · p. 3

1) Assumption 1 - Approximating Phase Voltages:

2) Assumption 2 - Approximating Angle Difference between Phase Currents:

The formulation approximates phase-current angle differences as constants estimated from a power flow with constant-impedance loads, while allowing other load types in the OPF model.

2) Assumption 2 - Approximating Angle Difference between Phase Currents:

  • Trigonometric terms involving phase-current angle differences significantly increase the complexity of the OPF problem.These nonlinearities arise when expanding the power-flow expressions.Source: 2) Assumption 2 - Approximating Angle… · p. 3
  • The method fixes each branch’s phase-current angle difference at an estimate obtained from power flow with constant-impedance loads.The constant-impedance model is used only to estimate the angle difference.Source: 2) Assumption 2 - Approximating Angle… · p. 3
  • The constant-impedance assumption does not restrict OPF load types, which can include constant-power and constant-current loads.The passage points to Section IV for further detail.Source: 2) Assumption 2 - Approximating Angle… · p. 3

3) Power Flow Equations: · C. Linear Three-Phase AC Power Flow Approximation · III. DISTRIBUTION SYSTEM EQUIPMENT MODELS

The paper develops three-phase power-flow equations combining linearized branch-flow and voltage-drop relations with a nonlinear quadratic constraint. It also introduces a linear approximation and equipment and load models designed to fit within the approximate equations.

3) Power Flow Equations:

  • Equations (7)–(9) express active and reactive branch flow and voltage drop linearly, while (10) imposes a nonlinear quadratic equality linking branch power flow, phase voltage, and currents.The expanded formulation contains 15×(n−1) variables, compared with 36×(n−1) in the original formulation.Source: 3) Power Flow Equations · pp. 3–4

C. Linear Three-Phase AC Power Flow Approximation

  • The linear approximation assumes branch losses are small relative to branch power flow, neglects their impact on power-flow and voltage-drop equations, and yields linearized flow and voltage-drop equations (12)–(13).Source: C. Linear Three-Phase AC Power Flow… · p. 4
  • The linearized AC power flow approximates actual feeder voltages by including load-driven power flow while omitting the impact of power losses on voltage drop.The passage attributes this approximation to losses being small relative to branch flow due to load demand.Source: C. Linear Three-Phase AC Power Flow… · p. 4

III. DISTRIBUTION SYSTEM EQUIPMENT MODELS

  • The equipment models cover capacitor banks, voltage regulators, smart inverters, and voltage-dependent customer loads, parameterized by control variables and voltage magnitude.Source: §III · p. 4
  • A new CVR-based load model makes power demand a function of voltage magnitude, using ZIP load coefficients to obtain equivalent CVR coefficients.Source: §III · p. 4
  • The proposed equipment and load models are designed to fit into approximate power-flow equations (7)–(13) without changing their type.Source: §III · p. 4

A. Voltage Regulator · B. Capacitor Banks

The sections model legacy voltage regulators through discrete tap controls and capacitor banks through voltage-dependent, binary-controlled reactive power. They specify the controls and electrical quantities used to represent each device in the optimization.

A. Voltage Regulator

  • The regulator model assumes 32 steps over ±10%, ignores its small series and shunt impedances [27], and represents tap ratios from 0.9 to 1.1 in 0.00625 pu increments.A regulator is modeled per phase, with an additional node introduced for the current equations.Source: A. Voltage Regulator · p. 4
  • Binary variables select among the regulator’s 32 step positions, represented by tap-ratio values from 0.9 to 1.1.The model defines a step-selection variable for each position and a vector of corresponding tap ratios.Source: A. Voltage Regulator · p. 4
  • The regulator relation is squared and reformulated to express it using squared voltage quantities.The passage identifies this reformulation as equation (15).Source: A. Voltage Regulator · p. 4

B. Capacitor Banks

  • The per-phase capacitor-bank model defines generated reactive power using an On/Off binary control, rated per-phase reactive power, and squared bus voltage.The bank is modeled as voltage-dependent.Source: B. Capacitor Banks · p. 4
  • When connected, a capacitor bank provides reactive power as a function of the squared bus voltage.Connection is represented by the capacitor-bank control variable being 1.Source: B. Capacitor Banks · p. 4
  • For a three-phase capacitor bank, a common control variable is defined for each phase.The control variable indicates the bank’s On/Off status.Source: B. Capacitor Banks · p. 4

C. Distributed Generation with Smart Inverters · D. Voltage-Dependent Model for Customer Loads

The sections model smart-inverter reactive support alongside distributed generation and develop a voltage-dependent customer-load representation based on CVR factors. The load model is reformulated to remain linear in squared voltage for inclusion in approximate power-flow equations.

C. Distributed Generation with Smart Inverters

  • Smart-inverter DGs are modeled as negative loads with forecast active generation, while their available reactive-power support depends on inverter rating and is constrained by a box constraint.Source: C. Distributed Generation with Smart… · p. 4

D. Voltage-Dependent Model for Customer Loads

IV. PROPOSED BI-LEVEL VOLT-VAR OPTIMIZATION · A. Level 1 - MILP Formulation for Coordinated Control

The proposed bi-level VVO addresses the MINLP of coordinating legacy devices and smart inverters by separating feeder-demand optimization from the nonlinear adjustment needed for feasible three-phase power flow.

IV. PROPOSED BI-LEVEL VOLT-VAR OPTIMIZATION

  • VVO coordinates voltage control to reduce energy consumption and system losses while regulating feeder voltages, but joint legacy-device and inverter control creates an MINLP.The bi-level approach is proposed to reduce complexity and ensure scalability.Source: §IV · p. 5

A. Level 1 - MILP Formulation for Coordinated Control

  • Level 1 minimizes feeder active-power consumption through a 15-minute MILP schedule for legacy-device controls and smart-inverter reactive-power set-points.The schedule controls voltage regulators, capacitor banks, and smart inverters while enforcing voltage limits.Source: §IV · p. 5, A. Level 1 - MILP Formulation for… · p. 5

IV. PROPOSED BI-LEVEL VOLT-VAR OPTIMIZATION

  • Level 2 revises the smart-inverter control schedule with an NLP using nonlinear power flow to obtain feasible three-phase solutions.Both levels develop revised 15-minute schedules, with Level 2 adjusting smart-inverter set-points for feasibility.Source: §IV · p. 5

A. Level 1 - MILP Formulation for Coordinated Control

  • Level 1 minimizes the sum of three-phase active power flowing from the substation, which equals net feeder demand for a radial distribution feeder.Its linear three-phase AC power-flow model yields a MILP because legacy-device controls introduce integer variables.Source: A. Level 1 - MILP Formulation for… · p. 5
  • Level 1’s constraints cover linear AC power flow, CVR-based load modeling, regulator and capacitor controls, inverter reactive-power dispatch, and feeder-voltage operating limits.These are specified in constraints (27)–(37).Source: A. Level 1 - MILP Formulation for… · p. 5

B. Level 2 - NLP Problem for Smart Inverter Control

Level 2 adjusts smart-inverter set-points to obtain an optimal, feasible three-phase nonlinear power-flow solution after Level 1’s linear approximation may violate feeder operating constraints. It solves an NLP with a linear objective and quadratic constraints while holding Level 1’s discrete control settings fixed.

B. Level 2 - NLP Problem for Smart Inverter Control

  • Level 2 adjusts smart-inverter control set-points to obtain an optimal and feasible three-phase nonlinear power-flow solution after Level 1’s linear solution may violate feeder constraints.The Level 2 problem optimizes reactive-power dispatch from smart inverters.Source: B. Level 2 - NLP Problem for Smart… · pp. 5–6
  • Optimal reactive-power dispatch set-points for smart inverters are obtained by solving the NLP defined in (38)-(50), which has a linear objective and quadratic constraints.Source: B. Level 2 - NLP Problem for Smart… · p. 6
  • Constraints (39)-(43) approximate nonlinear AC power flow, (44)-(45) define the CVR load model, and (50) sets feeder-voltage operating limits.Source: B. Level 2 - NLP Problem for Smart… · p. 6
  • Regulator and capacitor-bank control equations retain their Level 1 settings, while constraints (48)-(49) govern smart-inverter reactive-power dispatch.The regulator tap and capacitor settings are known from the Level 1 solution.Source: B. Level 2 - NLP Problem for Smart… · p. 6

V. RESULTS AND DISCUSSION · A. Verification of Approximate Power Flow Formulations

The paper validates approximate power-flow formulations against OpenDSS across three feeders, finding small peak-load voltage errors and phase-angle approximation deviations below 2°. It also tests the formulations under increased phase imbalance and describes the simulation and feeder setup for the broader VVO evaluation.

V. RESULTS AND DISCUSSION

  • The evaluation uses IEEE 13-bus, IEEE 123-bus, and PNNL 329-bus feeders, with the 329-bus system covering 329 physical and 860 single-phase nodes.The feeders include different voltage-control devices and distributed generation; daily load and generation profiles use 15-minute intervals, and the assumed CVR factors are 0.6 for active power and 3 for reactive power.Source: §V · p. 6
  • The study compares approximate power-flow models with OpenDSS, validates voltage-dependent load models against equivalent ZIP models, and validates MATLAB results against OpenDSS.Level 1 is solved as a MILP using CPLEX 12.7, and Level 2 as an NLP using MATLAB's fmincon.Source: §V · p. 6

A. Verification of Approximate Power Flow Formulations

  • At peak load, maximum voltage errors (linear/nonlinear) were 0.0096/0.0025 pu (13-bus), 0.0074/0.0016 pu (123-bus), and 0.002/0.0002 pu (329-bus).The nonlinear formulation includes losses and has lower error in both flow quantities and voltages; the linear formulation ignores losses, increasing flow-quantity errors while still approximating bus voltages well.Source: Table I · p. 7, A. Verification of Approximate Power… · p. 7
  • The nonlinear model's approximated voltage and current phase-angle differences each deviate from OpenDSS by less than 2°.The formulation assumes voltage phase-angle differences are close to 120° and branch-current phase-angle differences are close to those from a constant-impedance load model.Source: A. Verification of Approximate Power… · p. 7, Table II · p. 7
  • For IEEE-123, increasing phase-A apparent power raised power imbalance from 23.2% to 39.6% and maximum voltage unbalance from 3.2% to 5.7%.The feeder's inherent apparent-power imbalance was 23.2%; the increased phase-A loading produced the more unbalanced test case.Source: A. Verification of Approximate Power… · p. 8
  • For the 329-bus feeder, phase-B loading increases produced apparent-power imbalances of 40.72% and 57.39%, with maximum voltage unbalances of 2.8% and 4.5%.The paper describes the 329-bus system as balanced before phase-B load was increased to generate imbalance.Source: A. Verification of Approximate Power… · p. 8
  • The authors report that Table III supports using the nonlinear power flow for heavily unbalanced systems, including cases with voltage unbalance above ANSI C84.1's 3% maximum.The reported IEEE-123 case reaches 5.7% maximum voltage unbalance; the 329-bus cases reach 4.5%.Source: A. Verification of Approximate Power… · p. 8, Table III · p. 8

B. Validation of Proposed CVR-based Load Model · C. CVR using proposed VVO approach · 1) IEEE 13-bus test system:

The proposed load model tracks its equivalent ZIP model across the acceptable voltage range, while the coordinated VVO approach reduces net demand except at peak and keeps feeder voltages near 0.96 pu. The 13-bus results are checked against OpenDSS and across realistic load mixes.

B. Validation of Proposed CVR-based Load Model

C. CVR using proposed VVO approach

  • Average feeder voltage is close to 0.96 pu at both minimum and maximum load, with controls keeping voltages near the minimum limit while respecting voltage constraints.As load increases, the regulator tap moves from -13 to 14; the three-phase capacitor remains off, while the single-phase capacitor turns on at peak load and supplies reactive power on Phase C.Source: 1) IEEE 13-bus test system · p. 8, C. CVR using proposed VVO approach · p. 8, Table V · p. 9

1) IEEE 13-bus test system:

  • MATLAB control results closely match OpenDSS substation demand and minimum, maximum, and average node voltages for the 13-bus feeder.The controls optimized for minimum and maximum load conditions were implemented in OpenDSS.Source: 1) IEEE 13-bus test system · p. 9, Table V · p. 9
  • Net power demand falls with the proposed VVO strategy except during peak demand, with the largest reductions at low loading.The comparison is against autonomous capacitor and regulator control with DGs at unity power factor; Fig. 5 presents the 13-bus CVR benefits.Source: 1) IEEE 13-bus test system · p. 9, Figure 5 · p. 9
  • Load mixes with large commercial demand show a smaller reduction in active power demand because those loads are less voltage-sensitive.The comparison uses CVR factors derived from ZIP coefficients for residential, commercial, and large commercial loads.Source: 1) IEEE 13-bus test system · p. 9
  • Level-1 solutions take less than 5 seconds on average, Level-2 solutions finish within 9 seconds for the 13-bus system, and the longest Level-2 solve takes 14 seconds.Reported hardware is a dual-core i7 3.41 GHz processor with 16 GB RAM.Source: 1) IEEE 13-bus test system · p. 9
  • On the 123-node feeder, Reg1 moves from tap -13 at minimum load to -8 at maximum load, while Reg4 stays at tap 0 and Cap1 switches on at maximum load.At maximum load, DG3 absorbs reactive power on Phase B; that phase adjusts its set point as load increases while Reg3's tap remains unchanged.Source: 2) IEEE 123-bus test system · p. 9, Table VII · p. 9

2) IEEE 123-bus test system: · 3) 329-bus PNNL Taxonomy Feeder: · VI. CONCLUSION

The sections report CVR performance on IEEE 123-bus and PNNL 329-bus feeders, then conclude that the bi-level approach coordinates legacy voltage devices and smart inverters through linear MILP and nonlinear NLP stages.

2) IEEE 123-bus test system:

  • The 123-bus feeder operated at an average voltage near 0.96 pu for both load conditions.MATLAB substation-demand and voltage characteristics closely matched OpenDSS; the approach was also tested with ZIP models for residential, commercial, and large-commercial loads.Source: 2) IEEE 123-bus test system · p. 10, Table VIII · p. 10
  • Around 150 kW of net feeder active power demand was reduced on average on the 123-bus feeder, with the largest savings at minimum load.The comparison is against the case without VVO control.Source: 2) IEEE 123-bus test system · p. 10, Figure 6 · p. 10
  • On the 123-bus feeder, Level-1 solved in under 5 seconds and Level-2 averaged 2 minutes, with a maximum Level-2 time of 4 minutes.The test feeder has 123 buses and 267 single-phase nodes; both level solve times were within the 15-minute control interval, while Level-2 was identified as harder to scale.Source: 2) IEEE 123-bus test system · p. 10

3) 329-bus PNNL Taxonomy Feeder:

  • Reducing the 329-bus feeder to 184 buses cut Level-2 variables from 4233 to 2415 and limited its maximum solve time to 9 minutes.The exact radial-network reduction combined nodes without branches, loads, or voltage-control devices; the reduced model retained the same power-flow quantities.Source: 3) 329-bus PNNL Taxonomy Feeder · p. 10, 2) IEEE 123-bus test system · p. 10
  • On the 329-bus feeder, minimum phase voltages were 0.958 pu at minimum load and 0.955 pu at maximum load; average feeder voltages were 0.959 and 0.972 pu, respectively.The MATLAB substation-demand and voltage results closely matched OpenDSS. At minimum load the regulator was at -6 tap and capacitors were off; at maximum load it was at tap 1, the three-phase capacitor was on, and single-phase banks were off.Source: Table IX · p. 10, 3) 329-bus PNNL Taxonomy Feeder · p. 11
  • Around 200 kW of net feeder active power demand was reduced on average over 24 hours on the 329-bus feeder.The comparison is against operation without VVO control.Source: 3) 329-bus PNNL Taxonomy Feeder · p. 11, Figure 7 · p. 11

VI. CONCLUSION

  • The proposed approach coordinates legacy voltage-control devices and smart inverters for CVR on three test feeders: IEEE 13-bus, IEEE 123-bus, and PNNL 329-bus.Level-1 uses a linear three-phase OPF approximation in a MILP to set controls; Level-2 freezes legacy controls and adjusts DG setpoints through an approximate nonlinear OPF NLP.Source: §VI · p. 11
  • The conclusion reports reasonably accurate power-flow approximations, successful device coordination for CVR, and computational efficiency of both levels for real-time operation.Source: §VI · p. 11
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