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Options for Control of Reactive Power by Distributed Photovoltaic Generators

Petr Sulc, Konstantin Turitsyn, Scott Backhaus, Michael Chertkov

arXiv:1008.0878v1math-ph

TL;DR

High distributed-PV penetration can produce rapid voltage excursions that slow utility equipment cannot compensate, motivating reactive-power control through PV inverters. The paper compares centralized and local control schemes in simulations and finds that local schemes generally maintain acceptable voltage, while control choices trade voltage regulation against loss minimization.

  • Problem

    High PV penetration can cause rapid voltage variations that exceed the response of slowly responding utility equipment, while control architecture and input-variable choices remain open questions.

  • Method

    The paper uses distribution-flow modeling and simulations to compare centralized and local reactive-power control schemes and local control variables.

  • Results

    Local control schemes provide adequate voltage regulation, while including local real and reactive power flows improves control-system performance.

  • Takeaways & Limitations

    Reactive-power control can support voltage regulation and circuit-loss objectives, but these objectives require a tunable trade-off rather than one global optimum.

Abstract

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High penetration levels of distributed photovoltaic(PV) generation on an electrical distribution circuit present several challenges and opportunities for distribution utilities. Rapidly varying irradiance conditions may cause voltage sags and swells that cannot be compensated by slowly responding utility equipment resulting in a degradation of power quality. Although not permitted under current standards for interconnection of distributed generation, fast-reacting, VAR-capable PV inverters may provide the necessary reactive power injection or consumption to maintain voltage regulation under difficult transient conditions. As side benefit, the control of reactive power injection at each PV inverter provides an opportunity and a new tool for distribution utilities to optimize the performance of distribution circuits, e.g. by minimizing thermal losses. We discuss and compare via simulation various design options for control systems to manage the reactive power generated by these inverters. An important design decision that weighs on the speed and quality of communication required is whether the control should be centralized or distributed (i.e. local). In general, we find that local control schemes are capable for maintaining voltage within acceptable bounds. We consider the benefits of choosing different local variables on which to control and how the control system can be continuously tuned between robust voltage control, suitable for daytime operation when circuit conditions can change rapidly, and loss minimization better suited for nighttime operation.

I. INTRODUCTION

High PV penetration creates power-quality challenges but also lets utilities use inverter reactive power to regulate voltage and reduce losses. The paper frames control design around local measurements, communication architecture, inverter limits, and equitable dispatch.

  • Motivation: High PV penetration can make cloud-induced generation changes outpace slow utility equipment, causing voltage excursions and degraded power quality.The accumulated impact of many small PV generators becomes significant at higher circuit penetrations.
  • Motivation: PV inverters can inject or consume reactive power rapidly to help regulate voltage, although current interconnection standards do not permit this operation.The proposed approach uses latent inverter capacity as a fast voltage-regulation resource.
  • Design questions: Control design must determine how to divide reactive compensation, whether control is centralized or distributed, and which variables drive the algorithm.These choices affect communication needs, robustness, and achievable performance.
  • Opportunities: Reactive-power dispatch can improve voltage regulation and reduce distribution losses without excessive inverter dispatch or limiting PV generation.The utility opportunity is constrained by the need to avoid undue burdens on PV generators.
  • Modeling framework: The circuit model represents real and reactive flows, nodal consumption and generation, voltage changes, and inverter reactive-power control within a radial distribution network.The paper introduces LinDistFlow for intuition while using an AC solver for distribution-circuit quantities.
  • Inverter constraints: Inverter reactive power is bounded by apparent-power capability, and the available range shrinks as PV real-power generation approaches that capability.Only nodes with PV generation can provide reactive power in the model.

A. Distribution Loss Reduction vs Power Quality

Reactive-power control must balance voltage regulation against distribution-loss minimization because the circuit conditions that optimize these objectives differ. Rapid changes in PV generation make fast reactive-power adjustment important for keeping voltage within acceptable bounds.

  • Objective trade-off: Losses are minimized when Q_j = 0, whereas voltage variation is minimized when Q_j = −(r_j/x_j)P_j, creating an inherent control trade-off.The paper therefore does not expect one algorithm to optimize both objectives globally.
  • Voltage regulation: Rapid PV-generation changes can reverse segment flows and switch a circuit between voltage-drop and voltage-rise conditions.A circuit with a 0.05 p.u. voltage drop without PV may experience a 0.05 p.u. voltage rise during changing irradiance.
  • Voltage regulation: Rapidly modifying Q_j through inverter reactive power can keep voltage variation within acceptable bounds during changing irradiance.The control acts on the reactive-flow term that contributes directly to voltage variation.

B. Centralized versus Local Control

Centralized control can optimize feeder-wide objectives but requires communication that may introduce vulnerability and latency. The paper therefore emphasizes local schemes, which are faster and robust to communication loss, while examining their performance and equitable dispatch implications.

  • Centralized control: A centralized controller could infer segment flows from consumer measurements and optimize reactive-power dispatch using weighted losses and voltage deviations.Its potential performance advantage comes with greater communication dependence.
  • Centralized control: Communication requirements, vulnerability, and latency may outweigh centralized control’s performance benefits during rapid cloud-cover changes.Latency is especially problematic when circuit conditions vary quickly.
  • Local control: Local control avoids communication latency and vulnerability but cannot guarantee optimal control because it lacks segment flows P_j and Q_j.Local schemes operate primarily on local flows and measurements.
  • Comparative performance: A local scheme supplying local reactive consumption achieved almost 80% of the loss savings of centralized full optimization in a realistic distribution circuit.A heuristic further reduced losses in exporting or importing conditions but degraded when generation and load were closely balanced.
  • Equitable treatment: Local voltage-based dispatch can impose greater reactive-power duty on generators whose circuit voltage stays above or below 1 p.u., raising equitable-treatment concerns.The paper considers capacity-based limits so dispatch does not depend on location along the circuit.
  • Scope: The manuscript focuses on reactive-power dispatch and does not consider limiting PV generation.It notes that an external framework addresses equitable division of generation reductions.

III. MODELING DETAILS

The inverter model limits reactive-power output according to fixed apparent-power capability and instantaneous PV real-power generation. The assumed oversizing provides reactive-power flexibility while recognizing that capability is smallest near maximum real-power output.

  • Inverter capability: PV inverter reactive-power capability is bounded by fixed apparent power s_j and variable real-power generation p_j^(g).The allowable range is expressed by |q_j^(g)| ≤ sqrt(s_j^2 − (p_j^(g))^2).
  • Inverter capability: As PV real-power output approaches the inverter’s apparent-power capability, the available reactive-power range decreases toward zero.This is the operating limit illustrated by the inverter phasor relationship.
  • Model assumption: The model assumes s_j ≈ 1.1 p_j^(g),max, which prior work found sufficient for most distribution-loss reduction.The paper notes that inverter sizes are discrete and may be somewhat oversized relative to maximum PV output.

B. Description of the prototypical distribution circuit

The simulations use a 250-node distribution circuit with PV at 50% of nodes, sampling load, generation, and PV locations under contrasting irradiance conditions. AC power flow is solved with a modified Newton-Raphson implementation that supports voltage-dependent inverter reactive power.

  • Circuit configuration: The circuit has 250 nodes, with PV generators installed at 50% of nodes and neighboring nodes spaced 0.2 kilometers apart.The nominal phase-to-neutral voltage is 7.2 kV, and line impedance is constant at (0.33 + 0.38i) Ω/km.
  • PV configuration: Each PV-enabled node uses an inverter with capacity sj = 2.2 kVA and maximum PV generation p(g)max = 2.0 kW.Uniform maximum generation represents identical installations and spatially uniform irradiance.
  • Operating cases: The undergenerated case sets PV generation to zero and produces an average net real-power import of 1.25 kW per node.Loads are sampled uniformly from 0 to 2.5 kW under heavy cloud cover.
  • Operating cases: The overgenerated case sets PV generation to 2 kW per PV node and produces an average net real-power export of 500 W per node.Loads are sampled uniformly from 0 to 1 kW under clear-sky conditions.
  • Sampling and robustness: Reactive load is sampled to represent residential power factors from 0.955 to 0.98, while PV-enabled nodes and circuit loads are randomly selected across realizations.The contrasting cases probe robustness to rapidly changing irradiance and power-flow conditions.
  • Power-flow solution: Matpower solves the AC power-flow equations with Newton-Raphson, modified to account for inverter reactive power changing with voltage.The standard solver does not support voltage-dependent q(g)i directly.

IV. CONTROL SCHEMES

The paper compares local reactive-power control schemes based primarily on inverter connection voltage, including a smoothed version of a piecewise-linear reference control. Smoothing is introduced to improve AC-solver convergence while preserving the original control shape.

  • Reference local control: Reference proposes local reactive-power control using piecewise-linear relationships between inverter reactive power q(g)j and voltage Vj.The paper simplifies one proposed mode, ‘PV1’, for comparison.
  • Reference local control: The simplified PV1 control sets q(g)j = 0 at Vj = 1 p.u. and saturates at dynamically determined qmaxj for high and low voltages.Except for the dynamic qmaxj definition, the scheme is essentially proportional control because q(g)j depends linearly on Vj.
  • Smoothed implementation: The original piecewise control is smoothed with a sigmoid function because discontinuous first derivatives caused the AC solver to jump between solution regions.The smoothing parameter δ controls how closely the smoothed function follows the sharp transitions.
  • Smoothed implementation: δ = 0.04 is used to improve convergence while closely representing the reference piecewise-linear control.Figure 3 compares the dark-red simplified piecewise curve with the blue smooth curve from Equation (9).

B. Control on Local Flows Only

Local-flow control uses power-flow information rather than only connection voltage to balance loss reduction and voltage regulation. A single parameter K continuously interpolates between the two objectives.

  • Local-flow inputs: The local-flow scheme constructs inverter reactive-power control from local real and reactive power flows rather than directly from Vj.The scheme is homogeneous across the line, with bus dependence entering through each inverter’s dynamically determined qmaxj.
  • Local-flow inputs: Smart-meter data can provide local net real and reactive flows, which combine with inverter measurements to estimate the uncontrolled power flows near each inverter.These measurements are communicated to the local PV inverter in near real time.
  • Objective trade-off: Loss-oriented control supplies local reactive consumption up to inverter capacity limits and was shown to be effective at reducing losses.However, loss minimization does not ensure voltage regulation because the objectives compete.
  • Objective trade-off: Voltage-oriented control minimizes the absolute combined flow rjPj + xjQj to reduce voltage variations, rather than directly minimizing losses.For circuits with nearly constant rj/xj, this expression can be driven to zero under the stated node-flow condition.
  • Objective trade-off: K = 1 recovers loss reduction, K = 0 recovers voltage regulation, and intermediate K values provide a continuous compromise between them.The parameter allows the control scheme to adapt as circuit conditions change.

C. Hybrid Control

Hybrid control combines local-flow and local-voltage inputs to address weaknesses of flow-only heuristics when voltage deviates substantially from 1 p.u. Its particular blending rule is one of several possible designs.

  • Hybrid rationale: The hybrid scheme combines the voltage-based G control with the local-flow F control to correct flow-estimation errors when Vj moves significantly from 1 p.u.Without voltage information, the local-flow heuristic cannot detect such deviations directly.
  • Hybrid behavior: The hybrid control is designed so low voltage drives the inverter toward reactive-power support at qmaxj, with analogous corrective behavior for high voltage.The transition is implemented through a simple voltage-dependent blending function.
  • Design scope: The paper uses one particular way to blend G and F, while leaving other possible blending strategies for future study.Thus, the reported hybrid design does not exhaust the available combination rules.

V. SIMULATIONS: RESULTS AND DISCUSSIONS

The simulations evaluate node voltages and distribution-circuit losses for undergenerated and overgenerated operating cases. The no-control baseline produces substantially larger voltage variation during transitions between these conditions.

  • Simulation setup: The simulations calculate node voltages and distribution-circuit losses for both undergenerated and overgenerated cases.Figures 4–7 report voltages and losses for the two operating conditions.
  • Base case—no control: With q(g)_j = 0, the undergenerated case reaches about 0.07 p.u. voltage deviation below 1 p.u.This operating condition has P_j and Q_j in the same direction.
  • Base case—no control: With q(g)_j = 0, the overgenerated case reaches about 0.015 p.u. maximum voltage rise.The deviation occurs at or near the end of the distribution circuit.
  • Base case—no control: The transition between undergenerated and overgenerated conditions produces a voltage swing of about 0.085 p.u. during partly cloudy daylight.The paper describes this swing as uncomfortably close to allowable limits.
  • Base case—no control: Under higher load or PV-generation conditions, the voltage swing would easily exceed 0.1 p.u., demonstrating the need for reactive-power control.The losses from this no-control case are used to normalize losses for other control schemes.

B. Control on Local Voltage Only-G(V )

Local-voltage control substantially reduces voltage deviations but increases normalized losses, while the broader control comparison shows a trade-off between voltage regulation and loss reduction. Hybrid control improves voltage regulation by switching toward stronger voltage control when deviations grow, but its loss performance depends on the operating case and tuning parameter.

  • Local-voltage control: Local-voltage control reduces the undergenerated voltage drop to about 0.027 p.u. and the overgenerated voltage rise to 0.008 p.u.The resulting partly cloudy-day voltage swing is about 0.035 p.u.
  • Local-voltage control: Local-voltage control increases relative losses by about 5% in the undergenerated case and 20% in the overgenerated case.In the overgenerated case, inverter reactive-power consumption increases Q_j flows and dissipation as V_j rises above 1 p.u.
  • Scheme comparison: The figures compare no control, local-voltage control, local-power-flow control, and hybrid control using voltage deviation and normalized circuit losses.Figures 4 and 6 show maximum deviation from 1 p.u.; Figures 5 and 7 normalize losses to the q(g)_j = 0 baseline.
  • Tuning trade-off: Near K = 0, F(K) emphasizes voltage regulation; near K = 1, it reduces dissipation but has voltage regulation nearly as poor as q(g)_j = 0.The comparison makes the tuning trade-off explicit.
  • Tuning trade-off: No globally optimum K exists because voltage regulation and loss reduction compete, so engineers must choose K using application judgment and Fig. 8.The same engineering-judgment dependence is reported for hybrid control.
  • Hybrid control: Hybrid control switches smoothly from F(K) toward G(K) as V_j moves away from 1 p.u., and it outperforms both for voltage regulation when K < 1.For losses, hybrid control increases losses in the undergenerated case but significantly reduces them in the overgenerated case around K = 1.

E. Comparison of Control Schemes

The simulations compare control schemes using voltage swing during rapid transitions and average losses, revealing a trade-off between voltage regulation and circuit dissipation. Local real and reactive power-flow information improves performance over voltage-only control, while local schemes generally provide adequate voltage regulation.

  • The F(K) curve generally produces poorer voltage regulation and higher losses than G(K) and its scaled versions.For some K values F(K) performs better, but those cases approach voltage deviations of 0.1 p.u.
  • Hybrid H(K,V) controls generally outperform G(K) and scaled versions by incorporating local real and reactive power flows alongside local voltage.The H(K,V) and H(K,V)/2 curves lie below and to the left of G(K), indicating better combined performance in the plotted objectives.
  • The study evaluates robustness using maximum per unit voltage swing during over-to-undergenerated transitions and average dissipation across both conditions.These metrics collapse the over- and undergenerated results into a single comparison focused on rapid changes in circuit power flows.
  • Voltage regulation and loss minimization compete, so no control scheme achieves a global optimum for both objectives.The comparison therefore weighs voltage deviation against average relative losses rather than selecting a single universally optimal setting.
  • For the cases considered, control schemes using only local variables provide adequate voltage regulation.This supports distributed control as a viable option for handling rapid variations without requiring centralized measurements.
  • The study leaves nighttime reactive-power dispatch and the best control scheme for that setting as open questions.The paper focuses mainly on rapid loading transitions caused by changes in solar irradiance.
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