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A distributed control strategy for reactive power compensation in smart microgrids

Saverio Bolognani, Sandro Zampieri

arXiv:1106.5626v3math.OCeess.SY

TL;DR

The paper addresses optimal reactive-power compensation for reducing smart-microgrid distribution losses. It derives an approximate convex quadratic formulation and a randomized distributed algorithm based on partial knowledge and local measurements. The algorithm is analyzed for convergence and validated in simulations, while radial-network performance favors cooperation among electrically neighboring units.

  • Problem

    The paper asks how reactive power can be compensated to reduce microgrid distribution losses when agents have limited global information and centralized collection is impractical.

  • Method

    It derives an approximate power-flow model yielding a convex quadratic, linearly constrained problem, then applies a randomized, gossip-like distributed optimization algorithm to microgenerator commands.

  • Results

    Both clustering strategies converge to the same minimum, while edge-disjoint gossip performs better than star topology in the long-time regime.

  • Takeaways & Limitations

    Distributed reactive-power optimization can use local communication, local topology knowledge, partial state information, and local measurements without central supervision.

  • Takeaways & Limitations

    The analysis relaxes operational voltage and inverter reactive-power constraints and leaves network dynamics for future work.

Abstract

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We consider the problem of optimal reactive power compensation for the minimization of power distribution losses in a smart microgrid. We first propose an approximate model for the power distribution network, which allows us to cast the problem into the class of convex quadratic, linearly constrained, optimization problems. We then consider the specific problem of commanding the microgenerators connected to the microgrid, in order to achieve the optimal injection of reactive power. For this task, we design a randomized, gossip-like optimization algorithm. We show how a distributed approach is possible, where microgenerators need to have only a partial knowledge of the problem parameters and of the state, and can perform only local measurements. For the proposed algorithm, we provide conditions for convergence together with an analytic characterization of the convergence speed. The analysis shows that, in radial networks, the best performance can be achieved when we command cooperation among units that are neighbors in the electric topology. Numerical simulations are included to validate the proposed model and to confirm the analytic results about the performance of the proposed algorithm.

I. INTRODUCTION

The paper frames reactive power compensation in smart microgrids as a distributed networked-control problem with limited local information and physical-system measurements. It introduces an approximate power-flow model and a distributed optimization strategy for microgenerators.

  • I. INTRODUCTION: Networked control systems constrain distributed optimization because agents have limited parameter and state knowledge, while communication can occur through physical actuation and measurement.Decision-variable assignment is also constrained by the application rather than freely chosen by the designer.
  • I. INTRODUCTION: Reactive power flows contribute to line losses, voltage drops, and possible grid instability, motivating local reactive power production near users.The paper focuses on minimizing reactive power flows through compensation close to demand.
  • I. INTRODUCTION: Prior reactive-power studies mainly assume centralized access to grid parameters and state, few microgenerators, and commands from a central processor.Related distributed work considered large networks with mechanical synchronous generators or regional supervisors.
  • I. INTRODUCTION: The paper derives an approximate quadratic model, designs a distributed reactive-power command strategy, analyzes convergence and performance, and validates both through simulations.The model and algorithm are developed in successive sections covering optimization, convergence, communication strategy, and simulation validation.
  • I. INTRODUCTION: Smart microgrids integrate loads, microgenerators, and a PCC within an autonomously managed portion of the distribution network.Microgenerators connect through inverters that enable power injection into the microgrid.

B. Power losses minimization problem

The paper models active power losses using injected-power variables and formulates optimal reactive power compensation as a quadratic, linearly constrained problem. The formulation motivates distributed solution methods because centralized access to network and load information is impractical.

  • B. Power losses minimization problem: Active power losses on the lines are selected as the optimality metric for reactive power flows.The total losses are expressed through line-current magnitudes and network impedances.
  • B. Power losses minimization problem: Only compensator nodes are commanded, with reactive power q_v as the decision variable while active-power decisions follow imperative economic criteria.Uncommanded nodes retain their nominal reactive-power injections, and 1^T q = 0 enforces balance.
  • B. Power losses minimization problem: The exact current vector depends implicitly and nonlinearly on the commanded reactive powers through the power-flow equations.The optimization formulation uses the approximate model rather than directly solving the nonlinear state equations.
  • B. Power losses minimization problem: Voltage-range and inverter-capability constraints are relaxed to enable analytic performance analysis and control-strategy design.Inverter limits depend on inverter size and concurrent active-power production.
  • B. Power losses minimization problem: The approximate model makes power losses quadratic in injected power and decouples optimal active-power and reactive-power injection.The reactive-power problem is represented as a quadratic, linearly constrained optimization problem.
  • B. Power losses minimization problem: Centralized solution would require network parameters and all load reactive-power demands, creating communication, delay, robustness, and privacy concerns.These concerns motivate a distributed solution for large-scale systems.

IV. A DISTRIBUTED ALGORITHM FOR REACTIVE POWER DISPATCHING

The proposed algorithm repeatedly assigns local optimization subproblems to small, possibly overlapping compensator clusters. Using shared local states and voltage measurements, the clusters update reactive-power injections without central supervision.

  • A. Optimization problem decomposition: The decomposition is needed because the quadratic cost is not separable into individual-agent terms and feasibility must be maintained.The paper therefore does not directly apply common primal or dual decomposition methods.
  • A. Optimization problem decomposition: Compensators are partitioned into possibly overlapping clusters whose members can coordinate and share measurements through a local supervisor.A supervisor processes cluster data and returns results to participating compensators.
  • A. Optimization problem decomposition: At each iteration, one cluster is selected, solves the global-cost subproblem while other nodes hold their states fixed, and updates its reactive powers.The repeated local updates are designed to yield the solution of the original global optimization problem.
  • A. Optimization problem decomposition: The strategy uses local measurements, local topology knowledge, limited data processing, and limited communication instead of central supervision.The compensators determine feasible state updates from shared injected reactive powers and measured voltages.
  • A. Optimization problem decomposition: Cluster-specific subproblems operate on the controllable components of q while uncontrollable components remain fixed.The state and the matrix Re(X) are partitioned to distinguish these components.
  • A. Optimization problem decomposition: The cluster update is obtained from quadratic optimization using the gradient of J with respect to q_C.The construction uses subspace and pseudoinverse properties associated with each cluster.

B. Hessian reconstruction from local topology information

The method reconstructs local Hessian information from mutual effective impedances and local voltage measurements, under structural assumptions on the network and line parameters.

  • Hessian reconstruction: The identity ΩrReffΩr = −2ΩrMΩr links effective impedances to the Hessian block used by the cluster supervisor.The supervisor computes the required pseudoinverse offline because these coefficients depend only on topology and impedances.
  • Local information: Mutual effective impedances between compensators provide the local topology information needed to compute each cluster’s Hessian block.In radial networks, these impedances correspond to the impedance of the unique electric path between compensators.
  • Local measurements: Cluster supervisors can compute Kr(uC) using only voltage measurements from compensators within their cluster.The local map is related to the corresponding elements of the cost-function gradient.
  • Assumptions: The approximation assumes equal power-line inductance/resistance ratios and is considered reasonable for most practical cases.Its effects are evaluated later through numerical simulations.

D. Description of the algorithm

The algorithm repeatedly selects a cluster, gathers local voltage and reactive-power states, computes a local update, and changes only that cluster’s injections.

  • Algorithm structure: The proposed iterative algorithm solves successive optimization subproblems associated with clusters of compensators.The online procedure is represented as a switching discrete-time feedback system.
  • Online procedure: At each iteration, one cluster is randomly selected and its compensators send voltage measurements and reactive-power states to the supervisor.A Poisson-timer construction can generate the independent random cluster sequence.
  • Online procedure: The selected supervisor computes Kr(uC) and updates the reactive-power injection qh of each compensator in its cluster.Compensators outside the selected cluster keep their reactive-power injections constant.
  • Distributed implementation: Each cluster supervisor requires only local cluster measurements and updates only the decision variables belonging to that cluster.This localized control law enables partial state information and local actuation.
  • Distributed implementation: The system must be actuated after each optimization step so subsequent measurements reflect the updated state.The resulting feedback structure resembles iterative congestion-feedback control in communication networks.

V. CONVERGENCE OF THE ALGORITHM

The convergence analysis reduces the approximate algorithm to a random linear iteration and characterizes convergence through cluster connectivity and an exponential-rate metric.

  • System representation: The error dynamics are expressed as a discrete-time system whose state is the deviation x = qC − qoptC from the optimal reactive-power allocation.Convergence of q(t) to qopt is equivalent to convergence of x(t) to zero on ker 1T.
  • System representation: The matrices Fr are self-adjoint and orthogonal projection operators under the M-weighted inner product, preserving ker 1T.These properties support the spectral analysis of the random iteration.
  • Convergence condition: The algorithm has no nonzero common equilibrium error exactly when the cluster hypergraph is connected.Connectivity is therefore the structural condition ruling out persistent errors shared by all cluster updates.
  • Convergence condition: Under iid cluster selection with positive probabilities and a connected hypergraph, the iterative state converges in mean square to the global optimum.Exponential mean-square convergence also implies almost-sure convergence.
  • Convergence rate: R < 1 implies exponential mean-square convergence, while β is used as a computable bound for evaluating the convergence rate.The analysis studies β because the exact rate R is generally difficult to compute.

VI. OPTIMAL COMMUNICATION HYPERGRAPH FOR A RADIAL DISTRIBUTION NETWORK

For radial distribution networks, edge-disjoint communication clusters achieve the optimal convergence rate under the paper’s connectivity and modeling assumptions. Equal-sized clusters attain the bound when their associated weights are uniform.

  • The radial-network assumption makes the distribution graph a tree and enables the clustering optimality result.
  • Edge-disjoint clusters make the interaction matrices orthogonal, with E_rE_r′ = 0 for all r ≠ r′.
  • The bound is minimized by uniform weights ρ_r = 1/ℓ, and the resulting construction is optimal according to Theorem 9.
  • When all clusters have cardinality c, their count satisfies ℓc = m + ℓ − 1, supporting the optimal edge-disjoint hypergraph construction.

VII. SIMULATIONS

Simulations on a balanced 4.8 kV IEEE 37-inspired radial testbed validate the approximate model and distributed algorithm. Edge-disjoint gossip converges fastest and reaches losses close to the centralized optimum, while modeling assumptions explain the small residual gap.

  • The simulations use a balanced 4.8 kV testbed inspired by the standard IEEE 37 distribution network.
  • 61.6 kW losses, equal to 3.11% of delivered active power, occur before distributed reactive power compensation.
  • The approximate model’s error is negligible when its predicted network state is compared with the exact model.
  • 18.27% losses reduction lowers losses from 61589 W to 50338 W, compared with 18.41% and 50253 W for centralized optimization.
  • The residual gap to the centralized minimum is attributed to the large-nominal-voltage approximation and the assumption that θ is constant across the network.
  • Edge-disjoint gossip converges faster than the star topology in the long-time regime and achieves the fastest rate predicted by Theorem 9.
  • Both clustering strategies reach the same steady-state minimum, which is slightly above the minimum from the original nonconvex optimization problem.

VIII. CONCLUSIONS

The paper develops a quadratic model and a distributed randomized algorithm for reactive power compensation, using local communication and knowledge. Its analysis identifies short-range cooperation among electrically neighboring units as optimal, while future work will incorporate dynamics and compensator constraints.

  • The model casts optimal reactive power compensation as quadratic optimization and exploits system physics to design a distributed algorithm.This makes robust solvers available and renders performance analysis tractable.
  • The algorithm is distributed, leaderless, and randomized, requiring only local communication and local knowledge of network topology.
  • The performance metric provides a bound on the best achievable algorithmic performance and identifies the clustering choice attaining it.
  • Short-range cooperation among electrically neighboring units is optimal, unlike consensus methods that benefit from long-range communication.The result is also technologically relevant because distant inverter cooperation raises synchronization and power-line communication challenges.
  • Future work will add network dynamics and operational constraints reflecting limited and time-varying reactive-power injection capability.

APPENDIX A PROOF OF PROPOSITION 2

The appendix constructs a Taylor approximation of voltages and currents around a small parameter and applies the implicit function theorem after verifying invertibility of the relevant Jacobian.

  • The analysis introduces ϵ := 1/UN and represents currents and voltages as functions of ϵ with the PCC voltage phase fixed at zero.
  • Taylor expansions of u(ϵ) and i(ϵ) are substituted into the network equations to obtain a system involving Gv and Fv.
  • The complex equations are interpreted as real-vector functions, yielding mappings from R2n × R2n × R2 to R2.
  • At δ = 0, λ = 0, and ϵ = 0, the functions vanish; if the evaluated matrix is invertible, δ(ϵ) and λ(ϵ) have continuous derivatives at ϵ = 0.
  • The complex multiplication and conjugation operations are represented through real-coordinate derivatives to determine the matrix used in the approximation.
  • Evaluating the derivative expressions at ϵ = 0 yields the required matrix structure, whose invertibility enables Taylor’s theorem and bounded remainder functions.

APPENDIX B PROOF OF THEOREM 7

The appendix proves convergence-related properties by linking projection matrices, hypergraph connectivity, and spectral behavior of their average. These results establish a contraction bound away from the consensus eigenvalue.

  • The conditions that all projection matrices fix only the relevant subspace, that their spans equal ker 1T, and that the associated hypergraph is connected are equivalent.
  • Connectivity of the hypergraph is equivalent to connectivity of a weighted graph whose edge weights count shared cluster membership.
  • If P ≥ Q, then applying the operator L repeatedly preserves the ordering, giving Lk(P) ≥ Lk(Q) for every nonnegative integer k.
  • Under the connectivity assumption, Fave has real eigenvalues with absolute value at most 1, and eigenvalue 1 is the only unit-circle eigenvalue with multiplicity 1.
  • The eigenvalue-1 right and left eigenvectors are M −11 and 1, respectively.
  • Defining β as the largest absolute value among eigenvalues other than 1 yields the contraction bound used in Theorem 7.
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