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Distributed Optimal Power Flow for Smart Microgrids
Emiliano Dall'Anese, Hao Zhu, Georgios B. Giannakis
TL;DR
The paper tackles nonconvex OPF in unbalanced microgrids while targeting voltage regulation and either loss or supplied-power-cost minimization. It uses SDP relaxation and an ADMM-based distributed solver; numerical tests report rank-1 optima in most cases and faster convergence than a sub-gradient alternative.
Problem
Unbalanced microgrid OPF is nonconvex and centralized SDP solution may scale poorly for larger networks and rapidly changing operating conditions.
Method
The paper applies SDP relaxation to unbalanced microgrid OPF and solves the relaxed problem distributively using ADMM.
Results
The SDP optimum was always rank 1 in tested cases except the IEEE 37-node feeder when cs/c0 > 1.75; the proposed distributed solver converged considerably faster than the sub-gradient method.
Takeaways & Limitations
The approach offers the potential to recover globally optimal solutions while supporting scalability, communication-outage robustness, and data privacy and integrity.
Abstract
from arXiv · showhide
Optimal power flow (OPF) is considered for microgrids, with the objective of minimizing either the power distribution losses, or, the cost of power drawn from the substation and supplied by distributed generation (DG) units, while effecting voltage regulation. The microgrid is unbalanced, due to unequal loads in each phase and non-equilateral conductor spacings on the distribution lines. Similar to OPF formulations for balanced systems, the considered OPF problem is nonconvex. Nevertheless, a semidefinite programming (SDP) relaxation technique is advocated to obtain a convex problem solvable in polynomial-time complexity. Enticingly, numerical tests demonstrate the ability of the proposed method to attain the globally optimal solution of the original nonconvex OPF. To ensure scalability with respect to the number of nodes, robustness to isolated communication outages, and data privacy and integrity, the proposed SDP is solved in a distributed fashion by resorting to the alternating direction method of multipliers. The resulting algorithm entails iterative message-passing among groups of consumers and guarantees faster convergence compared to competing alternatives
I. INTRODUCTION
The paper addresses nonconvex OPF for unbalanced microgrids, where voltage regulation and efficient operation are complicated by unequal loads, conductor spacings, and distributed generation. It proposes SDP relaxation and a distributed ADMM solver to support global optimality, scalability, outage robustness, and privacy.
- OPF is needed to control voltage fluctuations, node over- and under-voltages, and power-factor drops while operating smart microgrids efficiently.
- Unbalanced distribution networks arise from unequal single-phase loads and non-equilateral three-phase conductor spacings, while single-phase DG can worsen imbalance.
- The proposed SDP relaxation brings global-optimality potential, quantifiable worst-case complexity, and accommodation of additional thermal and quality-of-power constraints.
- Centralized SDP can become computationally prohibitive for medium and large feeders and may respond too slowly to changing loads and intermittent renewable generation.
- The distributed approach uses ADMM to decompose SDP solution across local areas, extending prior decentralized OPF work to unbalanced distribution networks.
- The paper is organized around OPF modeling, centralized SDP, distributed optimization, numerical tests, and conclusions.
II. MODELING AND PROBLEM FORMULATION
The formulation models an unbalanced microgrid with phase-specific voltages, currents, loads, capacitors, lines, transformers, and DG units. It minimizes either distribution losses or supplied-power cost subject to network, voltage, and generation constraints, yielding a nonconvex OPF problem.
- The microgrid is represented as nodes and edges, with each line and node associated with its available phases.
- Line-to-ground voltages and injected currents are phase-specific complex variables, with PCC voltages serving as the phasorial reference.
- Distribution lines use π-equivalent models with phase impedance and shunt-admittance matrices, while transformers are modeled as series components dependent on connection type.
- The model includes wye-connected loads, capacitor susceptances, and DG units whose controllable active and reactive outputs become OPF variables.
- The two objectives are minimizing power losses or the cost of power drawn from the PCC and supplied by DG units; they coincide when c0 = 1 and cs = 1 for all s.
- The OPF includes voltage, conventional-DG box, and network constraints, but remains nonlinear and nonconvex because of load-flow equations and voltage constraints.
III. SDP-BASED CENTRALIZED SOLUTION
The centralized solver lifts the voltage formulation into an outer-product matrix and expresses network quantities linearly in that matrix. Dropping its rank-1 constraint produces a convex SDP whose rank-1 optimum recovers a globally optimal OPF solution.
- Voltages and injected currents satisfy Ohm’s law i = Yv, with Y assembled as a symmetric block matrix from the network phases and line models.
- The formulation expresses injected powers, line flows, and voltage magnitudes as linear functions of the outer-product matrix V := vvH.
- Lemma 1 establishes linear relations between V and voltage magnitudes, active powers, and reactive powers, enabling an equivalent reformulation of the OPF.
- Problem (P2) remains nonconvex because of its rank-1 constraint, while dropping that constraint yields the convex SDP problem.
- If the SDP optimum Vopt has rank 1, its recovered voltage vector is globally optimal for the original OPF, with identical optimal costs.
- The relaxed SDP can be solved in polynomial time, although sparsity and chordal structure are needed for substantial computational savings.
- If Vopt has rank greater than 1, randomization can produce a feasible rank-1 approximation with quantifiable error, but the resulting solution is generally suboptimal.
A. Constraints on line flows
The paper reformulates line power-loss and current-magnitude limits as semidefinite-program constraints, including protection of neutral cables in unbalanced microgrids.
- Line-loss constraints: Line real-power dissipation is bounded by adding a trace inequality to the relaxed SDP.The bound uses Tr(Φm→nV) + Tr(Φn→mV) ≤ ∆Pmn.
- Current constraints: Line currents are collected in imn and related to endpoint voltages through the inverse line impedance matrix.This voltage-current relation enables current constraints to be expressed using the outer-product matrix V.
- Current constraints: Lemma 2 provides an equivalent SDP-compliant reformulation of each phase-current magnitude constraint.The reformulation uses a Hermitian matrix and trace-based expressions.
- Current constraints: The same derivation yields constraints on injected-current magnitudes at network nodes.The paper states that these constraints follow using the corresponding power-flow relations.
- Neutral-cable protection: Neutral-cable currents are modeled through a neutral transformation matrix obtained by Kron reduction and constrained to protect against overheating.The transformation relates neutral currents to line currents as i^(ϕ)_mn = Tmnimn.
B. The rank conundrum
The SDP relaxation can enlarge feasible regions and produce higher-rank optima, but the paper’s unbalanced-network examples show cases where the original and relaxed problems share optimal solutions.
- Why rank is difficult: Balanced-network rank-1 guarantees rely on decoupled line flows, whereas unbalanced networks couple flows through voltage-angle relationships.Changing one line’s flow can affect angle differences on connected lines, so prior balanced-network results do not directly apply.
- Why rank is difficult: Analytically characterizing the unbalanced feasible-flow region is challenging because it involves many voltage angles and coupled line power flows.The paper uses examples to investigate why rank-1 solutions may nevertheless occur.
- Two-node example: For the two-node example, the feasible line-flow region is a perturbed ellipsoid formed from the Minkowski sum of per-phase regions.The Pareto points of the feasible region and its convex hull are compared to identify when relaxation introduces higher-rank solutions.
- Two-node example: Convex-hull points outside the feasible two-node flow region correspond to relaxed solutions with rank higher than 1.The example distinguishes the original feasible region from the additional region admitted by convexification.
- Three-node example: For the three-node example, the feasible injected-power region is nonconvex, yet its Pareto region is unchanged by taking the convex hull.The paper therefore concludes that the relaxed solution has rank 1 and is optimal for the original problem in this example.
- Implication: The examples suggest that the nonconvex OPF and its relaxation share an optimum when the relevant line-angle quantities are sufficiently small.This motivates analytical characterization of unbalanced distribution-system power-injection regions.
IV. DISTRIBUTED SOLUTION
The centralized SDP is decomposed across overlapping microgrid areas whose local voltage matrices remain coupled through shared entries. Chordal-graph conditions enable equivalent local PSD constraints, while ADMM coordinates distributed updates and converges to the centralized SDP solution.
- Area decomposition: The distributed formulation partitions the microgrid into L areas, extends each area to include neighboring nodes, and assigns each area to a local controller.Neighboring extended areas are identified through nonempty intersections, and local voltage vectors collect the nodes in each extended area.
- Area decomposition: The decomposed objective expresses the total cost as a superposition of local area costs while assigning network constraints on a per-area basis.The local cost can represent distribution losses or costs associated with substation and distributed-generation power, depending on the objective formulation.
- PSD coupling: Shared entries among local voltage submatrices prevent direct PSD decomposition, so neighboring areas must agree on their common matrix entries.Auxiliary variables are introduced to enable a fully distributed treatment of these coupling constraints.
- PSD coupling: If the macro graph is a tree and no nested extended areas exist, the induced micro graph is chordal and its local matrices are the maximal cliques.Matrix-completion results then make global V ⪰ 0 equivalent to enforcing V(l) ⪰ 0 for every area.
- ADMM algorithm: ADMM alternates per-area primal updates, auxiliary-variable updates, and dual-variable updates to solve the coupled distributed SDP.In the primal step, each area minimizes the partially augmented Lagrangian with other variables and multipliers fixed; dual variables use sub-gradient ascent.
- ADMM algorithm: The distributed iterations converge to the solution of the centralized SDP, with each limiting local matrix equal to the corresponding submatrix of the centralized optimum.The convergence result states lim_i→+∞ V(l)(i) = V(l)_opt for all areas l.
V. NUMERICAL TESTS
Numerical tests evaluate the SDP-based OPF and its ADMM-based distributed solver on an unbalanced IEEE 37-node feeder and a 10-node three-phase network. The relaxation attained rank-1 solutions in the reported tests, while the distributed solver converged faster than the sub-gradient alternative.
- Test networks: The SDP-based solver was tested on an unbalanced IEEE 37-node feeder and a 10-node three-phase network.The IEEE feeder is an actual 4.8 kV distribution-network portion with seven single-phase DG units; the second network has DG units at nodes 5 and 7.
- Centralized solution: The centralized SDP problems required 9.0 seconds and 0.3 seconds for the two test networks, respectively.These times were reported for SeDuMi solving the centralized relaxed problem and were lower than times reported for commercial nonlinear-program solvers.
- DG operation: The DG dispatch varied with cs/c0: units close to the PCC were unused when cs ≥c0, while DG 6 and 7 operated above 50% capacity.The passage also reports significantly higher power loss when DG units were not used to their maximum extent.
- Relaxation tightness: The optimal matrix Vopt was rank 1 in the reported tests, attaining globally optimal solutions of the original nonconvex OPF.For the IEEE 37-node feeder, rank(Vopt) exceeded 1 when cs/c0 > 1.75, but a rank-1 solution was obtained after raising the PCC voltage from 1 to 1.02 pu; additional IEEE 13-node tests also produced rank-1 matrices.
- Distributed convergence: The ADMM-based distributed solver converged considerably faster than the sub-gradient method on both tested network partitions.For the 10-node network, matrix gaps rapidly vanished at κ = 100 and κ = 10 after approximately 100 iterations; smaller areas reduced subproblem solution times.
VI. CONCLUDING REMARKS
The paper applies SDP relaxation to unbalanced microgrid OPF and develops an ADMM-based distributed solver. Numerical tests support globally optimal solutions, while the distributed method targets scalability, communication-outage robustness, and data privacy and integrity.
- SDP relaxation: The SDP relaxation converts unbalanced microgrid OPF into a convex problem with potential to attain globally optimal solutions of the original nonconvex formulation.The conclusion frames global optimality as the main contribution supported by numerical tests.
- Distributed optimization: The ADMM-based distributed SDP solver is designed to scale with microgrid size, tolerate isolated communication outages, and preserve data privacy and integrity.The distributed algorithm uses iterative message-passing among groups of consumers.
APPENDIX
The appendix establishes the matrix representations and convergence properties underlying the distributed SDP formulation. It relates network quantities to the outer-product voltage matrix and shows optimality of limit points under the stated assumptions.
- Matrix reformulation: Injected powers, line flows, and voltage magnitudes are expressed as linear functions of the outer-product matrix V := vvH.The appendix derives node injections from voltage-related matrix expressions and line flows from admittance-based current relations.
- Distributed updates: The appendix analyzes multiplier and local-variable updates that decouple across neighboring areas during distributed iterations.For the first iteration, the optimization decouples into subproblems involving pairs of neighboring areas.
- Convex subproblems: Schur’s complement is used to derive a linear matrix inequality that bounds the squared l2 norm of auxiliary residual vectors.The construction introduces auxiliary variables αj and constraints such as ∥rj,ℜ∥2^2 ≤αj.
- Convergence guarantee: Every limit point of the local matrix sequence is an optimal solution under the assumptions making the distributed and centralized SDPs equivalent.The proof invokes convexity and boundedness of the constraint set, then transfers optimality from (P6) to (P3).