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Decentralized Optimal Dispatch of Photovoltaic Inverters in Residential Distribution Systems
Emiliano Dall'Anese, Sairaj V. Dhople, Brian B. Johnson, Georgios B. Giannakis
TL;DR
Residential PV inverter control must address voltage regulation and network losses beyond maximum-power, unity-power-factor operation. The paper combines OPF, sparsity-promoting regularization, semidefinite relaxation, and ADMM to compute real and reactive setpoints across utilities and customer systems. The resulting decentralized framework distributes computation and communication while retaining convergence and optimality properties described in the paper.
Problem
Conventional PV inverter control targets maximum power at unity power factor but does not address objectives such as voltage regulation and network loss minimization.
Method
The paper decomposes an SDR- and sparsity-based optimal inverter dispatch problem into decentralized ADMM subproblems for utilities and customer-owned PV systems.
Results
At λ = 0.8, the number of participating inverters reaches a maximum of 7 at 13:00, while at λ = 0 all inverters are controlled from 8:00 to 18:00.
Takeaways & Limitations
The decentralized optimal inverter dispatch strategy shares computational burden and optimization objectives across the distribution network while supporting limited information exchange.
Abstract
from arXiv · showhide
Decentralized methods for computing optimal real and reactive power setpoints for residential photovoltaic (PV) inverters are developed in this paper. It is known that conventional PV inverter controllers, which are designed to extract maximum power at unity power factor, cannot address secondary performance objectives such as voltage regulation and network loss minimization. Optimal power flow techniques can be utilized to select which inverters will provide ancillary services, and to compute their optimal real and reactive power setpoints. Leveraging advances in semidefinite relaxation techniques and sparsity-promoting regularizations, such problems can be solved with reduced computational burden and with optimality guarantees. To enable large-scale implementation, a novel algorithmic framework is introduced - based on the so-called alternating direction method of multipliers - by which the optimal power flow problem in this setting can be systematically decomposed into sub-problems that can be solved in a decentralized fashion by the utility and customer-owned PV systems with limited exchanges of information. Since the computational burden is shared among multiple devices and the requirement of all-to-all communication can be circumvented, the proposed optimization approach scales to large distribution networks.
I. INTRODUCTION
Residential PV inverters can support voltage regulation and other network objectives, but centralized optimal inverter dispatch requires communication with all inverters. The paper develops decentralized ADMM-based approaches that distribute optimization across utilities and customer-owned PV systems.
- PV inverter secondary control can address overvoltages and voltage transients during high or rapidly varying generation.
- Reactive power compensation can regulate voltage but may reduce power factors, increase network currents, and raise power losses.
- The centralized OID framework selects critical PV inverters and computes their real and reactive power setpoints using OPF, sparsity regularization, and semidefinite relaxation.
- The proposed DOID framework decomposes OID into subproblems solved by utility energy managers and customer-owned PV systems with limited information exchange.
- Two decentralized setups support different communication patterns, including customer-to-utility, customer-to-CEM, and CEM-to-CEM exchanges.
- The framework uses SDR-based optimality guarantees, ADMM-based convergence, exact AC power flow, and joint active-reactive setpoint computation.
A. Network and PV-inverter models
The paper models a balanced residential distribution network with utility-pole and PV-home nodes, line admittance relationships, constant-PQ loads, and inverter operating regions that jointly constrain active curtailment and reactive power.
- The balanced network contains transformer, utility-pole, and residential-PV nodes connected by distribution lines.
- Line voltages and injected currents are related through the system admittance matrix formed from network topology and π-equivalent line circuits.
- Loads use a constant-PQ model with active demand Pℓ,h and reactive demand Qℓ,h at PV-inverter nodes.
- PV arrays have maximum available active power P av_h under given solar irradiation conditions.
- The inverter model jointly controls real and reactive power within an apparent-power-based operating region, optionally enforcing a lower power-factor bound.
B. Centralized optimization strategy
The centralized OID formulation jointly optimizes PV real and reactive power, selects participating inverters, and enforces network and inverter constraints. Sparsity regularization and semidefinite relaxation reduce combinatorial complexity while retaining optimality guarantees under rank-one conditions.
- Problem formulation: OID jointly optimizes active and reactive PV-inverter powers while selecting the subset providing ancillary services.Binary variables indicate dispatched inverters, while selected setpoints must lie in the OID operating region.
- Problem formulation: The formulation enforces power balance, voltage, inverter-operating, and cardinality constraints across the distribution network.At most K of the |H| inverters may provide ancillary services.
- Computational challenge: The OID problem is nonconvex because of power-flow and voltage constraints, while binary inverter-selection variables create combinatorially many subproblems.These features make globally optimal and efficient solution difficult.
- Sparsity-promoting reformulation: Group sparsity replaces binary selection by regularizing the paired active and reactive setpoints, which are jointly zero for undispatched inverters.The resulting regularized objective can trade achievable cost against the number of controlled inverters through λ.
- Semidefinite relaxation: Semidefinite relaxation expresses powers and voltage magnitudes linearly in V := vv^H and drops rank(V) = 1 to obtain a convex problem.When the relaxed optimum has rank 1, voltages, currents, and power flows are globally optimal for the given inverter setpoints.
- Guarantees and implementation: The relaxed inverter setpoints may be slightly sub-optimal because of regularizer shrinkage, and their optimality gap is computationally impractical to assess generally.The centralized implementation also requires gathering customer loads and available PV powers at a utility-managed processor.
III. DOID: UTILITY-CUSTOMER MESSAGE PASSING
The decentralized OID framework splits utility and customer objectives through auxiliary consensus variables and solves the resulting problem with ADMM. Utility-customer message passing converges to the centralized OID solution while sharing computation and limiting information exchange.
- Problem decomposition: The cost separates utility objectives, such as losses and voltage deviations, from customer curtailment costs or rewards.Customer terms can be modeled by convex functions, including quadratic curtailment costs.
- Problem decomposition: Utility copies of inverter setpoints and consensus constraints make the decentralized formulation equivalent to the relaxed centralized OID problem.Auxiliary variables are introduced to enable decomposition despite the coupling imposed by consensus.
- ADMM message passing: ADMM alternates utility, customer, auxiliary-variable, and dual updates, with utility and customers exchanging current real and reactive setpoint iterates.The auxiliary update decomposes into 2|H| scalar unconstrained quadratic programs with closed-form solutions.
- Algorithm simplification: Lemma 1 simplifies the ADMM iterations by equating paired dual variables and relating auxiliary variables to customer-side setpoints.These identities support the reduced utility-customer implementation in Algorithm 1.
- ADMM message passing: The utility solves a network-constrained regularized OID problem, while each customer updates its PV-inverter setpoints using local constraints and objectives.The utility relays updated desired setpoints to customers, which return their updated local setpoints.
- Convergence: For any κ > 0, the decentralized iterates converge, and the voltage and active-power copies converge to the optimal centralized OID solutions.The algorithm can terminate when the real- and reactive-power consensus error falls below a prescribed ε.
IV. DOID: NETWORK CLUSTER PARTITIONS
The network is partitioned into overlapping clusters so the semidefinite-relaxed dispatch problem can be decomposed into per-cluster subproblems. ADMM coordinates neighboring clusters and PV systems, yielding convergent decentralized optimization with lower worst-case complexity than the centralized formulation.
- Network partitioning: Clusters include internal nodes and boundary nodes connected to neighboring clusters, enabling local voltage representations while preserving inter-cluster coupling.For cluster a, the extended set ˜C_a adds nodes outside C_a that connect to nodes inside it.
- Network partitioning: The network objective, power-flow constraints, and PV-setpoint consensus constraints are decoupled on a per-cluster basis.Consensus ensures that cluster energy managers and customer-owned PV systems agree on inverter setpoints.
- Semidefinite decomposition: Under a tree-structured, non-nested cluster graph, the centralized semidefinite constraint can be replaced by positive-semidefinite constraints on reduced per-cluster matrices.The replacement is justified through matrix-completion results and chordal structure.
- Decentralized algorithm: ADMM alternates local CEM voltage and setpoint updates, auxiliary-variable updates, and dual ascent while exchanging boundary voltage submatrices and PV-setpoint copies.Neighboring CEMs exchange local submatrices, while CEMs and customer PV systems exchange local setpoint copies.
- Decentralized algorithm: The resulting iterates converge to a solution of the centralized OID and clustered OID problems, after which PV controllers implement the real and reactive setpoints.The algorithm is summarized as Algorithm 2 and uses limited exchanges between neighboring CEMs and customer-owned PV systems.
- Computational implications: The clustered formulation has markedly lower worst-case complexity than the centralized SDP and can exploit sparsity and chordal structure for additional computational savings.The centralized SDP complexity is expressed in terms of total constraints and variables, whereas clustering reduces the problem dimensions.
V. CASE STUDIES
The case study evaluates centralized and decentralized optimal inverter dispatch under residential PV and load conditions. Results show that the dispatch selects participating inverters according to the trade-off parameter and that the decentralized algorithms converge to compatible setpoints and voltages.
- Simulation setup: The test network models 12 houses with rooftop PV, July Minneapolis solar data, perturbed residential loads, and voltage limits of 0.917 pu and 1.042 pu.PV systems use a 0.77 dc-ac derating coefficient, 10% inverter oversizing, and a minimum power factor of 0.85.
- Simulation setup: The utility objective minimizes network power losses through C_utility(V, p̄_c) = Tr(LV), while customers incur the active-power-curtailment cost R_h(P_c,h) = 0.1P_c,h.The parameter λ controls the trade-off between the achievable objective and the number of controlled inverters.
- Centralized OID results: With λ = 0, all inverters are controlled and curtail active power from 8:00 to 18:00; with λ = 0.8, at most 7 inverters operate away from the business-as-usual point at 13:00.For λ = 0.8, participating inverters increase as solar irradiation increases.
- Decentralized convergence: For λ = 0.8 at 12:00, Algorithm 1 reaches the centralized solution by iteration 20, controlling only inverters at houses H7–H12 with matching active-power-curtailment setpoints.Problems (4) and (5) are equivalent, with (5) arranged for ADMM-based distributed solution schemes.
- Decentralized convergence: The set-point consensus error converges quickly, and similar convergence trajectories are obtained for reactive-power setpoints.The voltage discrepancies between the two CEMs also converge to agreement on the shared branch voltage, although the ADMM trajectory is bumpy.
VI. CONCLUDING REMARKS
The paper develops decentralized approaches for computing optimal real and reactive power setpoints for residential PV inverters. The framework shares computation and optimization objectives across the distribution network and supports customer participation in distribution-system markets.
- The proposed decentralized optimal inverter dispatch strategy computes optimal real and reactive power setpoints while sharing computational burden and optimization objectives across the distribution network.