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Distributed Energy Trading: The Multiple-Microgrid Case

David Gregoratti, Javier Matamoros

arXiv:1307.7622v5math.OCcs.MA

TL;DR

The paper addresses how islanded microgrids can minimize global operating cost while satisfying local demands and keeping local consumption and cost information private. It proposes an iterative distributed algorithm based on local subproblems and price-like variables, with numerical results confirming convergence after a reasonable number of iterations and cost gains over nonconnected microgrids. The iterations admit an economic interpretation in terms of energy trading and market clearing.

  • Problem

    Islanded microgrids need to minimize global operating cost while satisfying local demands, but centralized coordination conflicts with scalability and privacy requirements.

  • Method

    The paper proposes an iterative distributed algorithm that decomposes the problem into local microgrid computations while keeping local cost functions and consumption private.

  • Results

    Numerical results confirm convergence after a reasonable number of iterations and a gain over nonconnected microgrids that depends on energy demands and network topology.

  • Takeaways & Limitations

    The algorithm provides an economic interpretation in which local trading iterations lead toward agreement on prices and transferred energies, while reducing cost at each microgrid.

Abstract

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In this paper, a distributed convex optimization framework is developed for energy trading between islanded microgrids. More specifically, the problem consists of several islanded microgrids that exchange energy flows by means of an arbitrary topology. Due to scalability issues and in order to safeguard local information on cost functions, a subgradient-based cost minimization algorithm is proposed that converges to the optimal solution in a practical number of iterations and with a limited communication overhead. Furthermore, this approach allows for a very intuitive economics interpretation that explains the algorithm iterations in terms of "supply--demand model" and "market clearing". Numerical results are given in terms of convergence rate of the algorithm and attained costs for different network topologies.

I. INTRODUCTION

The paper studies energy trading among islanded microgrids, emphasizing distributed optimization that reduces total operational cost while preserving local cost information. Its algorithm uses iterative local bidding and distributed price adjustment, providing a supply–demand interpretation until prices and transferred energies agree.

  • Motivation and setting: Islanded microgrids can exchange energy over interconnected networks without exchanging energy with the main grid.This setting creates internal energy flows while the microgrids remain disconnected from the main grid.
  • Motivation and setting: Centralized power-flow solutions face scalability and privacy problems, while the general problem is often nonconvex.These issues motivate distributed approaches and abstract trading models focused on energy exchanges rather than detailed grid operations.
  • Problem formulation: The paper formulates an analytical energy-trading problem for M microgrids with generation costs, transfer costs, and local demand constraints.The objective is to choose exchanged energy amounts that minimize total production and transportation costs.
  • Proposed approach: Dual decomposition yields a distributed iterative algorithm that protects local cost functions and avoids a central coordinator.The approach decomposes the global problem into local computations requiring limited information about the rest of the system.
  • Proposed approach: Each iteration has a supply–demand interpretation: microgrids compute local buy, sell, and production decisions, then adjust prices according to demand.The process repeats until prices and transferred energies reach global agreement.

II. SYSTEM MODEL

The system model represents cooperating islanded microgrids that satisfy local demands through generation and bilateral energy transfers. Convex generation and transfer costs support a global cost-minimization formulation, with soft constraints available for practical capacity limits.

  • Microgrid model: The model contains M interconnected microgrids operating in islanded mode, each generating and consuming energy during a scheduling interval.Microgrids may sell energy to neighbors and buy energy from other neighbors.
  • Network and costs: An adjacency matrix specifies which directed microgrid links exist, and transfer costs account for energy movement between connected microgrids.Transfer costs may represent operator charges and line-congestion effects.
  • Optimization objective: The cooperating microgrids minimize total system cost while satisfying local energy-equilibrium constraints.The objective includes generation and transportation costs rather than separate individual-microgrid benefits or losses.
  • Cost-function assumptions: Generation and transfer cost functions are assumed positive, increasing, convex, and twice differentiable.Quadratic generator-cost models are cited as satisfying the stated assumptions.
  • Capacity modeling: Soft capacity limits can be encoded through steeply increasing costs, allowing nominal generation or transfer limits to be exceeded at substantial extra cost.This avoids adding explicit hard inequalities to the minimization problem.

III. ITERATIVE DISTRIBUTED MINIMIZATION

The minimization framework decomposes the global convex problem into local microgrid subproblems and solves the resulting dual problem iteratively. This reduces information requirements while retaining convergence to the minimum-cost solution.

  • Problem decomposition: The centralized formulation has a unique minimum but can be difficult to solve because it contains M(M −1) energy-transfer unknowns.A central controller would also need global system characteristics, increasing traffic and threatening privacy.
  • Problem decomposition: Dual decomposition relaxes coupling constraints and separates the problem into M reduced-complexity local subproblems.Each microgrid solves its local problem with little information about the rest of the system.
  • Dual formulation: The auxiliary variable ε(s)_i represents energy sold by microgrid i before being matched to energy bought by neighboring microgrids.Relaxing and subsequently enforcing these coupling relationships enables the decomposition.
  • Iterative algorithm: At each iteration, microgrids solve local Lagrangian minimizations, exchange neighboring energy information, and update Lagrange multipliers using a subgradient step.The sequence of dual variables converges to the optimal point under the stated convex formulation.
  • Iterative algorithm: The algorithm derives exchanged-energy vectors from local decisions and updates each microgrid’s multipliers after receiving information from neighboring microgrids.Algorithm 1 summarizes this distributed minimization procedure.

3) Interpretation—Market clearing:

The distributed approach solves local trading subproblems using exchanged prices and energy requests, while multiplier updates clear the market until aggregate supply matches demand. Its case analysis characterizes microgrid behavior under different price and marginal-cost relationships, with privacy and communication constraints preserved.

  • Distributed approach: All necessary data is computed locally, without requiring external centralized control.
  • Distributed approach: Microgrids exchange Lagrange multipliers and demanded energies only with corresponding neighbors, limiting both information exposure and traffic.
  • Market interpretation: Each local subproblem determines energy buying, selling, and generation from unitary prices and marginal generation and transportation costs.
  • Market interpretation: The multiplier update acts as market clearing by modifying prices until global energy demand matches energy offer.
  • Local cases: The six local cases partition the multiplier hyperplane, making their necessary conditions sufficient and yielding exact energy-flow values.

C. Interpretation and summary

The economic interpretation maps multipliers to selling and buying prices and explains local decisions through marginal costs. Under convex cost functions, the resulting cases characterize when microgrids buy, sell, generate, or remain self-contained, while all microgrids are willing to trade.

  • Economic interpretation: Lagrange multipliers represent unitary selling and buying prices, while derivatives of generation and transportation costs represent marginal costs.
  • Economic interpretation: A microgrid does not sell when its selling price is below marginal production cost, and buying is likewise unprofitable under the corresponding cost condition.
  • Trading outcome: Microgrids are always willing to trade because local cost without trading is higher than their net expenditure with trading.
  • Optimality: Convexity and monotonicity of the cost functions support the derived optimality result for the local subproblem.

IV. NUMERICAL RESULTS

Numerical experiments evaluate the distributed minimization algorithm for four microgrids across fully connected, ring, and line topologies. The algorithm converges after a reasonable number of iterations, while trading reduces costs and produces prices consistent with supply and demand.

  • Experimental setup: Four microgrids are evaluated using fully connected, ring, and line network topologies.The experiments use shared and topology-specific cost analyses.
  • Convergence: The convergence experiment tracks the duality gap, total system cost, and selling prices over iteration number.Figure 2 uses loads E(c) = [8, 11, 11, 6]T MWh in a fully connected system.
  • Convergence: The algorithm converges after a reasonable number of iterations, with an almost-null duality gap.The result is reported for the fully connected convergence experiment.
  • Price behavior: After convergence, higher local load corresponds to a higher selling price under the strictly increasing generation-cost function.Lower local loads allow lower-cost production of extra energy for sale.
  • Price behavior: Selling prices also reflect total demand, which combines internal local loads with external demand from other microgrids.The paper interprets this relationship through the law of demand and market clearing.
  • Cost outcomes: With the specified local cost, optimal trading always reduces costs for all microgrids relative to disconnected operation.Disconnected costs are shown as a benchmark in the topology experiments.

6 MWh that the gain becomes noticeable, reaches its maximum for E(c)

The topology and load distribution shape trading benefits and local costs. Fully connected, ring, and line cases show how marginal costs, transfer costs, intermediary roles, and bottlenecks determine energy flows and gains.

  • Fully-connected topology: The best trade-off between sold energy and unit price determines the maximum selling income.At higher loads, increased prices can improve gains even as sold energy decreases.
  • Fully-connected topology: At higher loads, matching selling- and buying-side prices creates a trade-off that benefits µG-4, while the convex transfer cost accentuates the earlier trend.The four microgrids operate in the nonlinear generation-cost region when E(c)4 > 6 MWh.
  • Ring topology: In the ring topology, µG-2 and µG-3 benefit as intermediaries by buying energy for internal needs and reselling it to µG-1.Their intermediary role substantially reduces their local costs.
  • Line topology: In the line topology, µG-3 is the bottleneck because all energy from µG-4 to µG-1 and µG-2 passes through it, benefiting µG-3.This topology-specific flow constraint changes the distribution of costs.

V. CONCLUSIONS

The paper addresses distributed energy exchange among islanded microgrids to minimize global operation cost while satisfying local demands. It proposes an iterative distributed algorithm combining local optimization with market clearing, and reports convergence and cost gains influenced by demands and topology.

  • The framework considers several microgrids exchanging energy through a network while satisfying local demands and minimizing total operation cost.
  • The proposed algorithm is scalable in the number of microgrids and preserves the privacy of local cost functions and consumption.
  • Each iteration performs a local minimization step, followed by market clearing that adjusts energy prices according to demand.
  • A closed-form local optimization solution gives the algorithm an economic interpretation in terms of energy bids and prices.
  • Every microgrid is willing to initiate trading regardless of its local demand because its net expenditure becomes lower than operating independently.
  • The algorithm converges after a reasonable number of iterations, while gains over nonconnected microgrids depend strongly on energy demands and network topology.

APPENDIX A

The appendix derives the six possible local microgrid operating states from the KKT conditions. For each state, it identifies price conditions and energy values that jointly characterize the local cost-minimizing solution.

  • The derivation uses KKT conditions, including complementary slackness, to analyze the local problem.
  • The local microgrid state is classified by whether it sells, buys, or generates energy, producing six possible operating cases.
  • For each case, the derivation computes relevant energy flows and identifies price constraints required for feasibility.
  • The six sets of necessary conditions partition the price hyperplane, making each case's conditions sufficient as well as necessary.
  • The resulting energy values minimize the local cost function for the given prices.
  • The inverse function Γ of the marginal generation cost derivative exists and is increasing under the continuity and convexity assumptions.
  • Case 2: In Case 2, a unique η solution determines energy bought from neighboring microgrids, with Fig. 6 illustrating the associated boundary condition.

C. Proof of Case 3

The proof of Case 3 applies KKT conditions to a microgrid that generates and buys energy. It derives unique price-dependent solutions for purchased and generated energy and identifies the relevant selling neighbors.

  • Case 3: Case 3 assumes the microgrid generates energy and buys positive energy from at least one neighboring microgrid.
  • Case 3: With zero Lagrange multiplier ω, the KKT conditions distinguish neighbors supplying positive energy from those supplying none.
  • Case 3: The KKT conditions impose λ_i < C′ as the first requirement for Case 3.
  • Case 3: A unique solution in η determines the energy bought from neighboring microgrids.
  • Case 3: Knowing η and using the inverse of C′ determines the microgrid's generated energy.
  • Case 3: The minimum neighbor price λ_min belongs to the set of supplying neighbors when Case 3 is optimal.

D. Proof of Case 4

The proof of Case 4 analyzes a microgrid that sells and generates energy without buying from neighbors. KKT conditions yield price requirements and characterize the generated and sold energy.

  • Case 4: Case 4 assumes the microgrid buys no energy, sells energy, and generates a positive amount.
  • Case 4: The first KKT condition requires λ_i > C′ for this operating case.
  • Case 4: The remaining KKT conditions require λ_j ≥ λ_i − γ′(0) for neighboring microgrids.
  • Case 4: The generated energy is obtained by inverting C′ under the Case 4 price conditions.
  • Case 4: For a generating and selling microgrid, purchased energy from an active neighbor satisfies E_j,i = Γ(λ_i − λ_j).
  • Case 4: The expression is meaningful when λ_j < λ_i − γ′(0), which exactly characterizes membership in the active-neighbor set S∗.

F. Proof of Case 6

Case 6 characterizes a microgrid operating state through KKT conditions, positive energy flows, and parameter inequalities. Its mutually exclusive and sufficient conditions uniquely solve the local minimization problem.

  • KKT conditions: The KKT conditions impose −γ′(0) − λ_j + µ_j = 0 and µ_j ≥ 0 for every j ∈ S0.
  • Operating-state sets: Case 6 partitions other microgrids into S* with positive flows and S0 with zero flows.S* contains indices j ≠ i with E_j,i > 0, while S0 contains those with E_j,i = 0.
  • Positive-flow solution: For every j ∈ S*, λ_i − λ_j − γ′(E_j,i) = 0 yields E_j,i = Γ(λ_i − λ_j), with Γ the inverse of γ.This expression requires λ_j < λ_i − γ′(0) for all j ∈ S*.
  • Energy quantities: The total generated and sold energies are expressed as sums over the positive-flow set, and meaningful positive quantities require the stated positivity conditions.The text also requires P_i > 0 and E_j,i > 0 for at least one index in the relevant case.
  • Case completeness: The case conditions are mutually exclusive and sufficient, so the local minimization problem is univocally solved.The paper states that all possible microgrid operating states have corresponding energy-flow values and feasibility conditions.
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