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Robust Energy Management for Microgrids With High-Penetration Renewables

Yu Zhang, Nikolaos Gatsis, Georgios B. Giannakis

arXiv:1207.4831v3math.OCeess.SY

TL;DR

Renewable-rich microgrids must schedule distributed generation, storage, and controllable demand despite intermittent RES and grid transactions. The paper formulates a robust net-cost problem with committed renewable energy and solves it through dual decomposition. The resulting distributed design incorporates local controllers and is evaluated numerically, while vertex enumeration can become exponentially complex as uncertainty-set dimensions grow.

  • Problem

    Scheduling grid-connected microgrids with high RES penetration requires accounting for random, nondispatchable renewable availability while maintaining supply-demand balance.

  • Method

    The paper introduces committed renewable energy, worst-case RES transaction costs, and a dual-decomposition algorithm solved by local controllers for DG, DS, and loads.

  • Results

    Numerical results corroborate the merits of the proposed robust scheduling and distributed designs.

  • Takeaways & Limitations

    The formulation jointly schedules supply and demand while incorporating DG, DS, elastic-load utility, and worst-case energy transactions.

  • Takeaways & Limitations

    Vertex enumeration has exponential complexity as the number of uncertainty-set variables and constraints increases, though one-time enumeration can be affordable for small sub-horizons.

Abstract

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Due to its reduced communication overhead and robustness to failures, distributed energy management is of paramount importance in smart grids, especially in microgrids, which feature distributed generation (DG) and distributed storage (DS). Distributed economic dispatch for a microgrid with high renewable energy penetration and demand-side management operating in grid-connected mode is considered in this paper. To address the intrinsically stochastic availability of renewable energy sources (RES), a novel power scheduling approach is introduced. The approach involves the actual renewable energy as well as the energy traded with the main grid, so that the supply-demand balance is maintained. The optimal scheduling strategy minimizes the microgrid net cost, which includes DG and DS costs, utility of dispatchable loads, and worst-case transaction cost stemming from the uncertainty in RES. Leveraging the dual decomposition, the optimization problem formulated is solved in a distributed fashion by the local controllers of DG, DS, and dispatchable loads. Numerical results are reported to corroborate the effectiveness of the novel approach.

NOMENCLATURE

The nomenclature defines the scheduling horizon, resource and load indices, uncertainty sets, decision variables, costs, utilities, and optimization constructs used throughout the paper.

  • Indices: T and t denote the number of scheduling periods and the period index, while M, N, Q, J, I, and S index system components and sub-horizons.These indices cover conventional DG, dispatchable loads, energy loads, DS units, RES facilities, and RES uncertainty sub-horizons.
  • Uncertainty: W and W_i denote aggregate and facility-specific RES power-output uncertainty sets, respectively.The nomenclature distinguishes uncertainty across all RES facilities from uncertainty associated with one facility.
  • Decision variables: P^t_Gm, P^t_Dn, P^t_Eq, P^t_Bj, and B^t_j represent conventional generation, load consumption, DS charging or discharging, and stored energy.The variables describe power outputs or consumption and storage state over each period.
  • Power and uncertainty variables: P^t_R is net power delivered from RES and storage, while P~^t_R is an auxiliary variable used in the formulation.W^t_worst denotes RES production yielding the worst-case transaction cost.
  • Objective and dual notation: C^t_m, U^t_Dn, U^t_Eq, H^t_j, and G denote conventional-generation cost, load utilities, DS cost, and worst transaction cost.L(x,z) and D(z) denote the Lagrangian and dual functions.

I. INTRODUCTION

The introduction frames distributed energy management for renewable-rich microgrids as a scheduling problem under intermittent RES, and presents a robust, distributed optimization approach covering supply and demand.

  • Microgrid context: Microgrids combine distributed energy resources and end-users, including DG, DS, renewable sources, and potentially controllable elastic loads.DG includes small-scale generators and RES, while DS includes batteries, flywheels, and pumped storage.
  • Motivation: The central scheduling challenge is accounting for the random and nondispatchable nature of RES across grid-connected or islanded microgrid operation.An MGEM coordinates DERs and controllable loads through local controllers and communications infrastructure.
  • Related work: Prior work addressed microgrid economic dispatch, unit commitment, and demand-side management without a robust formulation against RES uncertainty.Related approaches include wind-based risk minimization and stochastic or chance-constrained programs.
  • Paper approach: The paper develops robust grid-connected energy management that includes DG cost, elastic-load utility, penalized DS cost, and worst-case RES transaction cost.Committed renewable energy is introduced to maintain supply-demand balance under intermittent RES.
  • Contributions: The model includes multiple wind farms, two uncertainty models, detailed DS choices, and controllable loads with total-horizon energy requirements such as PHEV charging.Numerical tests illustrate scheduling decisions for DG, DS, and controllable loads.

B. Distributed Storage Model

The storage and RES-trading model represents battery dynamics and operating costs, then models renewable uncertainty and grid transactions through shortage, surplus, and worst-case costs.

  • Distributed storage: Each DS unit tracks stored energy over time, with charging or discharging power constrained by operating limits and efficiency.The storage state follows a dynamic equation and has bounded final energy for future scheduling horizons.
  • Storage cost: Storage costs can impose soft penalties that discourage large energy variations or encourage stored energy to remain above a specified depth of discharge.Higher weights encourage smaller variation, while very small weights permit greater power exchange.
  • RES uncertainty: Actual harvested renewable energy is collected into w, while RES availability is modeled as unknown within a polyhedral uncertainty set W.The framework supports facility-specific and joint uncertainty models with temporal sub-horizon bounds.
  • Uncertainty bounds: Deterministic lower and upper bounds, including sub-horizon totals, provide the required RES uncertainty information and can be inferred from historical data.The uncertainty models can incorporate geographical and meteorological factors.
  • Grid transactions: In grid-connected operation, shortage energy is purchased and surplus energy is sold, with prices α_t and β_t defining the worst-case net transaction cost.An auxiliary net-power variable supports supply-demand balancing between RES, storage, and the microgrid.
  • Robust versus stochastic modeling: The worst-case model is attractive when RES probability distributions are unavailable, whereas an expectation-based stochastic program can be used when accurate probabilistic models exist.The paper notes that worst-case optimization may be conservative relative to a stochastic formulation.

D. Microgrid Energy Management Problem

The microgrid energy-management problem minimizes social net cost subject to generation, load, storage, renewable, and balance constraints, with convexity enabling decentralized solution methods.

  • Generation model: Conventional DG costs are increasing and convex, typically modeled as piecewise-linear or smooth quadratic functions.These functions represent the operating costs of conventional generators.
  • Objective: The objective combines conventional DG cost, storage cost, worst-case RES transaction cost, and the utility of dispatchable loads.The formulation minimizes the microgrid social net cost.
  • Constraints: Constraints impose DG output, ramping, spinning-reserve, flexible-load, and committed-renewable limits, together with power supply-demand balance.The balance equation ensures total demand is satisfied by generation at every time.
  • Convexity: The constraints are linear, while generator, load, and storage cost or utility functions are convex, so overall convexity depends on the transaction-cost term.The transaction-cost convexity is established separately.
  • Convexity condition: When β_t ≤ α_t for all t, the worst-case transaction cost is convex and the energy-management problem is convex with no duality gap.This condition supports development of an efficient decentralized solver.

III. DISTRIBUTED ALGORITHM

The distributed algorithm reformulates the scheduling problem so strong duality and separability enable decentralized solution under the transaction-price condition.

  • A variable transformation rewrites (P1) into transformed problem (P2) for distributed solution.
  • If (P2) is feasible and βt does not exceed αt for every t, there is no duality gap.
  • Convexity and finite optimal value establish strong duality for the transformed scheduling problem.
  • Because the transformed problem is separable, Lagrangian relaxation and dual decomposition yield a decentralized algorithm coordinated by dual variables.

A. Dual Decomposition

Dual decomposition coordinates local scheduling decisions through multiplier updates, subgradients, averaging, and message exchange among microgrid controllers.

  • A. Dual Decomposition: The partial Lagrangian dualizes constraints coupling generators, loads, and renewable energy sources through multipliers µt, λt, and νt.
  • 1) Subgradient Iterations: A subgradient method iteratively updates the dual multipliers using subgradients of the dual function.
  • 1) Subgradient Iterations: The iterates converge to a neighborhood of the optimal multipliers, whose size is proportional to and controllable by the stepsize.
  • 1) Subgradient Iterations: When the primal objective is not strictly convex, running averages are used to obtain power schedules converging to a neighborhood of the optimal solution.
  • 2) Distributed Implementation: The MGEM broadcasts multiplier iterates to local controllers, which solve their subproblems and return quantities used to form subgradients.

B. Solving the LC Subproblems

Local-controller subproblems are solved using efficient convex programs for conventional components and a bundle method for the renewable-energy subproblem with worst-case transaction costs.

  • B. Solving the LC Subproblems: The first four local-controller subproblems are essentially linear programs or quadratic programs and can be solved efficiently.
  • B. Solving the LC Subproblems: The renewable-energy subproblem is convex and nondifferentiable because of the absolute value operator and maximization defining G.
  • B. Solving the LC Subproblems: The bundle method solves the renewable-energy subproblem and generates schedules converging to the optimal renewable-energy decisions.
  • B. Solving the LC Subproblems: Danskin’s Theorem provides the subgradient of the modified worst-case transaction cost needed by the bundle method.
  • B. Solving the LC Subproblems: The bundle update minimizes a polyhedral approximation with quadratic proximal regularization, while the proximity weight controls iterate stability.
  • B. Solving the LC Subproblems: The resulting quadratic program over a simplex in the dual space can be solved efficiently by practical optimization algorithms.

C. Vertex Enumerating Algorithms

The paper turns the robust RES uncertainty problem into a finite vertex-enumeration procedure, exploiting the structure of its uncertainty polytopes. The resulting worst-case optimization remains computationally practical when sub-horizons are small, with vertices listed once before optimization.

  • Motivation and vertex solution: The worst-case objective is generally an NP-hard convex maximization, but the problem’s special structure enables a computationally efficient solution.The global solution is obtained by evaluating the objective at the finitely many vertices of W.
  • Polytope vertex characterization: Propositions 3 and 4 characterize vertices of the structured uncertainty polytopes used for RES forecasting.Proposition 4 applies to Cartesian products formed from lower-dimensional polytopes, characterizing a global vertex blockwise.
  • Polytope vertex characterization: The geometric structure restricts vertices to surviving hyperrectangle vertices or intersections between the hyperrectangle and its defining hyperplanes.These intersections occur on edges of the hyperrectangle.
  • Vertex-enumeration procedures: Algorithms 2 and 3 generate vertices by enumerating sub-horizon vertices and concatenating them across RES facilities when required.For uncertainty set (6), concatenation across sub-horizons is unnecessary because the optimization decomposes across sub-horizons.
  • Complexity: Vertex enumeration has exponential complexity, but it is affordable when each sub-horizon contains relatively few time slots.For example, the paper cites partitioning 24 hours into four sub-horizons of six time slots and notes that the vertices of W need only be listed once.

IV. NUMERICAL TESTS

The numerical tests evaluate the robust distributed scheduler on a grid-connected microgrid with generators, dispatchable loads, storage, and wind facilities. Results show load shifting, price-responsive energy transactions and storage operation, and lower net cost as selling prices increase relative to purchase prices.

  • Test setup: The test microgrid includes 3 conventional generators, 6 class-1 loads, 4 class-2 loads, 3 storage units, and 2 wind facilities.Wind forecasts and fixed loads are rescaled from MISO data; the joint uncertainty model uses S = 1.
  • Power schedules: The optimal schedule shifts class-1 elastic demand opposite to fixed-load variation, dispatching less demand when fixed load is large from 6PM to 10PM.This reflects the load-shifting ability of the proposed design under limits on conventional generation and grid power.
  • Transaction-price effects: Case A purchases energy throughout the horizon because its purchase price is much lower than conventional-generation marginal cost.The resulting economic decision reduces conventional generation while purchasing more power to maintain supply-demand balance.
  • Transaction-price effects: Case B sells energy from 7PM to 9PM, encouraged by the highest selling prices, and consequently has higher conventional-generation and worst-case transaction costs than Case A.The higher transaction prices also produce larger distributed-generation output in Case B.
  • Storage schedules: All storage units discharge at 7PM, 8PM, and 9PM, then finish at their initial 5kWh stored energy by 12AM.Returning to 5kWh satisfies the minimum stored-energy requirement for the next scheduling horizon.
  • Cost results: The net cost decreases as the selling-to-purchase-price ratio βt/αt increases.A higher ratio increases the transaction revenue margin and reduces the worst-case transaction cost.

V. CONCLUSIONS AND FUTURE WORK

The paper develops a distributed energy-management approach for renewable-rich microgrids by combining robust renewable scheduling with distributed optimization. It concludes that future work should extend the model toward optimal power flow and unit commitment.

  • Conclusions: The approach models committed renewable energy to maintain supply-demand balance under intermittent renewable availability.The objective includes conventional-generation, adjustable-load, storage, and worst-case transaction costs.
  • Conclusions: Dual decomposition splits the optimization into smaller subproblems solved by local controllers of generators, dispatchable loads, storage units, and renewable facilities.This provides the distributed scheduling mechanism described in the paper.
  • Future work: Future work should re-investigate optimal power flow and unit commitment for microgrids with growing renewable-energy use.These are identified as extensions of the proposed model and approach.

APPENDIX I ENHANCING THE BUNDLE METHOD

The appendix derives vertex characterizations for uncertainty polytopes and uses them to formulate the associated optimization over feasible vertices. A block-diagonal structure allows vertices of the combined set to be formed by concatenating subproblem vertices.

  • Bundle formulation: An auxiliary variable r rewrites the robust optimization constraint into a linearized form suitable for dual analysis.The rewritten constraint bounds each affine expression by r.
  • Dual problem: Strong duality and the Lagrangian convert the reformulated problem into a dual quadratic program over the simplex in R^(ℓ+1).The appendix notes that this simplex-constrained QP can be solved efficiently.
  • Vertex characterization: A feasible point of a polytope is a vertex exactly when a full-rank active subsystem has that point as its unique feasible solution.This lemma supports vertex enumeration through linear subsystems.
  • Vertex enumeration: For the uncertainty set A, vertices are enumerated by solving the two possible full-column-rank subsystem forms described in the appendix.The forms consist of signed diagonal constraints or one all-ones row combined with signed coordinate rows.
  • Block structure: Because the combined uncertainty set B is block diagonal, each vertex is obtained by concatenating one vertex from every sub-horizon set.The result follows by selecting linearly independent rows separately from each block.
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