Mathematical Programming Methods for Microgrid

manuscript No. (will be inserted by the editor)

Mathematical Programming Methods for Microgrid Design and Operations: A Survey on Deterministic and Stochastic Approaches Guanglei Wang ¨ Hassan Hijazi

Received: date / Accepted: date

Abstract Planning and operating a power grid is a nontrivial exercise due to conflicting objectives, nonlinear constraints and uncertainties at multiple decision levels. Considerable research work has been dedicated to independently solve different aspects of the overall problem. This survey provides a detailed review of state-ofthe-art techniques in mathematical optimization trying to address challenges in this area. We also provide a set of open problems and research perspectives. Keywords Global Optimization ¨ Convex Relaxation ¨ Uncertainty ¨ Power Systems

1 Introduction Widespread adoption of renewable and distributed generation is disrupting the top-down structure of power systems. This is especially true for remote areas (such as archipelagos), rural zones, and communities willing to go off-grid. This shift is calling for the development of reliable decision support tools for the design and operations of such power grids. The key challenges to be addressed include defining the optimal energy technology mix, the optimal generation sizing, the optimal scheduling of energy storage solutions and the optimal layout of assets. Given the intermittent nature of renewable generation, handling uncertainty is critical for providing confidence in computed solutions, which can help increase electrification and renewable penetration levels around the world. With conflicting objectives, nonlinear constraints and uncertainty in the input data, building a power gird form scratch entails considerable challenges related to physical, spatial and temporal constraints. These include the physics of power flows, the spatial layout of assets and the scheduling of generation and storage G. Wang ¨ H. Hijazi The Australian National University, ACTON 2601, Canberra, Australia. E-mail: [email protected] H. Hijazi Los Alamos National Laboratory, New Mexico, USA. E-mail: [email protected]

2

Guanglei Wang, Hassan Hijazi

solutions. One also needs to deal with uncertainty due to external factors such as weather and demand variability. The overall problem can be regarded as a constrained, stochastic, and multiobjective optimization problem [2]. Given the complexity at hand, a compromise between optimality and computational performance has to be reached.

1.1 Current Focus While a considerable number of existing surveys [2, 13, 246, 72, 89, 130] have contributed to the analysis of computational tools for power grid design and operations, two aspects seem to be lacking attention. First, few focus on mathematical programming techniques in the presence of nonlinear physical constraints (e.g., AC power flows), which raises concerns both in terms of solution feasibility and optimality evaluation. In contrast to heuristic approaches (e.g., Genetic Programming, Tabu Search) whose performance is usually evaluated by simulations, a mathematical programming approach offers performance guarantees with proved lower and upper bounds. Furthermore, mathematical programming based methods benefit from off-the-shelf solvers that are being continually improved. Second, with increased penetration of renewables, there is a need to develop stochastic and robust optimization methods to enable reliable and efficient operations given the uncertainty due to external factors such as weather and demand variability. To fill these gaps, this paper aims at reviewing power grid design and operations models and algorithms based on mathematical programming, considering both deterministic and stochastic approaches. The rest of this paper is organized as follows. In Section 3, we review approaches tailored for optimal power flow, optimal transmission switching, power supply restoration, unit commitment, energy storage sizing and scheduling, and facility location problems respectively. Section 4 considers the stochastic variants of these problems. Section 5 provides a quick overview on the state-of-the-art computational performance for each reviewed problem. Perspectives and concluding remarks follow in Section 6.

1.2 An overview As will be seen later, the reviewed problems are not independent but closely related. In fact, they are linked in a hierarchical way as illustrated in Figure 1. The corresponding references are summarized in Table 3 in the Appendix. The length of the planning horizon varies among the surveyed problems and is illustrate in Figure 2.

Title Suppressed Due to Excessive Length

Fig. 1 Hierarchical Structure and Corresponding References

3

4


Fig. 2 Planning horizon of various problems

2 Preliminaries In this section, we briefly introduce some basic concepts and terminology related to mathematical optimization and algorithmic graph theory that will be used throughout the paper. Some associated notations will be introduced in this section. Other notations and concepts from power system analysis will be introduced as needed in the subsequent sections. A full list of notations are given in the Appendix.

2.1 General optimization frameworks We shall assume familiarity with basic knowledge in Mixed-Integer Nonlinear Programming (MINLP). We refer to [38, 18] for excellent reviews. Here we introduce some concepts and mathematical notations that will be used throughout the paper. A MINLP is an optimization of the following form min s.t.

f pxq gi pxq ď 0, i “ 1, . . . , m,

(MINLP)

n

xPX ĂR , where gi : x P Rn ÞÑ R, pi “ 1, . . . , mq, X “ tx P Rn : Ax ď b, xj P Z, j P J u. We do not assume any convexity property of functions gi but explicitly assume that X is a compact set in an n-dimensional real space. We further assume that the constraint set denoted by K :“ X X tx : gi pxq ď 0 pi “ 1, . . . , mqu is nonempty and compact, so that f pxq attains its minimum at some x P K. All optimization models investigated in this article can be expressed in the form of (MINLP). MINLP constitutes a very general family of problems and there are several ways to categorize them. Given a problem instance encoded by pK, f q we can categorize it by the convexity, linearity, and integrality of the problem. – By convexity: if f and all gi pi “ 1, . . . , mq are convex, we say that (MINLP) is a convex MINLP; otherwise, we say that (MINLP) is a nonconvex MINLP. – By linearity: if m “ 0, (MINLP) becomes a Mixed-Integer Linear Program (MILP). – By integrality: if X does not contain integrality constraints, (MINLP) becomes a continuous Nonlinear Program (NLP).


5

Methods that address convex MINLP include generalized Benders’ decomposition [91], Outer Approximation [68], Non-Linear Programming (NLP) based Branch-and-Bound (BB) algorithms [212]. We refer readers to excellent surveys [33, 100, 18] for more details. In contrast, for a nonconvex MINLP, dropping the integrality constraints leads to a nonconvex program, which remains a hard problem. Most optimization problems in this survey are nonconvex MINLPs. Convex relaxations are an efficient tool for handling such problems. One can replace nonconvex functions with their convex under-estimators (or over-estimators) to obtain a convex MINLP, which is more computationally tractable in practice. Convex relaxations provide optimality guarantees in the form of lower bounds on the objective of the original MINLP (for a minimization problem). To solve a nonconvex MINLP to global optimality, one can embed convex relaxations as a building block in global optimization algorithms, e.g., spatial branch and bound (sBB) frameworks (first proposed by McCormick [175]). SBB methods recursively partition the search space to reduce the gap between upper and lower bounds until convergence is reached.

2.2 Network representation We denote by G “ pN, E q a power system, where N represents the set of all buses and E Ă N ˆ N represent the set of all lines. We shall let pi, j q P E represent the directed line from bus i to bus j. We say that i and j are adjacent if pi, j q P E or pj, iq P E. In addition to the topological properties, each bus has terminals to connect different power units, e.g., generators, storage equipments and transformers. We also assume that the network is in single phase and thus one conductor is attached to each bus. A bus i has an associated voltage denoted by Vi P C and a power injection Si P C, where C represents the complex field. We use superscript p¨qg and p¨qd to differentiate generation and load quantities. For instance Sig represents the AC power generation at bus i and Sid is the power demand parameter. To distinguish variables from parameters, bold font is used for parameters. A line pi, j q has an admittance matrix Yij “ Gij ` jBij that is composed by the real part Gij and imaginary part Bij , where j represents the imaginary unit. In addition, it has power flow Sij “ pij ` jqij , where pij denotes the apparent power flow and qij is the reactive power flow. We will also review some techniques exploiting the structure of networks which builds upon graph theory. Thus “network” and “graph”, “bus” and “node”, “line” and “edge” are interchangeable. A graph is called complete if every pair of vertices are adjacent. A clique of a graph is an induced subgraph which is complete, and a clique is maximal if its vertices do not constitute a proper subset of another clique. A sequence of nodes tv0 , v1 , v2 , . . . , vk , v0 u Ď V is a cycle of length k ` 1 if vi´1 and vi are adjacent for all i P t1, . . . , ku and vk , v0 are adjacent. A graph is said to be chordal if every cycle of length greater than or equal to 4 has a chord (an edge joining two nonconsecutive vertices of the cycle). Given a graph G “ pV, E q, we say that a graph GF “ pV, F q is a chordal extension of G if GF is chordal and E Ď F .

6


3 Deterministic Models In this section, we review and discuss six types of problems associated with power grid design and planning. These problems are: optimal power flow, optimal transmission switching, power supply restoration, unit commitment, energy storage sizing and scheduling, and capacitated facility location. For each problem, we discuss state-of-the-art modeling and solution methods in depth. Some qualitative comparisons between techniques are also provided.

3.1 Optimal Power Flow The Optimal Power Flow (OPF) problem is a fundamental optimization problem in power systems operations that was first introduced in the 1960s [44]. OPF is used to match the supply and demand in real-time. There are two challenges in the solution of OPF. First, it is an operational level problem that has to be solved every few minutes. Second, it is a non-convex optimization problem which is computationally challenging in terms of providing optimality guarantees. ACOPF AC network flow constraints are established based on complex quantities for current I, voltage V , admittance Y , and power S, which are linked by the physical properties of Kirchoff’s Current Law ÿ

Iig ´ Iid “

Iij

pi,j qPE YE R

Ohm’s Law, i.e., Iij “ Yij pVi ´ Vj q and the definition of AC power: ˚ , Sij “ Vi Iij

where p¨q˚ denotes the conjugate transpose. Combining these three properties yields the AC Power Flow equations, i.e., Sig ´ Sid “

ÿ

Sij i P N

(1a)

pi,j qPE YE R

Sij “ Yij˚ |Vi |2 ´ Yij˚ Vi Vj˚ pi, j q P E Y E R .

(1b)

where |¨| represents the magnitude; Sig represents the total power generator at bus i; Sid is the power demand. In addition to physical power flow constraints, power systems optimization problems share a set of common operational constraints. As mentioned in [107, 60], these include bus voltage constraints (2a), generator’s production constraints (2b),


7

thermal limit constraints (2c) for each line and Phase Angle Difference (PAD) constraint (2d): vil ď |Vi | ď viu , i P N, Sigl

ď

Sig

|Sij | ď ´

M θij

ď

Sigu ,

u Sij ,

(2a)

i P NG ,

(2b) R

(2c)

pi, j q P E ,

(2d)

pi, j q P E Y E , ˚

ď =pVi Vj q ď

M , θij

where p¨ql and p¨qu denote the lower and upper bounds; and =p¨q represents the angle that a complex number makes in the complex plane. The objective is to minimise the cost of generation and can be expressed as ÿ

min

c2i ppgi q2 ` c1i pgi ` c0i ,

(3)

iPNG

where c2i , c1i , c0i are the respective quadratic, linear, and constant cost coefficient associated with generator i P NG and NG is a subset of buses N that are attached with generators. The full AC Optimal Power Flow (ACOPF) formulation is presented in Model 1.

Model 1 The AC Optimal Power Flow Problem (ACOPF) variables: Sig pi P N q, Vi pi P N q, Sij pi, jq P E Y E R ÿ c2i ppgi q2 ` c1i pgi ` c0i minimize: iPN

ÿ

s.t.: Sig ´ Sid “

Sij i P N,

pi,jqPEYE R

Sij “ Yij˚ |Vi |2 ´ Yij˚ Vi Vj˚ pi, jq P E Y E R vil ď |Vi | ď viu , i P N, Sigl ď Sig ď Sigu , i P N, u |Sij | ď Sij , pi, jq P E Y E R , M M ´ θij ď =pVi Vj˚ q ď θij , pi, jq P E.

One can represent the complex constraints (2b) in real variables as g gu pgl i ď pi ď pi , i P N

(5a)

qigl

(5b)

ď

qig

ď

qigu ,

iPN

and (2c) as 2 u 2 p2ij ` qij ď pSij q

(6)

8


Observe also that the PAD constraints (2d) can be implemented as a linear relation of the real and imaginary components of Vi Vj˚ M ˚ ˚ M ˚ tanp´θij q< Vi Vj ď = Vi Vj ď tanpθij q< Vi Vj

˘

`

˘

`

`

˘

pi, j q P E,

(7)

where =p¨q and