Chance-constrained optimal capacity design for a renewable-only islanded microgrid

https://doi.org/10.1016/j.epsr.2020.106564Get rights and content

Highlights

  • We consider the capacity design of renewable-only islanded microgrid and explicitly pose the problem as a chance-constrained optimization problem, which is solved using a probabilistically robust method.

  • An affine control policy is designed to dispatch battery power under uncertain renewable generation and load. The policy is integrated into the chance-constrained optimization problem.

  • In order to address conservativeness inherent in the robust method, we develop two approaches to set reshaping that reduce the volume of the robust set, thereby enabling less conservative designs.

Abstract

Microgrids offer a promising opportunity for achieving greater use of renewable generation. In this paper, we consider optimal capacity design for an islanded microgrid supplied by a wind turbine, solar panel and battery system. The objective is to reduce plant cost while ensuring energy sufficiency, taking into account stochasticity of renewable generation and load. An affine control policy is designed to dispatch battery power under uncertain renewable in-feed and load. The policy is integrated into a stochastic chance-constrained optimization problem, which is solved using a probabilistically robust method. In order to address conservativeness inherent in the robust method, we develop two approaches to set reshaping that reduce the volume of the robust set, thereby enabling less conservative designs.

Introduction

Microgrids are defined as small-scale power systems that group a variety of distributed energy resources (DERs), such as renewable resources, storage systems and loads, to provide high reliability. A microgrid can operate in either grid-connected mode or islanded mode. This paper considers islanded microgrids supplied solely by renewable resources, where backup power supply from the main power grid is not available. Sufficient capacity must be available within the microgrid to safely supply loads, while excessive capacity should be avoided to minimize the overall cost. This implies that in rare cases where renewable generation is extremely limited, for example cloudy breezeless days, loads with low priority are subject to load shedding, as a last resort. Optimal design of the capacity of DERs within islanded microgrids must therefore consider the trade-off between energy sufficiency and economics.

Numerous aspects of optimal microgrid capacity design have been considered previously. Optimal sizing of a microgrid with a wind turbine, solar photovoltaic (PV) and fuel cell is studied in [1] using an evolutionary computation technique. Sizing a variety of DERs in a microgrid is considered in [2] where the focus is on satisfying regulatory constraints imposed by government. Much work has been done on incorporating multiple objectives into the design of microgrids. A multi-objective optimization problem is formulated in [3] to consider both cost and power availability. A genetic algorithm is applied to solve the optimization problem, however a trade-off is required between the multiple feasible solutions. Simultaneous minimization of levelized cost of energy and CO2 emissions is considered in [4] through application of a strength pareto evolutionary algorithm. Design criteria considered in [5] include minimum cost, CO2 emissions and maximum reliability. Markov models for wind generation, solar PV and loads are generated in [6] and [7].

Most literature handles stochastic variables using Monte Carlo simulation and heuristic algorithms, which cannot provide probability guarantees in a tractable way. Recently there has been substantial interest though in chance-constrained (CC) formulations. CC problems are generally difficult to solve analytically, except for rare cases where the uncertain variables are uniformly or normally distributed [8]. However, under certain light assumptions, randomized optimization methods [9] can solve CC problems and provide solutions with a priori probability guarantees and appealing tractability properties. A CC formulation for the optimal power flow (OPF) problem was first proposed in [10] using a solution methodology that does not scale well, while the scenario approach [9] was applied in [11] to solve the CC OPF problem.

In this paper, we explicitly pose microgrid capacity design as a stochastic optimization problem with chance constraints, and solve it using a randomized optimization method. The scenario approach [9] may require a large number of scenarios, depending on the number of decision variables. Considering the high dimension of our problem with respect to the decision variables, we have instead adopted a related randomized optimization technique, the probabilistically robust method [12], in which the required number of scenarios depends on the number of uncertain variables. This approach [12] initially constructs a CC problem to establish a robust set for the uncertain variables, then a robust counterpart of the original problem is solved with uncertain variables confined to the computed robust set.

To apply [12], we need to choose the shape of the robust set that encloses the randomly selected scenarios. We have found that a hyper-rectangular robust set, as proposed in [12], can be overly conservative. This paper therefore introduces two methods for reshaping the robust set to reduce conservativeness. Firstly, a cutting-based reshaping method is proposed which takes into account the physical characteristics of wind and solar generation and load. This method is efficient and does not incur many extra scenarios for maintaining the desired probability guarantee. Secondly, a reshaping method that exploits principal component analysis (PCA) [13] is proposed. PCA identifies the principal directions that capture the most variation in data and thereby provides a way to reduce data dimensionality. Its applications in power systems are largely related to wind speed forecasting [14] and data compression [15]. In this paper, we use PCA to remove unnecessary parts in the robust set and hence improve the solution for the original CC problem.

To facilitate the use of randomized methods we need a formulation where certain decision variables are defined as a function, i.e., a “control policy”, of the uncertain variables. In our microgrid design problem, an affine control policy [16], [17] is proposed to dispatch battery power under uncertain renewable generation and load. In contrast, a nonlinear control policy is designed for load shedding control. These control policies are integrated into the stochastic CC problem, the solution to which provides the optimal policy parameters, DER capacities, the upper bound on the load shedding ratio, and a forecast dispatch for the battery power. Note that both the affine policy for battery dispatch and the nonlinear policy for load shedding are purely to aid the design process and should not be interpreted as determining the actual operating strategy.1 In an operational setting, the battery in a renewable-only islanded microgrid has to continually compensate for power shortages due to insufficient renewable generation, while load shedding is rather an emergency action scheme.

The paper is organized as follow: Section 2 describes the microgrid structure and presents the problem description. Control policies for battery power dispatch and load shedding are proposed in Section 3, and the stochastic CC optimization problem is formulated in Section 4. Section 5 introduces the robust reformulation as well as the two methods for refining the robust set. Robust reformulation of the original CC problem is provided in Section 6. Shrinking horizon implementation of the control policy is proposed in Section 7. Numerical results and validation tests are provided in Section 8 and conclusions are given in Section 9.

Section snippets

Microgrid structure

The paper considers an islanded microgrid system that is built around a central electrical bus. Various DERs, including a wind turbine, solar PV and energy storage, together with load are connected to the central bus. Fuel-based power plants are excluded on the basis of their environmental impact, and grid connection is not considered due to the high cost of rural area electrification. We assume a load shedding scheme is available to cope with rare weather conditions when renewable generation

Control policies

To handle the stochasticity introduced by renewable resources and load, control policies are designed to generate trajectories for battery charging/discharging and load shedding under arbitrary generation and load conditions.

Chance-constrained problem formulation

The overall objective is to design the capacities for DERs and load shedding in a microgrid to guarantee energy sufficiency with a priori probability guarantee while minimizing the net present cost (NPC) of the microgrid system.

Robust set formulation

A standard scenario-based approach [9] can be used to solve the CC problem (P1) for a given probability guarantee. However, a large number of scenarios are required to achieve a sufficiently high confidence level. This results in a heavy computational burden. Furthermore, it may be challenging to obtain a sufficiently large data set. Instead, we resort to a robust reformulation approach proposed in [12], which firstly constructs a CC problem (P2) to search for a hyper-rectangular robust set for

Robust reformulation

With the refined robust set computed from Section 5, we now reformulate the original CC problem (P1) into its robust counterpart. The stochastic constraints in (11) can be reformulated using affine policy (7), (8). The expression for p˜b over the entire time horizon in vector form is given by,p˜b=pbf+AwΔp˜w0+ApvΔp˜pv0AdΔp˜d,where Awdiag(dw)Dw, Apvdiag(dpv)Dpv, and Addiag(d)D. The notation diag(·) expands a vector into a matrix with the vector lying on its diagonal.

Based on battery energy

Shrinking horizon implementation

At the off-line microgrid design phase, we obtain optimal capacity design for DERs and load shedding, a forecast battery trajectory pbf, and control policy parameters for dispatching battery power pb, over the horizon t=0,,T1. In this section we discuss on-line implementation of the designed battery control policy, where the microgrid has been constructed according to the optimal capacity design.

Direct on-line implementation of the proposed policy uses day-ahead weather forecasts and

Renewable generation and load data

Wind and solar are the two renewable resources under consideration. Aggregated wind generation data with five-minute resolution are drawn from Bonneville Power Administration (BPA)’s website [20], and averaged within each hour to give hourly resolution data. Due to the pure renewable microgrid setting under consideration, we target regions with adequate wind resource. Thus, we make the assumption that the minimum hourly wind power is no less than 1% of its rating, and that wind goes above 10%

Conclusion

This paper considers the optimal capacity design for DERs in a renewable-only islanded microgrid. This is a challenging problem due to the stochasticity of renewable resources and load. To address the stochasticity explicitly, we formulate the problem as a stochastic chance-constrained (CC) optimization problem. An affine policy for dispatching battery power is proposed and integrated into the optimization formulation. A probabilistically robust method is adopted to solve this CC problem by

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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