Sizing Energy Storage Systems to Dispatch Wind Power Plants

Abstract

Integrating wind power plants into the electricity grid poses challenges due to the intermittent nature of wind energy generation. Energy storage systems (ESSs) have shown promise in mitigating the intermittent variability associated with wind power. This paper presents a distributionally robust optimization (DRO) model for sizing energy storage systems to dispatch wind power plants. The variable wind power is formulated as a moment-based ambiguity set. Dispatchability is described by the expected value of the insufficient power of wind power relative to the dispatch command, which is a sum of nonlinear functions and is taken as the optimal index. A deterministic semi-definite positive model is derived to solve the problem effectively. Numerical studies are conducted to demonstrate the effectiveness and advantages of the proposed method.

1. Introduction

Renewable energy stands at the forefront of the global shift toward a sustainable and clean energy future [1]. Wind power, in particular, offers a compelling solution due to its potential for continuous operation, day and night. Despite this, the variable nature of wind presents a reliability challenge, casting wind power as unpredictable [2]. As their grid penetration deepens, ensuring power system stability becomes increasingly complex, necessitating sophisticated dispatching strategies.

Unlike the predictability of traditional coal power plants, wind power generation is inherently intermittent, subject to the vagaries of wind speed and direction. Energy storage systems (ESSs) have emerged as a reliable approach to mitigate this issue, providing services such as wind power smoothing and peak shaving [3]. Research on ESS sizing has explored various objectives, from compensating for wind power forecasting errors [4,5] to enhancing frequency stability [6,7], and optimizing cost–benefit outcomes [8,9]. Others have investigated ESS capacity allocation using spectral analysis [10]. Yet, there is a relative scarcity of research on transforming intermittent renewables into dispatchable resources akin to conventional power plants.

A “dispatchable” wind power plant, with the aid of ESSs, would have the capability to adhere to schedule commands. This concept was explored in [11], which determines ESS capacity to ensure a stable power output, albeit under the assumption of a known wind profile. Ref. [12] proposed a dual-battery ESS scheme for dispatchable wind farms, where two BESSs are utilized to cope with the positive and negative power mismatches, respectively. The model predictive control for managing ESS charge/discharge rates was presented in [13,14], where, however, the sizing question was not addressed. Refs. [15,16] addressed the size and lifetime problems of ESSs by assuming that the dispatched command is the average wind power. However, this way, the intermittent feature of wind power has not been really dealt with.

To cope with the stochastic nature of renewable energy sources, there are three primary optimization methods, i.e., stochastic optimization (SO), robust optimization (RO), and distributionally robust optimization (DRO). SO, which integrates randomness into models and employs Monte Carlo simulations, has been applied to reserve commitment and optimal power flow [17,18,19]. RO seeks solutions that perform well under worst-case scenarios [20,21], while DRO combines elements of SO and RO to provide robustness across various scenarios, though it presents significant analytical and computational challenges [22,23]. DRO has recently been employed in power systems to manage the unpredictability of renewable energy. Examples include a two-stage optimization method for energy and reserve dispatch [24], a double-norm-constraint DRO method to address the uncertain renewable energy in cluster optimization [25], and a distributionally robust chance constraint to optimize ESS investment costs [26]. Despite these advancements, the complex mathematical transformations required in DRO remain a barrier.

This paper addresses a fundamental question for dispatchability: To what extent can a wind power plant, aided by ESSs, comply with a given dispatch command? This issue has not been explored in the existing literature. Most reported works focus on minimizing the wind power curtailment, i.e., utilizing the wind power output as much as possible. In [24], the optimal index combining the generation costs, reserve power cost and wind power curtailment penalty cost has been investigated to obtain an optimal dispatch command. In [26], a linear optimal index for minimizing the renewable power curtailment is presented to size the energy storage. In these works, power flow constraints are necessary to enforce the reduction in renewable power output. On the contrary, the dispatchability issue allows wind power curtailment to meet the output command while penalizing the power shortage that the sum of wind power output and ESS output is less than the command. Since the power curtailment is allowed and its value is not cared for, the resulting index is nonlinear, which is quite different from existing results. Moreover, the sizing problem is over multiple time intervals, which, together with the nonlinear index, leads to a rather challenging problem. In order to focus on the main issue, the paper simplifies the fundamental problem by setting aside internal and external constraints, such as ramping rates and power flow equations, which in some sense are irrelevant to the sizing problem based on hourly scales.

The main contributions of this paper can be summarized as follows:

• The problem of wind power dispatchability with the aid of ESSs is first presented in the framework of a DRO, which is different and novel from the widely studied problem of minimizing the wind power curtailment.
• The DRO formulation based on the moment-based ambiguity set to size ESSs has been converted to several sufficient finite-dimensional convex optimization problems.
• The problem formulation is straightforward. The SOC dynamics and power limits of ESSs have been removed; both can be realized after the solutions of the proposed optimal problems.

The remainder of this paper is organized as follows. Section 2 formulates the problem, modeling wind power uncertainties with a moment-based ambiguity set, and establishes the optimal sizing problem. Section 3 presents our main results, including an equivalent finite-dimensional model and a more practical conservative model supplemented by two corollaries. Section 4 discusses the extension of considering discharging and charging losses. Section 5 contains numerical studies that compare our approach to SO and RO methods. We conclude in Section 6, with detailed proofs provided in the Appendix A.

2. Problem Formulation

Consider the integration of a wind power plant into the power grid, where the wind power plant is equipped with an ESS to effectively manage the fluctuating wind energy, as depicted in Figure 1.

2.1. Power Balancing

Figure 2 illustrates the wind speed data collected over one month, with each data point representing one hour. The data exhibit a stochastic nature. The active power of a wind turbine has the form of

$$P_W=\left\{\begin{array}{cc}0,\hfill & v_w<v_c \mathrm{or} v_w>v_f\hfill \\ p_r,\hfill & v_r\le v_w\le v_f\hfill \\ p_r\times \frac{v_w-v_c}{v_r-v_c},\hfill & v_c\le v_w<v_r\hfill \end{array}\right.,$$

(1)

where $P_W$ is the active power generated by wind turbines, $v_w$ is the wind speed, $v_r$ denotes the rated speed for wind turbines, and $v_c$ and $v_f$ are the cut-in and cut-off wind speeds, respectively. Certainly, $P_W$ has also a probability distribution, which is different from the probability distribution of wind due to the nonlinear map between them.

The output power $P_O$ of the wind power plant is the sum of the power of wind turbines and the power of the ESS:

$$P_O=P_W+P_B,$$

(2)

where $P_B$ is the output power of the ESS. A positive value of $P_B$ indicates that the ESS is in a discharging state. Conversely, a negative value means that the ESS is being charged with excess energy.

2.2. Energy Constraints

Besides the power requirements, the ESS has also faced energy constraints, which are often depicted by the value of state of charge (SOC):

$$\mathrm{SoC}=C_p/C_r$$

(3)

where $C_p$ and $C_r$ denote the present and rated charge capacities of the ESS, respectively. $C_r$ will decrease with time or usage cycles. In the case of a fixed $C_r$, the change in SOC can be depicted by $C_p$.

Neglecting the charging and discharging losses, one has

$$C_p(k+1)=C_p\left(k\right)+P_B·T_s$$

(4)

where k denotes the sample instant and $T_s$ is the time interval. Since SOC cannot be negative, there are minimum and maximum limits for the present capacity.

2.3. Ambiguity Set

The key technique of this paper is based on distributionally robust optimization methods, which aim to identify the optimal solution in various scenarios, considering the uncertainty associated with a probability distribution that can be characterized by ambiguity sets.

Here, the time horizon to be considered is one day, a generic natural and human life cycle. The uncertainty arises from the active power $P_w$. Assume there are n samples of $P_W$ in one day and the resulted stochastic vector $P_W\in \mathcal{X}$ has a probability distribution F belonging to the following ambiguity set:

$$\mathcal{D}(\mathcal{X},{\mu }_0,{\Sigma }_0)=\{F\in \mathcal{F}|\mathrm{E}\left(P_W\right)={\mu }_0,\mathrm{Var}\left(P_W\right)={\Sigma }_0\}$$

(5)

where $\mathcal{F}$ is the set of all probability measures, $\mathcal{X}\subset {\mathcal{R}}^n$ is a closed convex set containing the support of F, ${\mu }_0\in {\mathcal{R}}^n$ is the mean value, and ${\Sigma }_0\in {\mathcal{S}}_{+}$ is the covariance matrix with ${\mathcal{S}}_{+}$ denoting the set of non-negative semi-definite matrices. With n samples taken throughout one day as a daily vector sample, it is relatively straightforward to calculate the mean value ${\mu }_0$ and the covariance matrix ${\Sigma }_0$ from historical data.

Remark 1. 

Due to the correlation between wind speeds at adjacent time intervals, the covariance matrix ${\Sigma }_0$ is not diagonal. This causes the direct application of the distributions for specific time intervals, such as the Weibull distribution, to become infeasible.

2.4. Dispatching Objective

It is well known that the thermal power plant can follow dispatching commands well and is friendly to power balancing. The objective for wind power plants associated with ESSs is to achieve a similar dispatch level as that of the thermal power plant, that is,

$$min\sum _{k=1}^n|P_{Ok}-P_{Lk}|$$

(6)

where $P_O\in {\mathcal{R}}^n$ is the vector representing the actual daily power output of the wind power plant, $P_L\in {\mathcal{R}}^n$ denotes the dispatched command, and subscript k denotes the kth element representing the value at the kth instant. Similarly, let $P_B\in {\mathcal{R}}^n$ be the vector of daily power output of the ESS. Using (2) yields

$$\underset{P_{Bk}}{min}\sum _{k=1}^n|P_{Bk}+P_{Wk}-P_{Lk}|.$$

(7)

Here, the decision variables are the output power of the ESS at every instant, $P_{Bk}$, and cannot be directly obtained because the wind power $P_{Wk}$ is a random variable.

For a sizing problem of ESS, two parameters should be determined: the rated power $P_r$ and electrical capacity $C_r$. In the process of solving the optimization problem (7), both are omitted and consequently, neither upper nor lower limits are set for $B_k$. These limits and parameters will be obtained after the sequence of $P_{Bk}$ is obtained, as shown later. However, the capacity cycle constraint $C_p\left(1\right)=C_p\left(n\right)$ exists, which equals to

$$\sum _{k=1}^n{P}_{Bk}=0.$$

(8)

Since the wind turbine can reduce its output power whenever necessary, the dispatching commands $P_{Lk}$ can always be executed if $P_{Bk}+P_{Wk}>P_{Lk}$. In this consideration, the optimal index (7) becomes

$$\underset{P_{Bk}}{min}\sum _{k=1}^n{(P_{Lk}-P_{Bk}-P_{Wk})}^{+}$$

(9)

where ${\left(x\right)}^{+}=max(x,0)$. This nonlinear index is different from existing results and, moreover, is not easy to directly solve. For notation simplification, define

$$X^{[+]}=\sum _{k=1}^n{X}_k^{+},\mathrm{with}{X}_k=P_{Lk}-P_{Bk}-P_{Wk}.$$

Due to the presence of random variable $P_{Wk}$, the optimal value would be an expected value. Moreover, our goal is to minimize the worst-case performance over the set of plausible distribution $\mathcal{D}$. This is a distributionally robust optimization problem that can be formulated by

$$\begin{array}{c}\mathbf{Pr}\mathbf{0}:\underset{P_{Bk}}{min}\underset{F\in \mathcal{D}}{sup}\mathbb{E}\left(X^{[+]}\right)\mathrm{s}.\mathrm{t}.\left(8\right).\end{array}$$

(10)

Remark 2. 

If the wind power plant is not integrated into the grid, but provides energy to some load, $P_L$ can be regarded as the load demand. In this case, the optimal index (9) is just to minimize the expected total daily power loss.

3. Main Results

The optimal problem Pr0 is hard to solve partly because of the inclusion of the nonlinear function ${\left(x\right)}^{+}$. This section will transform it to a conservative but manageable form that is easy to work with.

The ambiguity set constraints $F\in \mathcal{D}$ equals to

$$\begin{array}{c}f\left(P_W\right)\ge 0\end{array}$$

(11)

$$\begin{array}{c}{\int }_{\mathcal{X}}f\left(P_W\right)d{P}_W=1,\end{array}$$

(12)

$$\begin{array}{c}{\int }_{\mathcal{X}}{P}_{W}f\left(P_W\right)d{P}_W={\mu }_0,\end{array}$$

(13)

$$\begin{array}{c}{\int }_{\mathcal{X}}(P_W-{\mu }_0){(P_W-{\mu }_0)}^{T}f\left(P_W\right)d{P}_W={\Sigma }_0,\end{array}$$

(14)

where $f\left(P_W\right)$ is the probability density function of $P_W$ and $d{P}_W=d{P}_{W1}d{P}_{W2}\cdots d{P}_{Wn}$. Furthermore, (12)∼(14) can be rewritten in a compact form

$${\int }_{\mathcal{X}}\left[\begin{array}{c}{P}_W\\ 1\end{array}\right]{\left[\begin{array}{c}{P}_W\\ 1\end{array}\right]}^{T}f\left(P_W\right)d{P}_W=\left[\begin{array}{cc}{\Sigma }_0+{\mu }_0{\mu }_0^T& {\mu }_0\\ {\mu }_0^T& 1\end{array}\right]\triangleq \Gamma .$$

(15)

First, consider the inner upper bound problem that aims to find the worst scenario. It has the form of

$$\begin{array}{c}\mathbf{Pr}:\underset{f\left(P_W\right)\ge 0}{sup}\mathbb{E}\left(X^{[+]}\right)\mathrm{s}.\mathrm{t}.\left(15\right).\end{array}$$

(16)

Its Lagrangian functional is

$$\mathcal{L}(f,M)=\underset{f\ge 0}{sup}{\int }_{\mathcal{X}}{X}^{[+]}f\left(P_W\right)d{P}_W+〈M,\Gamma -{\int }_{\mathcal{X}}\left[\begin{array}{c}{P}_W\\ 1\end{array}\right]{\left[\begin{array}{c}{P}_W\\ 1\end{array}\right]}^{T}f\left(P_W\right)d{P}_W〉,$$

(17)

where $M\in {\mathcal{S}}^n$ is a Lagrangian multiplier, $\langle \rangle$ denotes the inner product with $\langle A,B\rangle :=\mathrm{tr}\left(A{B}^T\right)$. Problem Pr can be transformed into an equivalent unconstrained problem as follows:

$$\underset{f\ge 0}{sup}\underset{M\in {\mathcal{S}}^n}{inf}\mathcal{L}(f,M).$$

(18)

Due to the strong duality, it equals to

$$\underset{M\in {\mathcal{S}}^n}{inf}\underset{f\ge 0}{sup}\mathcal{L}(f,M).$$

(19)

The Lagrangian functional can be rewritten as

$$\mathcal{L}(f,M)={\int }_{\mathcal{X}}\left(X^{[+]}-q_M\left(P_W\right)\right)f\left(P_W\right)d{P}_W+\mathrm{tr}(M\Gamma ),$$

(20)

where $q_M\left(P_W\right)$ is a quadratic function defined by

$$q_M\left(P_W\right)=〈M,\left[\begin{array}{c}{P}_W\\ 1\end{array}\right]{\left[\begin{array}{c}{P}_W\\ 1\end{array}\right]}^T〉=\left[\begin{array}{cc}{P}_W& 1\end{array}\right]M\left[\begin{array}{c}{P}_W^T\\ 1\end{array}\right].$$

(21)

Noting that all constraints on the probability density function have been casted into (15) besides $f\left(P_W\right)\ge 0$, i.e., no upper limits on $f\left(P_W\right)$, one has

$$\underset{f\ge 0}{sup}\mathcal{L}(f,M)=\left\{\begin{array}{cc}\mathrm{tr}(M\Gamma )\hfill & \mathrm{if}{X}^{[+]}\le q_M\left(P_W\right)\forall P_W\in \mathcal{X}\hfill \\ \infty \hfill & \mathrm{otherwise}\hfill \end{array}\right..$$

(22)

The dual function (19) reads

$$\mathbf{PrA}:\underset{M\in {\mathcal{S}}^n}{inf}\mathrm{tr}(M\Gamma ),\mathrm{s}.\mathrm{t}.\sum _{k=1}^n{X}_k^{+}\le q_M\left(W\right).$$

(23)

It looks like only one inequality constraint exists in the above optimal problem. However, due to the presence of nonlinear function ${\left(\right)}^{+}$, there are actually $2^n$ cases, which poses a big challenge in solving the problem.

Before coping with this challenge, some notations are made. Define index set $\mathcal{N}=\{1,2,\cdots ,n\}$. The set of all non-empty subsets of $\mathcal{N}$ is denoted by ${\mathcal{U}}_{\mathcal{N}}:=\{U:U\subseteq \mathcal{N}\}$. Noting that the empty set is excluded, the number of elements in ${\mathcal{U}}_{\mathcal{N}}$ is $|{\mathcal{U}}_{\mathcal{N}}|=2^n-1$. With them, two induced functions are given:

$$X_{\mathcal{U}}=\sum _{k\in \mathcal{U}}{X}_k,X_{\mathcal{U}}^{[+]}=\sum _{k\in \mathcal{U}}{X}_k^{+}.$$

(24)

Lemma 1. 

The optimal problem PrA is equivalent to

$$\mathbf{PrB}:\underset{M\in {\mathcal{S}}_{+}^n}{inf}\mathrm{tr}(M\Gamma ),\mathrm{s}.\mathrm{t}.{\mathcal{X}}_{\mathcal{U}}\le q_M\left(P_W\right),\forall \mathcal{U}\subseteq {\mathcal{U}}_N.$$

(25)

A simple example with $n=2$ is illustrated to show the difference between these two problems. For PrA, one has

$$\underset{M\in {\mathcal{S}}^n}{inf}\mathrm{tr}(M\Gamma ),\mathrm{s}.\mathrm{t}.\left\{\begin{array}{cc}0\le q_M\hfill & \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{l}\mathrm{l} \mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{s}\hfill \\ X_1\le q_M\hfill & if{X}_1>0,X_2<0\hfill \\ X_2\le q_M\hfill & if{X}_1<0,X_2>0\hfill \\ X_1+X_2\le q_M\hfill & if{X}_1>0,X_2>0\hfill \end{array}\right.;$$

while for PrB, it is

$$\underset{M\in {\mathcal{S}}^n}{inf}\mathrm{tr}(M\Gamma ),\mathrm{s}.\mathrm{t}.\left\{\begin{array}{c}0\le q_M\hfill \\ X_1\le q_M\hfill \\ X_2\le q_M\hfill \\ X_1+X_2\le q_M\hfill \end{array}\right.,\mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{l}\mathrm{l} \mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{s}.$$

Both of them have $2^n$ inequality constraints that hold simultaneously. However, in PrA, the inequalities are conditioned, which is often further converted into unconditioned inequalities by using the Lagrangian multiplier. Different inequalities require distinct Lagrangian multipliers, denoted as ${\tau }_i$, making the standard trick of $M/{\tau }_i\mapsto \overline{M}$ [27] no longer applicable.

Lemma 1 removes the nonlinear factor in PrA, but still suffers from the dimensional disaster. Let $p_A^{*}$ and $p_B^{*}$ denote the optimal value of PrA and PrB, respectively. A relaxed result is presented below to solve this issue.

Lemma 2. 

Given the following optimal problem,

$$\mathbf{PrC}:p_C^{*}\left(\alpha \right)=\underset{M\in {\mathcal{S}}_{+}^n}{inf}\mathrm{tr}(M\Gamma ),\mathrm{s}.\mathrm{t}.\alpha X_k\le q_M,\forall k\in \mathcal{N},$$

there is a constant $1\le {\alpha }_0<n$ such that

$$p_A^{*}=p_C^{*}\left({\alpha }_0\right).$$

(26)

Compared to PrB, the number of inequalities has been reduced from $2^n-1$ to n in PrC. The corresponding optimal solution $X_k^{*}\left({\alpha }_0\right)$ and $M^{*}\left({\alpha }_0\right)$ can be used as the solution of the original problem Pr0. Unfortunately, ${\alpha }_0$ does exist but is unknown. However, for an engineering application, one can choose a middle one located between the conservative solution with $\alpha =n$ and the relaxed solution with $\alpha =1$.

The constraints in PrC can be written as $\alpha (P_{Lk}-P_{Bk}-P_{Wk})\le q_M\left(P_W\right)$, which has a compact form of

$${\left[\begin{array}{c}{P}_W\\ 1\end{array}\right]}^{T}M\left[\begin{array}{c}{P}_W\\ 1\end{array}\right]\ge \alpha {\left[\begin{array}{c}{P}_W\\ 1\end{array}\right]}^T\left[\begin{array}{cc}0& -0.5e_k\\ -0.5e_k^T& P_{Lk}-P_{Bk}\end{array}\right]\left[\begin{array}{c}{P}_W\\ 1\end{array}\right],$$

where $e_k$ denotes the kth row of the unit matrix. Now, we are ready to present our main results below.

Theorem 1. 

The DRO problem Pr0 can be converted into the following finite-dimension problem for some constant scalar ${\alpha }_0$:

$$\begin{array}{c}\mathbf{Pr}\mathbf{1}:\underset{M\in {\mathcal{S}}^n,P_B\in {\mathcal{R}}^n}{inf}\mathrm{tr}(M\Gamma ),\mathrm{s}.\mathrm{t}.,\end{array}$$

$$\begin{array}{c}M\ge 0,\end{array}$$

(27)

$$\begin{array}{c}M-{\alpha }_0\left[\begin{array}{cc}0& -0.5e_k\\ -0.5e_k^T& P_{Lk}-P_{Bk}\end{array}\right]\ge 0,\forall k\in \mathcal{N},\end{array}$$

(28)

$$\begin{array}{c}\sum _{k=1}^n{P}_{Bk}=0.\end{array}$$

(29)

The solution of the optimal problem Pr1 will give the optimal output power sequence of the ESS, $P_B^{*}={[P_{B1}^{*},P_{B2}^{*},\cdots ,P_{Bn}^{*}]}^T$, by which the two targeted parameters can be obtained by

$$P_r=\underset{k}{max}\left|P_{Bk}^{*}\right|,$$

(30)

and

$$C_r=\frac{{max}_i{C}_{pi}^{*}-{min}_i{C}_{pi}^{*}}{{\mathrm{SoC}}^{up}-{\mathrm{SoC}}^{low}},\mathrm{with}{C}_{pi}^{*}=\sum _{k=1}^i{P}_{Bk}^{*}·T_s.$$

(31)

It can be seen that $C_{pi}$, $i\in \mathcal{N}$, is the energy sequence associated with the optimal power sequence $P_{Bk}^{*}$. The difference between the maximum and minimum values represents the minimum amount of energy required. For an energy storage system, the optimal variation region of SOC is often $[0.2,0.8]$, with which ${\mathrm{SoC}}^{up}-{\mathrm{SoC}}^{low}=0.6$.

Remark 3. 

Commonly, SOC limits ${\mathrm{SoC}}^{up},{\mathrm{SoC}}^{low}$ and power limits $P_r$, together with SOC dynamics, will be contained in the optimization models for ESS sizing problems [20,24,28]. All of them are neglected but are indirectly made by the cycle constraint (29) in Theorem 1. The obtained result is simple and straightforward in the sense that the ESS can be directly sized by (30) and (31).

Remark 4. 

The obtained ESS sizing Equations (30) and (31) are based on a fixed power sequence $P_B{k}^{*}$, that is, each time the power of the ESS is fixed. However, in real applications, $P_{Bk}$ can be any permissible value to ensure the dispatched command profile $P_{Lk}$. This implies that the ESS allows greater flexibility than the optimal power sequence. Furthermore, since DRO aims to identify the best worst-case performance, the relaxed solution with ${\alpha }_0=1$ is preferred. It is speculated that this relaxed solution is sufficient to realize the obtained optimal power losses, calculated as $\mathrm{tr}(M^{*}\Gamma )$.

Two extensions can be directly made.

Corollary 1. 

Given a threshold γ for expected power losses, there is a constant scalar ${\alpha }_0$ such that the feasible solution of the following problem over $(M,P_B)$

$$\begin{array}{c}\mathbf{Pr}\mathbf{2}:\mathrm{tr}(M\Gamma )\le \gamma ,M\ge 0,\sum _{k=1}^n{P}_{Bk}=0,\\ M-{\alpha }_0\left[\begin{array}{cc}0& -0.5e_k\\ -0.5e_k^T& P_{Lk}-P_{Bk}\end{array}\right]\ge 0,\forall k\in \mathcal{N},\end{array}$$

guarantees${sup}_{F\in \mathcal{D}}\mathbb{E}\left(X^{[+]}\right)\le \gamma$.

Corollary 2. 

If the ESS is sized with $(p_r,C_r)$ and the expected power loss γ is given, then the best load profile $P_L$ can be obtained by the following problem for a constant scalar ${\alpha }_0$ over variables $(M,P_B,P_L,\underline{C},\overline{C})$:

$$\begin{array}{c}\mathbf{Pr}\mathbf{3}:\underset{P_{Lk}\ge 0}{max}\sum _{k=1}^n{L}_k,\mathrm{s}.\mathrm{t}.,\\ \mathrm{tr}(M\Gamma )\le \gamma ,M\ge 0,\\ M-{\alpha }_0\left[\begin{array}{cc}0& -0.5e_k\\ -0.5e_k^T& P_{Lk}-P_{Bk}\end{array}\right]\ge 0,\forall k\in \mathcal{N},\\ -p_r\le P_{Bk}\le p_r,\forall k\in \mathcal{N},\sum _{k=1}^n{P}_{Bk}=0,\\ \underline{C}\le C_{pi}=\sum _{k=1}^n{P}_{Bk}{T}_s\le \overline{C},\forall i\in \mathcal{N},\overline{C}-\underline{C}\le C_r.\end{array}$$

4. Disscussion

The above results are based on an ideal case that both the charging and discharging losses of the ESS are not considered. Since charging and discharging have different power directions, their losses are formulated separately in the SOC dynamics, which is generally given by

$$SoC(k+1)=SoC\left(k\right)+T_s\left(C_k{\eta }_c-\frac{D_k}{{\eta }_d}\right),$$

(32)

where $D_k\ge 0$ and $C_k\ge 0$ are the charging and discharging powers at time k, ${\eta }_c$ and ${\eta }_d$ are their efficiencies, respectively. In this case, the capacity cycle constraint (8) turns to

$$\sum _{k=1}^n\left(C_k{\eta }_c-\frac{D_k}{{\eta }_d}\right)=0,$$

(33)

and the net output power of battery storage systems $P_{Bk}$ satisfies

$$P_{Bk}=D_k-C_k.$$

(34)

Considering the charging and discharging losses, the results developed in Section 3 still hold with the capacity cycle constraint replaced by (33) and some extra constraints like (34). For example, problem Pr1 in Theorem 1 becomes

$$\begin{array}{c}\mathbf{Pr}\mathbf{1}\mathbf{L}:\underset{M\in {\mathcal{S}}^n,D,C\in {\mathcal{R}}^n}{inf}\mathrm{tr}(M\Gamma ),\mathrm{s}.\mathrm{t}.,\\ M\ge 0,\\ M-{\alpha }_0\left[\begin{array}{cc}0& -0.5e_k\\ -0.5e_k^T& P_{Lk}-D_k+C_k\end{array}\right]\ge 0,\forall k\in \mathcal{N},\\ \sum _{k=1}^n\left(C_k{\eta }_c-\frac{D_k}{{\eta }_d}\right)=0,\\ D_k\ge 0,C_k\ge 0.\end{array}$$

Two corollaries can be handled with in a similar way. Noting that $0<{\eta }_c,{\eta }_d<1$ and using $C_k=D_k+P_{Bk}$, one can obtain ${\sum }_k{P}_{Bk}>0$. This further implies that the second matrix inequalities above require M to have a larger value of $\mathrm{tr}\left(M\right)$, so that the optimal value of Pr1L is larger than that of Pr1.

However, in practice, the discharging and charging states cannot be simultaneously activated for energy storage systems, that is, for any time k, either $D_k=0$ or $C_k=0$. Generally, binary variables depicting either-or situations will be used to yield a mixed integer programming problem [28,29]. This unfortunately is not compatible with the DRO framework developed above and will be left for future research.

5. Numerical Study

Simulation tests on the proposed distributionally robust optimization model for sizing energy storage systems will be conducted in this section. For an explicit expression, we refer to Pr1 as the relaxed distributionally robust optimization model (R-DROM) and PrC as the exact distributionally robust optimization model (E-DROM). In addition, for comparison, the sample-average approximation model (SAAM) to size ESS is defined as

$$\begin{array}{c}\mathbf{Pr}\mathbf{4}:\underset{P_{Bk}}{min}\sum _{i=1}^m\sum _{k=1}^n{(P_{Lk}-P_{Bk}-P_{Wi,k})}^{+},\mathrm{s}.\mathrm{t}.,\\ \sum _{k=1}^n{P}_{Bk}=0,\end{array}$$

(35)

where $P_{Wi,k}$ represents the actual power output of the wind power plant at the kth instant of the ith day, and m denotes the number of samples that are selected at random. The dependence of the SAAM solution on the samples is evident. Different samples yield different solutions. The simulations are based on Julia programming language and the JuMP modeling tool kit, performed on a computer with AMD R3550H CPU and 16 GB memory. Mosek 10.0 solver is used to solve optimization problems. In the simulations, all calculations are performed on per unit values.

5.1. The Case of $n=4$

In the case of $n=4$, four points each day, we schedule ESS at 0:00, 6:00, 12:00, and 18:00 over a month, i.e., 31 days in the month of December. Note that the wind power and dispatched power at each instant are generated randomly, but the latter is the same every day. Figure 3a shows the wind power and dispatched power at each instant from 1 December to 4 December. In the verifying step, the ESS will be run in two modes, fixed $P_{Bk}$ and variable $P_{Bk}$. In the fixed $P_{Bk}$ mode, the output of the ESS at each instant is equal to the $P_{Bk}$ obtained by solving the optimization problem. In the variable $P_{Bk}$ mode, the ESS charges when there is an excess of wind energy and discharges when there is a deficit of wind energy to execute the dispatched task, while complying by the maximum power $P_r$ and capacity $C_r$, which are calculated as (30) and (31), respectively.

The comparison of the results obtained by the R-DROM and the E-DROM is shown in

Table 1. Comparison of results between relaxed distributionally robust optimization model (R-DROM) and exact distributionally robust optimization model (E-DROM).

Model	R-DROM			E-DROM	No ESS
${\alpha }_0$	1	2.5	4	–	–
Optimal value	0.086	0.215	0.343	0.105	–
ESS power $P_{B1}$	−0.154	−0.154	−0.154	−0.153	0
ESS power $P_{B2}$	0.19	0.19	0.19	0.2	0
ESS power $P_{B3}$	0.24	0.24	0.24	0.246	0
ESS power $P_{B4}$	−0.277	−0.277	−0.277	−0.294	0
ESS rated power $P_r$	0.277	0.277	0.277	0.294	0
ESS capacity $C_r$	0.718	0.718	0.718	0.744	0
Total dispatch shortage with fixed $P_{Bk}$	2.872	2.872	2.872	2.811	13.558
Total dispatch shortage with variable $P_{Bk}$	0.474	0.474	0.474	0.186	–

, where the total dispatch shortage is the sum of the dispatch shortage of each day in the month. It can be seen that the optimal value of the E-DROM multiplied by 31 is 3.255, which is greater than the total dispatch shortage with fixed $P_{Bk}$, reflecting the robustness of the proposed ESS sizing approach. The optimal value of the E-DROM is less than the optimal value of R-DROM when ${\alpha }_0$ is set to 1 and greater than it when ${\alpha }_0$ is 4, confirming the theoretical analysis of Lemma 2. The solution obtained by R-DROM with different ${\alpha }_0$ is the same, except for the optimal value which in fact shows a linear relation to the set ${\alpha }_0$. The dispatch shortages with ESS equipped that are sized by E-DROM or R-DROM are relatively close and are greatly reduced in comparison to the situation without ESS equipped. Compared to the fixed $P_{Bk}$ mode, the total dispatch shortage with variable $P_{Bk}$ is greatly improved.

Figure 3a–d show the system operation under variable $P_{Bk}$ scheduling mode using the E-DROM to size ESS when $n=4$ from 1 December to 4 December and throughout December, respectively. It can be seen that at the 8th moment, which is 18:00 on 2 December, the wind power was higher than the dispatched command; therefore, the dispatch shortage was 0. Since the ESS was not in a fully charged state at that time, the wind power also charged the ESS in addition to supplying the load. Charging power was limited to the maximum power $P_r$, and the ESS’s SOC increased. On 3 December at 18:00, which is the 12th moment, the wind power was less than the dispatched value, and the ESS will work to compensate for the deficiency. As a result, dispatch shortages do not happen and the SOC of the ESS decreases.

5.2. The Case of $n=24$

For $n=24$, each point is an hour, and simulation tests are performed using actual wind power data from a wind power plant located in Hinggan League, Inner Mongolia, China, for the month of December 2021, as shown in Figure 4c. There are 31 data in total, each of which is a 24-dimensional vector, corresponding to the wind power values from 0 o’clock to 23 o’clock of the entire day. Note that in this scenario, the E-DROM contains $2^{24}$ semi-definite constraints, which is too large to make a E-DROM computation. On the other hand, R-DROM containing 24 semi-definite constraints is still easily solvable. Figure 4a–d show the operation of the system under the variable $P_{Bk}$ scheduling method using the R-DROM to size ESS when $n=24$ on 1 December and throughout December, respectively. Likewise, the ESS charges when sufficient wind power is available and discharges when wind power is insufficient, achieving both temporal and spatial transfers of wind energy, and thus effectively reducing total dispatch shortages.

The solutions of the sample-average approximation model depend on the chosen historical samples. In the simulation for SAAM, we increase the number of samples sequentially and run 50 independent experiments for each sample size. Each experiment randomly selects samples from historical data, i.e., the actual wind power data. Figure 5a shows the total dispatch shortages with different sample sizes.

The solid blue line in the figure represents the average dispatch shortage of 50 experimental results, while the shaded blue area corresponds to the 20% and 80% quantiles of the dispatch shortage. The dashed yellow line represents the dispatch shortage using an ESS sized by R-DROM. It can be observed that the SAAM results are significantly impacted by the selection of samples, especially when the sample size is small. As the number of samples increases, the fluctuation range of dispatch shortage decreases. The optimal SAAM, with a sample size equal to the total number of samples (31 in this case), finds the optimal $P_{Bk}^{*}$.

Table 2. Comparison of results between relaxed distributionally robust optimization model (R-DROM) and optimal sample-average approximation (SAAM).

Model	R-DROM	Optimal SAAM	Without ESS
(a) Over the span of December 2021
ESS rated power $P_r$ (p.u.)	0.656	0.687	0
ESS capacity $C_r$ (p.u.)	5.761	5.328	0
Total dispatch shortage with variable $B_k$ (p.u.)	22.272	22.61	90.823
(b) Over the span of 2021
ESS rated power $P_r$ (p.u.)	0.558	0.553	0
ESS capacity $C_r$ (p.u.)	4.91	4.987	0
Total dispatch shortage with variable $B_k$ (p.u.)	912.896	911.018	1649.123

a shows ESS rated power, ESS capacity, and total dispatch shortage with variable $P_{Bk}$ using R-DROM and optimal SAAM to size ESS. To ensure a consistent and reliable decision, i.e., $P_{Bk}$, it is necessary to increase the sample size or solve the SAAM many times and take the average value as the solution, both of which lead to an increase in computational complexity and time. However, R-DROM obtains decisions that are very close to the results of the SAAM and requires only one solution process based on the mean vector and covariance matrix of historical data rather than specific sample data. We also conduct SAAM and R-DROM simulations for the whole year; the results are shown in Figure 5b and Table 2b, and the same conclusions can be drawn as for the single-month data.

5.3. Optimal Dispatch Power

Corollary 2 shows that the optimal dispatch power sequence $P_{Lk}$ can be computed according to Pr3 given the ESS parameters ($p_r$, $C_r$) and expected dispatch shortage $\gamma$. To this end, the simulation is configured by setting the values of $p_r$ and $C_r$ to be 0.65 and 5.7, respectively, and taking different ${\alpha }_0$ and $\gamma$ values. The results are shown in

Table 3. The results for optimal load profile in December.

${\mathit{\alpha }}_0$	Expected Dispatch Shortage $\mathit{\gamma }$	The Optimal Value ${\sum }_{\mathit{k}}{\mathit{P}}_{\mathbf{Lk}}$ (p.u.)	Average Daily Dispatch Shortage (p.u.)
12	7	15.152	1.348
	10	22.141	4.711
	15	32.64	13.761
	20	42.822	23.943
24	7	2.079	0
	10	9.153	0.263
	15	16.41	1.771
	20	22.141	4.713

. The average daily dispatch shortage is obtained by dividing the number of days (31 in this case) by the total dispatch shortage, solved using the ESS parameters and the optimal dispatch power sequence in the variable $P_{Bk}$ mode. As can be seen from the table, when ${\alpha }_0=24$, the average daily dispatch shortage is less than $\gamma$. This is because the optimization model is most conservative when ${\alpha }_0=n$. In addition, when ${\alpha }_0$ is set to 12, the average daily dispatch shortage is less than $\gamma$ when $\gamma$ is lower than 15; but it is greater than $\gamma$ when $\gamma$ is 20. The value of ${\alpha }_0$ may be determined based on the conservatism requirements of the results in engineering applications.

6. Conclusions

The ESS sizing problem to make wind plants dispatchable has been studied by optimizing the expected value of the deficient power of the wind plant relative to the dispatched command. A moment-based DRO model is presented to deal with the uncertainty of wind power. An equivalent deterministic finite-dimension model is then derived to solve the DRO problem. The finite-dimension model is difficult to solve but reveals the principles involved. A conservative but easy-to-solve model is further presented, by which two corollaries are then developed for the upper bound of the expected value and the optimal dispatching command profile, respectively.

Future investigation includes considering the charging and discharging losses and extending to data-based ambiguity sets, such as the Wasserstein metric, to obtain a less conservative and more applicable result.