Robust optimization vs. stochastic programming incorporating risk measures for unit commitment with uncertain variable renewable generation

Abstract

Unit commitment seeks the most cost effective generator commitment schedule for an electric power system to meet net load, defined as the difference between the load and the output of renewable generation, while satisfying the operational constraints on transmission system and generation resources. Stochastic programming and robust optimization are the most widely studied approaches for unit commitment under net load uncertainty. We incorporate risk considerations in these approaches and investigate their comparative performance for a multi-bus power system in terms of economic efficiency as well as the risk associated with the commitment decisions. We explicitly account for risk, via Conditional Value at Risk (CVaR) in the stochastic programming objective function, and by employing a CVaR-based uncertainty set in the robust optimization formulation. The numerical results indicate that the stochastic program with CVaR evaluated in a low-probability tail is able to achieve better cost-risk trade-offs than the robust formulation with less conservative preferences. The CVaR-based uncertainty set with the most conservative parameter settings outperforms an uncertainty set based only on ranges.

Appendix A: Solution algorithm

We write $y_1$ for the UC variables that are independent of uncertainty; i.e., the unit commitment, start-up and shut-down cost variables. Also, we write $y_2$ for the dispatch variables (recall that the dispatch variables may depend on the values of uncertain parameters, e.g. power output, phase angle, production cost). The uncertain parameters (in our problem, the hourly net load) can vary on a set denoted as ${\mathcal {U}}$. Suppose that the RUC is summarized as

$$\begin{aligned} \min \limits _{y_1,y_2} \quad&\left( c^Ty_1+ \max \limits _{u \in {\mathcal {U}}} \ b^Ty_2(u) \right)&\end{aligned}$$

(26a)

$$\begin{aligned} s.t. \quad&\mathbf{F}y_1 \le \mathbf{f},&\end{aligned}$$

(26b)

$$\begin{aligned}&\mathbf{H}y_2(u) \le \mathbf{h},&u \in {\mathcal {U}} \end{aligned}$$

(26c)

$$\begin{aligned}&\mathbf{A}y_1 + \mathbf{B}y_2(u) \le \mathbf{g},&u \in {\mathcal {U}} \end{aligned}$$

(26d)

$$\begin{aligned}&\mathbf{E}y_2(u) = u,&u \in {\mathcal {U}} \end{aligned}$$

(26e)

$$\begin{aligned}&y_1 \in \mathbb {R}^{n_1}\times \{0,1\}^{p_1}, \end{aligned}$$

(26f)

$$\begin{aligned}&y_2(u) \in \mathbb {R}^{n_2},&u \in {\mathcal {U}}. \end{aligned}$$

(26g)

In Problem (26), the objective function (26a) minimizes a combination of unit commitment costs, such as start-up and shut-down costs, and the worst case of the dispatch costs, such as production and shortage costs. Constraint (26b) only defines feasibility of the unit commitment variables. Constraint (26c) involves both the unit commitment decisions and dispatch variables, such as ramp-up and ramp-down constraints. In constraint  (26d), we only constraint dispatch variables, such as power balance equations. Note that dispatch variables depend on the uncertain parameter u. Finally, we have restrictions on decision variables: unit commitment variables are mixed-integer where $n_1$ is the number of continuous variables and $p_1$ is the number of binary decisions, and dispatch variables are $n_2$ continuous variables.

We can associate these constraints to those of our formulation in Sect. 2: Constraint (26b) contains constraints (123)–(8), Constraint (26c) contains (10), (11), (19), Constraint (26d) contains (14)–(18), Constraint (26e) contains (12), (13), Constraint (26f) contains (9), and Constraint (26g) contains (20)–(22).

In this formulation, the second term of the objective function represents the worst case of the dispatch cost. By including this second term, we ensure that the unit commitment problem remains feasible, thus robust, under any realization of uncertainty.

Note that the dispatch constraints depend on both the unit commitment variable $y_1$ and the uncertain parameter u. Hence, we write ${\varOmega }(y_1,u)$ as a feasible set defined by the dispatch constraints. We let

$$\begin{aligned} {\varOmega }(y_1,u) = \{y_2: (\hbox {26c}), (\hbox {26d}), (\hbox {26e}) \text { and } (\hbox {26g}) \text { are satisfied for fixed } y_1 \text { and } u \}. \end{aligned}$$

Problem (26) can be equivalently reformulated as

$$\begin{aligned} \begin{array}{rl} \min \limits _{y_1} \quad &{} c^Ty_1+ \max \limits _{u \in {\mathcal {U}}} \min \limits _{y_2 \in {\varOmega }(y_1,u)} b^Ty_2 \\ s.t. &{} \text {Constraints } (\hbox {26b}), (\hbox {26f}). \end{array} \end{aligned}$$

(27)

One may observe that $\min _{y_2 \in {\varOmega }(y_1,u)} b^Ty_2$ is actually the dispatch problem for a fixed unit commitment decision $y_1$ and uncertain parameter u. Now, by maximizing the optimal cost of the dispatch problem over all possible $u \in {\mathcal {U}}$, the worst case dispatch decision is obtained.

To solve Problem (27), we reformulate it as follows:

$$\begin{aligned} \begin{array}{rll} \min \limits _{y_1,\gamma } \quad &{} c^Ty_1+ \gamma &{} \\ s.t.\quad &{} (\hbox {26b}), (\hbox {26f}) &{} \\ &{} \gamma \ge S(y_1,u), &{}\quad \forall u \in {\mathcal {U}}, \end{array} \end{aligned}$$

(28)

where

$$\begin{aligned} S(y_1,u) = \min \limits _{y_2 \in {\varOmega }(y_1,u)} b^Ty_2. \end{aligned}$$

(29)

We write $R(y_1)$ as the worst case of the dispatch problem:

$$\begin{aligned} R(y_1) = \max \limits _{u \in {\mathcal {U}}} S(y_1,u). \end{aligned}$$

(30)

Note that in our problem formulation, since $R(y_1)$ represents the worst case dispatch cost, we write $\gamma \ge 0$ without loss of optimality. This problem can be then reformulated as

$$\begin{aligned} \begin{array}{rll} \min \limits _{y_1,\gamma } \quad &{} c^Ty_1+ \gamma &{} \\ s.t.\quad &{} (\hbox {26b}), (\hbox {26f}) &{}\\ &{} \gamma \ge R(y_1), &{} \\ &{} \gamma \ge 0. &{} \end{array} \end{aligned}$$

(31)

Problem (31) is solved using a Benders decomposition approach applying a branch and cut approach. For more details on how to solve subproblem (30) and master problem (31), we refer to [23].