Multistage robust optimization for the day-ahead scheduling of hybrid thermal-hydro-wind-solar systems

Abstract

The integration of large-scale uncertain and uncontrollable wind and solar power generation has brought new challenges to the operations of modern power systems. In a power system with abundant water resources, hydroelectric generation with high operational flexibility is a powerful tool to promote a higher penetration of wind and solar power generation. In this paper, we study the day-ahead scheduling of a thermal-hydro-wind-solar power system. The uncertainties of renewable energy generation, including uncertain natural water inflow and wind/solar power output, are taken into consideration. We explore how the operational flexibility of hydroelectric generation and the coordination of thermal-hydro power can be utilized to hedge against uncertain wind/solar power under a multistage robust optimization (MRO) framework. To address the computational issue, mixed decision rules are employed to reformulate the original MRO model with a multi-level structure into a bi-level one. Column-and-constraint generation (C &CG) algorithm is extended into the MRO case to solve the bi-level model. The proposed optimization approach is tested in three real-world cases. The computational results demonstrate the capability of hydroelectric generation to promote the accommodation of uncertain wind and solar power.

Appendix

Proof of Theorem 1

We first focus on the first claim. The inequality $Z^{\left| \mathcal {U}\right| } \ge \tilde{Z}^{\left| \mathcal {U}\right| }$ holds due to the fact that a worse optimal objective value can never be obtained after removing some constraints to obtain a large feasible region. Additionally, since function $f_{j,t}$ is a single-valued mapping, implying that once all the non-intermediate variables in $\varvec{Y}_{\left[ T\right] }$ are fixed, the value of intermediate variable $y_{j,t}$ can be uniquely determined. Therefore, the following relationship is obtained:

$$\begin{aligned} y_{j,t}=f_{j,t}\left( \varvec{Y}_{\left[ t\right] }\right) =f_{j,t} \left( \varvec{w}_t+\varvec{W}_{j^{\prime },t}\varvec{u}_{\left[ t\right] }\right) , \end{aligned}$$

where $\varvec{w}_{j^{\prime },t}=\left( \left\{ w_{j,t}\right\} _{y_{j \in \mathcal {J}^{NI}, \tau \le t}} \right)$ and $\varvec{W}_{j^{\prime },t}=\left( \left\{ \varvec{W}_{j,t}\right\} _{y_{j \in \mathcal {J}^{NI}, \tau \le t}} \right)$ denote the vectors of the coefficients associated with the decision rules of the variables in $\varvec{Y}_{\left[ t\right] }$. Since $\varvec{w}_t+\varvec{W}_{j^{\prime },t}\varvec{u}_{\left[ t\right] }$ represents unique mappings from historical parameters $\varvec{u}_{\left[ t\right] }$ to non-intermediate variables, the value of $y_{j,t}$ can also be uniquely determined by $\varvec{u}_{\left[ t\right] }$, thus the decision rules for non-intermediate variables can simultaneously ensure the non-anticipativity constraints for both intermediate and non-intermediate variables.

Then we discuss the second claim. Suppose that function $f_{j,t}$ is nonlinear. If the decision rule for $y_{j,t}$ is imposed as a constraint, we will have

$$\begin{aligned} f_{j,t}\left( \varvec{Y}_{\left[ t\right] }\right) =w_{j,t}+ \varvec{W}_{j,t}^{\top }\varvec{u}_{\left[ t\right] }, \varvec{u} \in \mathcal {U}. \end{aligned}$$

(28)

It will be shown that constraint (28) is not always lazy and indeed has restrictions to the feasible region of (19) in some cases. Mathematically speaking, under any $\left\{ w_{j^{\prime },t}, \varvec{W}_{j^{\prime },t}\right\} _{y_{j^{\prime },\tau } \in \mathcal {J}^{NI}, \tau \le t }$, there does not always exist some $w_{j,t}\in \mathbb {R}$ and $\varvec{W}_{j,t} \in \mathbb {R}^ {\left( |H_p|+|\varOmega _R|\right) \times t}$ such that Eq. (28) holds. Consider the following two cases:

• (1) The gradient of function $f_{j,t}$ exists for any $\varvec{Y}_{\left[ t\right] } \in \text {int}\,\mathcal {Y}_t$, where $\text {int}$ denotes the interior of some set. Computing the gradient with respect to $\varvec{u}_{\left[ t\right] }$ on the both sides of (28) yields the following equation:

  $$\begin{aligned} \varvec{W}_{j,t}&=\sum _{j^{\prime } \in \mathcal {J}^{NI}_c} \sum _{\tau =1}^{t} \frac{\partial f_{j^{\prime },t}}{\partial y_{j^{\prime },\tau }} \nabla y_{j^{\prime },\tau }\left( \varvec{u}_{\left[ t\right] }\right) \nonumber \\&=\sum _{j^{\prime } \in \mathcal {J}^{NI}_c} \sum _{\tau =1}^{t} \frac{\partial f_{j^{\prime },t}}{\partial y_{j^{\prime },\tau }} \varvec{W}_{j^{\prime },\tau },\quad \varvec{u} \in \mathcal {U}. \end{aligned}$$

  (29)

  Since function $f_{j,t}$ is nonlinear, there exists at least one $\varvec{Y}_{\left[ t\right] } \in \mathcal {Y}_{j,t}$ such that the gradient of function $f_{j,t}$ at $\varvec{Y}_{\left[ t\right] }$ is not a vector whose components are all constants. That is to say, there exists some $\frac{\partial f_{j^{\prime },t}}{\partial y_{j^{\prime },\tau }}$ whose value is dependent of $\varvec{u}$, where $j^{\prime } \in \mathcal {J}^{NI}_c$ and $\tau \le t$. Therefore, there does not exist $\varvec{W}_{j,t}$ and $\left\{ \varvec{W}_{{j^{\prime },\tau }}\right\} _{y_{j^{\prime },\tau } \in \mathcal {J}^{NI}, \tau \le t }$ such that (29) holds for any $\varvec{u} \in \mathcal {U}$, which completes the proof in this case.

• (2) The gradient of function $f_{j,t}$ does not always exist for any $\varvec{Y}_{\left[ t\right] } \in \text {int}\,\mathcal {Y}_t$. Note that (29) still holds under this assumption, which implies that $\left\{ w_{j^{\prime },t}, \varvec{W}_{j^{\prime },t}\right\} _{y_{j^{\prime },\tau } \in \mathcal {J}^{NI}, \tau \le t }$ is restricted over the following set:

  $$\begin{aligned} \left\{ \left( \left\{ w_{j^{\prime },t}, \varvec{W}_{j^{\prime },t}\right\} _{y_{j^{\prime },\tau } \in \mathcal {J}^{NI}, \tau \le t }\right) : {\left\{ \begin{array}{ll} \varvec{Y}_{\left[ t\right] }=\varvec{w}_{j^{\prime },t}+\varvec{W}_{j^{\prime },t}\varvec{u}_{\left[ t\right] }\\ \nabla f_{j,t}\left( \varvec{Y}_{\left[ t\right] }\right) \;\text {exists} \end{array}\right. } \right\} , \end{aligned}$$

  (30)

That is to say, the linear decision rule for each non-intermediate variable in $\varvec{Y}_{\left[ t\right] }$ will never map $\varvec{u}_{\left[ t\right] }$ to a point where the gradient of $f_{j,t}$ with respect to $\varvec{Y}_{\left[ t\right] }$ does not exist. This implies that the feasible region shrinks due to the aforementioned restriction resulting from imposing decision rule (28), and completes the proof in this case. $\square$

Proof of Corollary 1

Suppose that function $f_{j,t}$ is linear with the following formulation:

$$\begin{aligned} y_{j,t}=\varvec{\pi }^{\top }\varvec{Y}_{\left[ t\right] }=\varvec{\pi }^{\top } \left( \varvec{w}_{j^{\prime },t}+\varvec{W}_{j^{\prime },t}\varvec{u}_{\left[ t\right] }\right) . \end{aligned}$$

(31)

Let $\varvec{w}_{j^{\prime },t}^*$ and $\varvec{W}_{j^{\prime },t}^*$ denote the optimal values of $\varvec{w}_{j^{\prime },t}$ and $\varvec{W}_{j^{\prime },t}$ in the modified model, and $w_{j,t}^*:=\varvec{\pi }^{\top }\varvec{w}_{j^{\prime },t}^*$ and $\varvec{W}_{j^{\prime },t}^*:=\varvec{W}_{j^{\prime },t}^*$. It can be verified that $w_{j,t}^*$ and $\varvec{W}_{j^{\prime },t}^*$ satisfy constraint (31), thus they are feasible with respect to model (20). Since this feasibility holds for any $\varvec{w}_{j^{\prime },t}^*$ and $\varvec{W}_{j^{\prime },t}^*$, it has been shown that the decision rule obtained from the modified model is also feasible for the original model (20). In Theorem 1, it has been shown that $Z^{\left| \mathcal {U}\right| } \ge \tilde{Z}^{\left| \mathcal {U}\right| }$. We can thus conclude that $Z^{\left| \mathcal {U}\right| } = \tilde{Z}^{\left| \mathcal {U}\right| }$ when function $f_{j,t}$ is linear, and completes the proof.$\square$

Proof of Lemma 1

Since $\mathcal {Y}_{j,t}^0$ is nonconvex, there exist some $\varvec{Y}_{\left[ t\right] }^1$ and $\varvec{Y}_{\left[ t\right] }^2$ belonging to $\mathcal {Y}_{j,t}^0$ such that $\lambda \varvec{Y}_{\left[ t\right] }^1+\left( 1-\lambda \right) \varvec{Y}_{\left[ t\right] }^2 \notin \mathcal {Y}_{j,t}^0$, where $0<\lambda <1$. In other words, the following equality holds:

$$\begin{aligned} \lambda \varvec{Y}_{\left[ t\right] }^1+\left( 1-\lambda \right) \varvec{Y}_{\left[ t\right] }^2&=\lambda \left( \varvec{w}_t+\varvec{W}_{t}^{\top }\varvec{u}_{\left[ t\right] }^1\right) +\left( 1-\lambda \right) \left( \varvec{w}_t+\varvec{W}_{t}^{\top }\varvec{u}_{\left[ t\right] }^2\right) \nonumber \\&=\varvec{w}_t+\varvec{W}_{t}^{\top }\left( \lambda \varvec{u}_{\left[ t\right] }^1+ \left( 1-\lambda \right) \varvec{u}_{\left[ t\right] }^2\right) \notin \mathcal {Y}_{j,t}^0. \end{aligned}$$

(32)

Equation (32) implies that $\lambda \varvec{u}_{\left[ t\right] }^1+\left( 1-\lambda \right) \varvec{u}_{\left[ t\right] }^2 \notin \mathcal {U}^{0}_{j,t}$. Since $\varvec{u}_{\left[ t\right] }^1$ and $\varvec{u}_{\left[ t\right] }^2$ belong to $\mathcal {U}^{0}_{j,t}$ and $\mathcal {U}^{0}_{j,t}$ is convex, the convex combination of any two points belonging to $\mathcal {U}^{0}_{j,t}$ must also belong to this set, which produces a contradiction and completes the proof.$\square$

Proof of Theorem 2

The proof of the first claim is similar to the one of Theorem 1, which follows that function $y_{j,t}=f_{j,t}\left( \varvec{Y}_{\left[ t\right] }\right)$ is enough to ensure that decision associated with variable $y_{j,t}$ is made without using any information after time period t. In the following, it will be shown that there indeed exists cases where the optimal objective value can be improved ($Z^{\left| \mathcal {U}\right| } > \tilde{Z}^{\left| \mathcal {U}\right| }$) if at least one of the sets $\mathcal {Y}_{j,t}^0$ and $\mathcal {Y}_{j,t}^1$ is nonconvex

Note that equation $w_{j,t}+\varvec{W}_{j,t}^{\top }\varvec{u}_{\left[ t\right] } = 0$ can be regarded as a hyperplane in $\mathbb {R}^ {\left( |H_p|+|\varOmega _R|\right) \times t}$. Therefore, the decision rule (17) for a binary variable can be interpreted as a hyperplane separating sets $\mathcal {U}^{0}_{j,t}$ and $\mathcal {U}^{1}_{j,t}$. It will be shown that such hyperplane does not always exist for any $\left\{ w_{j^{\prime },t}, \varvec{W}_{j^{\prime },t}\right\} _{y_{j^{\prime },\tau } \in \mathcal {J}^{NI}, \tau \le t }$ via contradiction method.

Suppose that there exists a hyperplane $w_{j,t}+\varvec{W}_{j,t}^{\top }\varvec{u}_{t} = 0$ separating sets $\mathcal {U}^{0}_{j,t}$ and $\mathcal {U}^{1}_{j,t}$, and let set $\mathcal {U}^{0}_{j,t}$ be nonconvex. Thus, there must exist $\varvec{u}^1_{\left[ t\right] }$ and $\varvec{u}^2_{\left[ t\right] } \in \mathcal {U}^{0}_{j,t}$ satisfying $\lambda \varvec{u}^1_{\left[ t\right] }+\left( 1-\lambda \right) \varvec{u}^2_{\left[ t\right] } \notin \mathcal {U}^{0}_{j,t}$ for some $\lambda \in \left( 0,1\right)$. In addition, $\mathcal {U}^{0}_{j,t} \subset \mathcal {U}$ and set $\mathcal {U}$ is convex, thus $\lambda \varvec{u}^1_{\left[ t\right] }+\left( 1-\lambda \right) \varvec{u}^2_{\left[ t\right] } \in \mathcal {U}$. Since $y_{j,t}$ is a binary variable that either equals to 0 or 1, we have $\mathcal {U}^{0}_{j,t} \cup \mathcal {U}^{1}_{j,t} = \mathcal {U}$. Therefore, $\lambda \varvec{u}^1_{\left[ t\right] }+\left( 1-\lambda \right) \varvec{u}^2_{\left[ t\right] } \in \mathcal {U}^{1}_{j,t}$. Suppose that $w_{j,t}+\varvec{W}_{j,t}^{\top }\varvec{u}_{t} < 0$ for $\forall \varvec{u}_t \in \mathcal {U}^{1}_{j,t}$ and $w_{j,t}+\varvec{W}_{j,t}^{\top }\varvec{u}_{t} \ge 0$ for $\forall \varvec{u}_t \in \mathcal {U}^{0}_{j,t}$. Then we have

$$\begin{aligned}&w_{j,t}+\varvec{W}_{j,t}^{\top }\left( \lambda \varvec{u}^1_{\left[ t\right] }+\left( 1-\lambda \right) \varvec{u}^2_{\left[ t\right] }\right) < 0, \end{aligned}$$

(33a)

$$\begin{aligned}&w_{j,t}+\varvec{W}_{j,t}^{\top }\varvec{u}^1_{\left[ t\right] } \ge 0, \end{aligned}$$

(33b)

$$\begin{aligned}&w_{j,t}+\varvec{W}_{j,t}^{\top }\varvec{u}^2_{\left[ t\right] } \ge 0. \end{aligned}$$

(33c)

Multiplying both sides of (33b) and (33c) with $\lambda$ and $1-\lambda$ respectively and summing this two equalities yields:

$$\begin{aligned} w_{j,t}+\varvec{W}_{j,t}^{\top }\left( \lambda \varvec{u}^1_{\left[ t\right] }+ \left( 1-\lambda \right) \varvec{u}^2_{\left[ t\right] }\right) \ge 0, \end{aligned}$$

(34)

which contradicts with (33a). This completes the proof that such a hyperplane does not always exist. Note that we can also assume that set $\mathcal {U}^{0}_{j,t}$ is nonconvex and complete the proof following a similar manner.

As for the case where $\mathcal {U}_{j,t}^0$ and $\mathcal {U}_{j,t}^1$ are both convex, it is clear that the hyperplane separating sets $\mathcal {U}^0_{j,t}$ and $\mathcal {U}^1_{j,t}$ always exists due to the separation theorem of two convex sets. Therefore, constraint $\epsilon - My_{j,t} \le w_{j,t}+\varvec{W}_{j,t}^{\top }\varvec{u}_{\left[ t\right] } \le M\left( 1-y_{j,t}\right)$ has no effects on the feasible region, which completes the proof under this case. $\square$

Proof of Corollary 2

By adopting Lemma 1, it can be concluded that $\mathcal {U}_{j,t}^0$ and $\mathcal {U}_{j,t}^1$ are also convex when both sets $\mathcal {Y}_{j,t}^0$ and $\mathcal {Y}_{j,t}^1$ are convex. Note that the convexity of $\mathcal {U}_{j,t}^0$ and $\mathcal {U}_{j,t}^1$ do not depend on the values of $\varvec{w}_{j^{\prime },t}$ and $\varvec{W}_{j^{\prime },t}\varvec{u}_{\left[ t\right] }$. Therefore, there always exists a hyperplane separating $\mathcal {U}_{j,t}^0$ and $\mathcal {U}_{j,t}^1$. It can thus be concluded that the optimal decision rules for the modified model are also feasible for the original model, which completes the proof.$\square$