Clearing the Day-Ahead Market with a High Penetration of Stochastic Production

Abstract

This chapter motivates, develops, and explains a market-clearing algorithm for the day-ahead market, intended for electric energy markets with a significant number of stochastic producers. To adequately mimic the real-world decision-making process, a two-stage stochastic programming model with recourse is presented. Market outcomes include both production and consumption quantities, and energy-only clearing prices. These prices embody desirable properties such as revenue adequacy in expectation and cost recovery, also in expectation. Complementarily, and as alternative to the two-stage stochastic programming approach, this chapter also introduces a dispatch method for energy and reserve that copes with uncertain generation using adaptive robust optimization. A number of clarifying examples illustrate the theoretical interest and practical relevance of the proposed market-clearing algorithms.

Appendices

Appendix 1: Settlement Scheme Properties

The properties of the settlement scheme in Sect. 3.3.1 are formally stated below in the form of theorems. For this purpose, we define first the following indices:

s(i) :

Index of the bus where conventional unit i is located.

s(j) :

Index of the bus where load j is located.

s(q) :

Index of the bus where stochastic production unit q is located.

Theorem 3.1 (Revenue adequacy in expectation).

Consider the market-clearing procedure (3.12), built on a stochastic programming framework, and the resulting sets of dual variables $\left\{{\lambda^{\rm D}}_{\!\!\!n}, \forall n \in N \right\}$ and $\left\{\left({\lambda^{\rm B}}_{\!\!\!\!n \omega}, {\pi}_{\omega}\right)\right.$, $\left.\forall n \in N, \forall \omega \in \Omega\right\}$. The settlement scheme (3.13) is revenue adequate in expectation.

Proof

Mathematically, the settlement scheme (3.13) is revenue adequate in expectation if, at the optimum, it holds

$$\begin{aligned} & \sum_{n \in N}{{\lambda_{n}^{\rm D*}} \left(\!\sum_{i \in \Phi_{n}^{I}}{{P}_{i}^{*}} + \sum_{q \in \Phi_{n}^{Q}}{{W}_{q}^{\rm S*}} - \sum_{j \in \Phi_{n}^{J}}{{L}_j}\!\right)} + \sum_{\omega \in \Omega}\sum_{n \in N}{\pi}_{\omega}{\lambda_{n\omega} ^{\rm B*}}\bigg[\!\sum_{i \in \Phi_{n}^{I}}\!\left({r^{\rm U*}_{i\omega}}-{r^{\rm D*}_{i\omega}}\right)&& \nonumber\\ &\qquad -\sum_{q \in \Phi_{n}^{Q}}{\left({W}_{q}^{\rm S*}+{W}_{q}^{\rm spill*}-{W}_{q\omega}\right)+\sum_{j \in \Phi_{n}^{J}}{L_{j\omega}^{\rm shed*}}}\bigg] \leq 0,&&\end{aligned}$$

(3.22)

where $\left\{{\lambda^{\rm B*}_{n\omega}} = \gamma_{n\omega}^{*}\big/{\pi}_{\omega}, \forall n \in N, \forall \omega \in \Omega\right\}$ are the probability-removed balancing prices and superscript “ $*$ ” denotes optimal values.

Using the power balance equations (3.12b) and (3.12c), expression (3.22) can be equivalently rewritten as follows:

$$\begin{aligned} & \sum_{n \in N}{{\lambda^{\rm D*}_{n}} \left[\sum_{\ell \in \lambda|o(\ell) = n}{b_{\ell}\left({\delta}^{0*}_{o(\ell)}-{\delta}^{0*}_{e(\ell)}\right)}-\sum_{\ell \in \lambda|e(\ell) = n}{b_{\ell}\left({\delta}^{0*}_{o(\ell)}-{\delta}^{0*}_{e(\ell)}\right)}\right]}&&\nonumber\\ &\quad-\sum_{\omega \in \Omega}\sum_{n \in N}{\pi}_{\omega}{\lambda^{\rm B*}_{n\omega}} \bigg[\sum_{\ell \in \lambda|o(\ell) = n}{b_{\ell}\left({\delta}^{0*}_{o(\ell)}-{\delta}^{*}_{o(\ell)\omega}-{\delta}^{0*}_{e(\ell)}+{\delta}^{*}_{e(\ell)\omega}\right)}&&\nonumber\\ &\quad-\sum_{\ell \in \Lambda|e(\ell) = n}{b_{\ell}\left({\delta}^{0*}_{o(\ell)}-{\delta}^{*}_{o(\ell)\omega}-{\delta}^{0*}_{e(\ell)}+{\delta}^{*}_{e(\ell)\omega}\right)}\Bigg] \leq 0.&&\end{aligned}$$

(3.23)

Let us consider the following partial Lagrangian function of problem (3.12):

$$\begin{aligned} {\mathcal{L}} &= \sum_{i \in I}{\left({C}_i P_i {+}{C_i^{\rm RU}} {R_i^{\rm U}}+{C_i^{\rm RD}} {R_i^{\rm D}}\right)}+ \sum_{q \in Q}{{C}_{q} {W_{q}^{\rm S}}}{+} \sum_{\omega \in \Omega} {\pi}_{\omega}\Bigg[\sum_{i \in I}{\left({C_{i}^{\rm U}} {r_{i\omega}^{\rm U}} - {C_{i}^{\rm D}}{r_{i\omega}^{\rm D}}\right)} &&\nonumber\\ & + \sum_{q \in Q}{{C}_{q} \left({W}_{q\omega} - {W_{q}^{\rm S}}{-} {W_{q\omega}^{\rm spill}}\right)} + \sum_{j \in J}{{V_{j}^{\rm LOL}}{L_{j\omega}^{\rm shed}}}\Bigg] - \sum_{n \in N}{\lambda_{n}^{\rm D}}\Bigg[\!\sum_{i \in \Phi_{n}^{I}}{{P}_i}+\! \sum_{\!q \in \Phi_{n}^{Q}}\!{W_{q}^{\rm S}}&&\nonumber\\ &\quad-\sum_{j \in \Phi_{n}^{J}}{{L}_{j}}- \sum_{\ell \in \Lambda|o(\ell) = n}{b_{\ell}\left({\delta_{o(\ell)}^{0}}-{\delta_{e(\ell)}^{0}}\right)} + \sum_{\ell \in \Lambda|e(\ell) = n}{b_{\ell}\left({\delta_{o(\ell)}^{0}}-{\delta_{e(\ell)}^{0}}\right)}\Bigg]&&\nonumber\\ &\quad-\sum_{\omega \in \Omega}\sum_{n \in N}\gamma_{n\omega} \Bigg[\sum_{i \in \Phi_{n}^{I}}{\left({r_{i\omega}^{\rm U}}-{r_{i\omega}^{\rm D}}\right)} + \sum_{j \in \Phi_{n}^{J}}{L_{j\omega}^{\rm shed}}+\sum_{q \in \Phi_{n}^{Q}}{\left({W}_{q\omega}-{W_{q}^{\rm S}}-{W_{q\omega}^{\rm spill}}\right)}&&\nonumber\\ &\quad+ \sum_{\ell \in \Lambda|o(\ell) = n}{b_{\ell}\left({\delta_{o(\ell)}^{0}}-{\delta}_{o(\ell)\omega}-{\delta_{e(\ell)}^{0}}+{\delta}_{e(\ell)\omega}\right)}&&\nonumber\\ &\quad- \sum_{\ell \in \Lambda|e(\ell) = n}{b_{\ell}\left({\delta_{o(\ell)}^{0}}-{\delta}_{o(\ell)\omega}-{\delta_{e(\ell)}^{0}}+{\delta}_{e(\ell)\omega}\right)}\Bigg].&&\end{aligned}$$

(3.24)

Since problem (3.12) is linear and thus convex, ${\mathcal{L}}$ is minimized subject to the rest of constraints, i.e., constraints (3.12d) –(3.12s), at the optimum. Note that by moving the power balance equations (3.12b) and (3.12c) to the objective function (3.12a) to form the partial Lagrangian function ${\mathcal{L}}$, the resulting optimization problem {minimize (3.24), subject to (3.12d) –(3.12s) } can be decomposed into appropriate minimization subproblems for any given set of Lagrange multipliers $\{{\lambda^{\rm D}_{n}}, \forall n \in N; \gamma_{n\omega}, \forall n \in N, \forall \omega \in \Omega\}$. In particular, the summation of the following terms, extracted from (3.24),

$$\begin{aligned} &\sum_{n \in N}{\lambda^{\rm D}_{n}} \left[\sum_{\ell \in \Lambda|o(\ell) = n}{b_{\ell}\left({\delta_{o(\ell)}^{0}}-{\delta_{e(\ell)}^{0}}\right)} - \sum_{\ell \in \Lambda|e(\ell) = n}{b_{\ell}\left({\delta_{o(\ell)}^{0}}-{\delta_{e(\ell)}^{0}}\right)}\right]&&\nonumber\\ &\quad-\sum_{\omega \in \Omega}\sum_{n \in N}\gamma_{n\omega} \Bigg[\sum_{\ell \in \Lambda|o(\ell) = n}{b_{\ell}\left({\delta_{o(\ell)}^{0}}-{\delta}_{o(\ell)\omega}-{\delta_{e(\ell)}^{0}}+{\delta}_{e(\ell)\omega}\right)}&&\nonumber\\ &\quad- \sum_{\ell \in \Lambda|e(\ell) = n}{b_{\ell}\left({\delta_{o(\ell)}^{0}}-{\delta}_{o(\ell)\omega}-{\delta_{e(\ell)}^{0}}+{\delta}_{e(\ell)\omega}\right)}\Bigg]&&\end{aligned}$$

(3.25)

is minimized subject to constraints (3.12d) –(3.12i) at the optimum.

A solution such that ${\delta}_{n}^{0} = 0, \forall n \in \Omega$ ; ${\delta}_{n\omega} = 0, \forall n \in N, \forall \omega \in \Omega$, is feasible for the minimization subproblem {minimize (3.25), subject to (3.12d) –(3.12i) }, as long as the capacity of transmission lines is non-negative, i.e., $C_{\ell}^{\rm max} \geq 0$, $\forall \ell \in \Lambda$. This solution allows us to set the upper bound of expression (3.25) to zero. Therefore, inequality (3.23) holds, and this concludes the proof.

Theorem 3.2 (Cost recovery in expectation).

Consider the market-clearing procedure (3.12), built on a stochastic programming framework, and the resulting sets of dual variables $\left\{{\lambda^{\rm D}_{n}}, \forall n \in N \right\}$ and $\left\{\left({\lambda^{\rm B}_{n\omega}}, {\pi}_{\omega}\right)\right.$, $\left.\forall n \in N\right.$, $\left.\forall \omega \in \Omega\right\}$. The settlement scheme (3.13) guarantees cost recovery for all market participants in expectation.

Proof

The settlement scheme (3.13) ensures that both conventional and stochastic producers recover their energy production costs in expectation. Mathematically, this is expressed as follows:

$$\begin{aligned} C_i P_i^{*} +{C_i^{\rm RU}} {R_i^{\rm U*}} +{C_i^{\rm RD}} {R_i^{\rm D*}} &+ \sum_{\omega \in \Omega}{\pi}_{\omega}\left({C_i^{\rm U}} {r_{i\omega}^{\rm U*}}-{C_i^{\rm D}}{r_{i\omega}^{\rm D*}}\right)-{\lambda_{s(i)}^{\rm D*}} P_{i}^{*}\nonumber\\ &- \sum_{\omega \in \Omega} {\pi}_{\omega}{\lambda_{s(i)\omega}^{\rm B*}}\left({r_{i\omega}^{\rm U*}}- {r_{i\omega}^{\rm D*}}\right) \leq 0, \forall i \in I;\end{aligned}$$

(3.26)

and

$$\begin{aligned} \sum_{\omega \in \Omega} \!\!{\pi}_{\omega}{C}_q \left({W}_{q\omega}\!-{W}_{q\omega}^{\rm spill*}\right)- {\lambda^{\rm D*}_{s(q)}}{W}_{q}^{\rm S*} \notag \\ &\quad- \sum_{\omega \in \Omega}\!\! {\pi}_{\omega}{\lambda^{\rm B*}_{s(q)\omega}}\left({W}_{q\omega}-{W}_{q}^{\rm S*}-{W}_{q\omega}^{\rm spill*}\right)\leq 0, \forall q \in Q,\end{aligned}$$

(3.27)

where superscript “ $*$ ” denotes optimal values and ${\lambda^{\rm B*}_{n\omega}} = \frac{\gamma^{*}_{n\omega}}{{\pi}_{\omega}}, \forall n \in N, \forall \omega \in \Omega$.

Let us consider again the partial Lagrangian function (3.24). At the optimum, this function is minimized subject to constraints (3.12d) –(3.12s). As stated in the proof for revenue adequacy in expectation, the optimization problem {minimize (3.24), subject to (3.12d) –(3.12s) can be decomposed into appropriate minimization subproblems for any given set of shadow prices $\{{\lambda^{\rm D}_{n}}, \forall n \in N; \gamma_{n\omega}, \forall n \in N, \forall \omega \in \Omega\}$. Specifically, the series of terms extracted from (3.24)

$$\begin{aligned} C_i P_i &+{C_i^{\rm RU}} {R_i^{\rm U}} +{C_i^{\rm RD}} {R_i^{\rm D}} + \sum_{\omega \in \Omega} {\pi}_{\omega} \left({C_{i}^{\rm U}} {r_{i\omega}^{\rm U}} - {C_{i}^{\rm D}}{r_{i\omega}^{\rm D}}\right)- {\lambda_{s(i)}^{\rm D}}{P}_i \notag \\& -\sum_{\omega \in \Omega} \gamma_{s(i)\omega}\left({r_{i\omega}^{\rm U}}- {r_{i\omega}^{\rm D}}\right),\end{aligned}$$

(3.28)

and

$$\sum_{\omega \in \Omega} {\pi}_{\omega} C_q \left({W}_{q\omega}-{W_{q\omega}^{\rm spill}}\right)- {\lambda_{s(q)}^{\rm D}}{W_{q}^{\rm S}} - \sum_{\omega \in \Omega} \gamma_{s(q)\omega}\left({W}_{q\omega}-{W_{q}^{\rm S}}-{W_{q\omega}^{\rm spill}}\right),$$

(3.29)

are minimized, for all $i \in I$ and for all $q \in Q$, subject to the set of constraints {(3.12k) –(3.12p),(3.12s) } and {(3.71j), (3.12r), (3.12s) }, respectively.

The collection of decision variables such that $P_{i} = {R^{\rm U}_i} = {R^{\rm D}_i} = 0, \forall i \in I$ (here, we appeal to assumption A7, according to which conventional producers are fully dispatchable) and ${r^{\rm U}}_{\!\!\!\!\!i\omega} = {r^{\rm D}}_{\!\!\!\!\!i\omega} = 0$, $\forall i \in I, \forall \omega \in \Omega$, constitutes a feasible solution to the minimization subproblem made up of the objective function (3.28) and the group of constraints (3.12k) –(3.12p), and (3.12s). Likewise, the set of decision variables such that ${W^{\rm S}_{q}} = 0$, $\forall q \in Q$, and ${W^{\rm spill}_{q\omega}} = W_{q\omega}$, $\forall q \in Q$, $\forall \omega \in \Omega$, is a feasible solution to the minimization subproblem composed of the objective function (3.29) and constraints (3.71j), (3.12r), and(3.12s). This pair of solutions sets the upper bound of expressions (3.28) and (3.29) to zero. Consequently, inequalities (3.26) and (3.27) hold, which concludes the proof.

It is important to underline that the decomposition-based reasoning employed to prove Theorem 3.2 cannot be used, however, to prove cost recovery per scenario due to the day-ahead dispatch variables P _i and ${W^{\rm S}}_{\!\!\!q}$, which link all the scenarios together. This is so because the settlement scheme (3.13) allows power producers to incur economic losses in some scenarios as long as they recover their production costs in expectation, i.e., in the long run and under similar conditions.

Appendix 2: Worst-Case Realization of Uncertain Production in Robust Optimization

In this appendix, we prove that the worst-case uncertainty realization for the adaptive robust optimization problem (3.21) occurs at an extreme point of the polyhedral uncertainty set $\mathcal W$ for the deviation of wind power production from its conditional mean forecast.

Let us consider the inner max-min problem in (3.21):

$$\begin{aligned} &\underset{\Delta W}{\text{Max}} \underset{\Xi^{\rm B}}{\rm min} \left[\sum_{i \in I}{\left({C_{i}^{\rm U}} {r_i^{\rm U}} - {C_i^{\rm D}} {r_i^{\rm D}} \right)} + \sum_{q \in Q}{{C}_{q} \left(\Delta W_q - {W_q^{\rm spill}} \right)} + \sum_{j \in J}{{V_j^{\rm LOL}} {L_j^{\rm shed}}} \right]\end{aligned}$$

(3.30a)

$$\begin{aligned} &\qquad\;\;\text{s.t.} \sum_{i \in \Phi_{n}^{I}}{\left({r_i^{\rm U}} - {r_i^{\rm D}} \right)} + \sum_{j \in \Phi_{n}^{J}}{L_j^{\rm shed}} + \sum_{q \in \Phi_{n}^{Q}}{\left(\Delta W_q - {W_q^{\rm spill}} \right)} \nonumber\\ &\qquad\quad\qquad + \sum_{\ell \in \Lambda|o(\ell) = n}{b_{\ell} \left({\delta_{o(\ell)}^{0}} - {\delta}_{o(\ell)} - {\delta_{e(\ell)}^{0}} + {\delta}_{e(\ell)}\right)} \nonumber\\ &\qquad\quad\qquad - \sum_{\ell \in \Lambda|e(\ell) = n}{b_{\ell} \left({\delta_{o(\ell)}^{0}} - {\delta}_{o(\ell)} - {\delta_{e(\ell)}^{0}} + {\delta}_{e(\ell)} \right)} = 0 \quad: \lambda_n, \forall n \in N,\end{aligned}$$

(3.30b)

$$\begin{aligned} &\qquad\qquad\;b_{\ell} \left({\delta}_{o(\ell)} - {\delta}_{e(\ell)} \right) \leq C_{\ell}^{\rm max} \quad: \sigma_\ell^{\rm{U}}, \forall \ell \in \Lambda,\end{aligned}$$

(3.30c)

$$\begin{aligned} &\qquad\qquad\;- b_{\ell} \left({\delta}_{o(\ell)} - {\delta}_{e(\ell)} \right) \leq C_{\ell}^{\rm max} \quad: \sigma_\ell^{\rm{D}}, \forall \ell \in \Lambda,\end{aligned}$$

(3.30d)

$$\begin{aligned} &\qquad\qquad\;{\delta}_1 = 0 \quad: \nu,\end{aligned}$$

(3.30e)

$$\begin{aligned} &\qquad\qquad\;{r_i^{\rm U}} \leq{R_i^{\rm U}} \quad: \mu_i^{\rm{U}}, \forall i \in I,\end{aligned}$$

(3.30f)

$$\begin{aligned} &\qquad\;{r_i^{\rm D}} \leq{R_i^{\rm D}} \quad: \mu_i^{\rm{D}}, \forall i \in I,\end{aligned}$$

(3.30g)

$$\begin{aligned} &\qquad\qquad\;{L_j^{\rm shed}} \leq{L}_j \quad: \epsilon_j^{\rm{shed}}, \forall j \in J,\end{aligned}$$

(3.30h)

$$\begin{aligned} &\qquad\; {W_q^{\rm spill}} \leq \widehat W_q + \Delta W_q \quad: \epsilon_q^{\rm{spill}} \forall q \in Q,\end{aligned}$$

(3.30i)

$$\begin{aligned} &\qquad\qquad\;{r_i^{\rm U}}, {r_i^{\rm D}} \geq 0, \enskip \forall i \in I; {W_q^{\rm spill}} \geq 0, \enskip \forall q \in Q; {L_j^{\rm shed}} \geq 0, \enskip \forall j \in J,\end{aligned}$$

(3.30j)

$$\begin{aligned} &\qquad{s.t.}\ {| \Delta W_q | \leq \Delta W_q^{\rm{max}}, \forall q \in Q,}\end{aligned}$$

(3.30k)

$$\begin{aligned} &\qquad{\sum_{q \in Q} \frac{| \Delta W_q |}{\Delta W_q^{\rm{max}}} \leq \Gamma.}\end{aligned}$$

(3.30l)

Notice that we indicated the dual variables for the inner minimization problem on the right-hand side of the corresponding constraints, preceded by a colon.

First of all, we can reformulate inequalities (3.30k) and (3.30l) as follows:

$$\begin{aligned} & - \Delta W_q^{\rm{max}} \leq \Delta W_q \leq \Delta W_q^{\rm{max}},\qquad\forall q \in Q,\end{aligned}$$

(3.31a)

$$\begin{aligned} & \Delta W_q = \Delta W_q^+ - \Delta W_q^-, ~~~~~~~~~~\qquad \forall q \in Q,\end{aligned}$$

(3.31b)

$$\begin{aligned} & \sum_{q \in Q} \frac{\Delta W_q^+ + \Delta W_q^-}{\Delta W_q^{\rm{max}}} \leq \Gamma,\end{aligned}$$

(3.31c)

$$\begin{aligned} & \Delta W_q^+, \Delta W_q^- \geq 0, ~~~~~~~~~~~~~~~~~~~\qquad \forall q \in Q.\end{aligned}$$

(3.31d)

It is worth to point out that reformulation (3.31) is linear, on the contrary of (3.30k)–(3.30l).

Denoting the set of dual variables of the inner problem (3.30a) –(3.30j) with $\Xi'$, we can replace the inner minimization problem with its dual (see Appendix B of the book). This renders the following problem:

$$\begin{aligned} &\underset{\Delta W}{\text{Max}} \underset{\Xi'}{\rm max} \sum_{n \in N} \Bigg[\sum_{\ell \in \Lambda|o(\ell) = n} b_{\ell} \left(- {\delta^{0}_{o(\ell)}} + {\delta^{0}_{e(\ell)}} \right) - \sum_{\ell \in \Lambda|e(\ell) = n}b_{\ell} \left(- {\delta^{0}_{o(\ell)}} + {\delta^{0}_{e(\ell)}} \right) \nonumber\\ & \qquad\qquad - \sum_{q \in \Phi^Q_{n}} \Delta W_q \Bigg] \lambda_n + \sum_{\ell \in \Lambda} C_{\ell}^{\rm max} \left(\sigma_\ell^{\rm{U}} + \sigma_{\ell}^{\rm{D}} \right) + \sum_{i \in I} \left(R^{\rm U}_i \mu^{\rm{U}}_i + R^{\rm D}_i \mu^{\rm{D}}_i \right) \nonumber\\ & \qquad\qquad + \sum_{j \in J} L_j \epsilon^{\rm{shed}}_j + \sum_{q \in Q} \left[\left(\widehat W_q + \Delta W_q \right) \epsilon^{\rm{spill}}_q + C_q \Delta W_q \right]\end{aligned}$$

(3.32a)

$$\begin{aligned} & \qquad\;\text{s.t.} \lambda_{s(i)} + \mu^{\rm{U}}_i \leq C^{\rm U}_i, \forall i \in I,\end{aligned}$$

(3.32b)

$$\begin{aligned} & \qquad\; - \lambda_{s(i)} + \mu^{\rm{D}}_i \leq - C^{\rm D}_i, \forall i \in I,\end{aligned}$$

(3.32c)

$$\begin{aligned} & \qquad\quad\;\;\,\lambda_{s(j)} + \epsilon^{\rm{shed}}_j \leq V^{\rm LOL}_j, \forall j \in J,\end{aligned}$$

(3.32d)

$$\begin{aligned} & \qquad\;\quad- \lambda_{s(q)} + \epsilon^{\rm{spill}}_q \leq - C_q, \forall q \in Q,\end{aligned}$$

(3.32e)

$$\begin{aligned} & \qquad\quad\;\,\nu - \left(\sum_{\ell \in \Lambda|o(\ell) = 1} b_\ell + \sum_{\ell \in \Lambda|e(\ell) = 1} b_\ell \right) \lambda_1 + \sum_{\ell \in \Lambda|e(\ell) = 1} b_\ell \lambda_{o(\ell)} \nonumber\\ & \qquad\qquad + \sum_{\ell \in \Lambda|o(\ell) = 1} b_\ell \lambda_{e(\ell)} + \sum_{\ell \in \Lambda|o(\ell) = 1} b_\ell \left(\sigma^{\rm{U}}_\ell - \sigma^{\rm{D}}_\ell \right) \nonumber\\ & \qquad\qquad - \sum_{\ell \in \Lambda|e(\ell) = 1} b_\ell \left(\sigma^{\rm{U}}_\ell - \sigma^{\rm{D}}_\ell \right) = 0,\end{aligned}$$

(3.32f)

$$\begin{aligned} & \qquad\qquad - \left(\sum_{\ell \in \Lambda|o(\ell) = n} b_\ell + \sum_{\ell \in \Lambda|e(\ell) = n} b_\ell \right) \lambda_n + \sum_{\ell \in \Lambda|e(\ell) = n} b_\ell \lambda_{o(\ell)} \nonumber\\ & \qquad\qquad + \sum_{\ell \in \Lambda|o(\ell) = n} b_\ell \lambda_{e(\ell)} + \sum_{\ell \in \Lambda|o(\ell) = n} b_\ell \left(\sigma^{\rm{U}}_\ell - \sigma^{\rm{D}}_\ell \right) \nonumber\\ & \qquad\qquad - \sum_{\ell \in \Lambda|e(\ell) = n} b_\ell \left(\sigma^{\rm{U}}_\ell - \sigma^{\rm{D}}_\ell \right) = 0, \forall n \in N \setminus \{1\},\end{aligned}$$

(3.32g)

$$\begin{aligned} & \qquad\;\;\quad\sigma^{\rm{U}}_\ell, \sigma^{\rm{D}}_\ell \leq 0, \enskip \forall \ell \in \Lambda; \mu^{\rm{U}}_i, \mu^{\rm{D}}_i \leq 0, \enskip \forall i \in I;\end{aligned}$$

(3.32h)

$$\begin{aligned} & \qquad\;\;\quad\epsilon^{\rm{shed}}_j \leq 0, \enskip \forall j \in J; \epsilon^{\rm{spill}}_q \leq 0, \enskip \forall q \in Q,\end{aligned}$$

(3.32i)

$$\begin{aligned} & \text{s.t.}\ -\Delta W_q^{\rm{max}} \leq \Delta W_q \leq \Delta W_q^{\rm{max}}, \forall q \in Q,\end{aligned}$$

(3.32j)

$$\begin{aligned} & \qquad\Delta W_q = \Delta W_q^+ - \Delta W_q^-, \forall q \in Q,\end{aligned}$$

(3.32k)

$$\begin{aligned} & \qquad\sum_{q \in Q} \frac{\Delta W_q^+ + \Delta W_q^-}{\Delta W_q^{\rm{max}}} \leq \Gamma,\end{aligned}$$

(3.32l)

$$\begin{aligned} & \qquad\Delta W_q^+, \Delta W_q^- \geq 0, \forall q \in Q.\end{aligned}$$

(3.32m)

The following observations on model (3.32) are in order.

• 1. The two max operators can be merged. Therefore, (3.32) is a single maximization problem.

• 2. Constraints (3.32b) –(3.32m) are linear.

• 3. Objective function (3.32a) is bilinear, owing to the cross products between variables $\Delta W_q$ and λ _n as well as $\epsilon^{\rm spill}_q$.

In view of the observations above, if we fix the variables in $\Xi'$ at their optimal value, model (3.32) boils down to a linear programming problem in the decision variables $\Delta W_q$, constrained by (3.32j)–(3.32m). For any linear program, at least one of the solutions (if it exists) is a vertex of the feasible set. Since the feasible set $\mathcal W$ is compact, at least an optimal solution of the bilinear program (3.32) is a vertex of  $\mathcal W$.

Exercises

3.1.

Reformulate the auction in Example 3.3 to include two time periods. Enforce ramping limits on the thermal generation units and analyze numerically the impact of such limits on market outcomes. Hint: the reader is advised to consult Sect. 5.3.3.

3.2.

Consider multiple Gaussian distributed wind power production scenarios in the problem of Example 3.3. Analyze numerically the impact of increasing the number of scenarios on market outcomes. Compare these outcomes with those obtained considering solely the average value scenario.

3.3.

Consider just two extreme scenarios (very-high wind production and no wind production), and analyze the outcomes of the auction in Example 3.3. Compare these outcomes with the outcomes obtained considering solely the average value scenario. What happens as scenarios become increasingly extreme?

3.4.

Analyze the market-clearing algorithm in Example 3.3 in a case in which only wind producers are available. Study the behavior of prices, both day-ahead prices and balancing prices.

3.5.

Consider the market-clearing algorithm in Example 3.3, but involving thermal plants with significantly high start-up costs. What happens with the clearing prices (both day-ahead and balancing) in such situation? Hint: you can get inspiration on how to model the start-up cost of a thermal power plant from Sect. 8.2.1 in the book.

3.6.

Consider the market-clearing algorithm in Example 3.3, but involving thermal plants with minimum power outputs. What happens with the clearing prices (both day-ahead and balancing) in such situation? Hint: you can get inspiration from Sect. 5.3.2 for the modeling of capacity limits.

3.7.

Consider the auction in Example 3.3, and solve it for a wide range of values of lost load. Study how market outcomes change as a result of an increasingly high unserved-energy value.

3.8.

Consider wind production offers at non-zero price in the auction of Example 3.3. Analyze numerically how market outcomes change as wind offering prices increase.

3.9.

Reformulate the auction in Example 3.3 to include two time periods involving highly different load levels, and a pumped storage plant. Is the availability of such pumped storage plant beneficial? Analyze how the impact of the pumped storage plant on market outcomes changes as the efficiency of the pumping-turbine cycle increases. Hint: Sect. 5.5 provides insight into how to model a pumped-storage power plant.

3.10.

Reformulate the auction in Example 3.3 to include two time periods involving highly different load levels, and a pumped storage plant. Consider that the transmission line has such a low capacity that often leads to transmission bottleneck. Analyze the ability of the pumped storage plant to alleviate the detrimental effect of transmission bottlenecks.

3.11.

Solve the robust optimization problem (3.20) considered in Example 3.6 for different values of the budget of uncertainty in (3.17). Start by enumerating the vertices of the polyhedral uncertainty set. What is the effect of increasing the uncertainty budget on the amount of dispatched reserve?

3.12.

Include constraints of the following type

$$ | \Delta W_1 - \Delta W_2 | \leq \rho$$

in the definition of the uncertainty set for the dispatch model based on robust optimization presented in Example 3.6. Determine the uncertainty set and enumerate its vertices, then solve the dispatch problem.

3.13.

Reformulate the robust optimization model (3.20) to include two time periods and a pumped storage plant. Consider the two-node system of Example 3.3, which includes only one wind power plant. Introduce intervals for the deviation of wind power production during each time period, and a budget of uncertainty to limit the total deviation of energy production over the two periods, similarly to (3.16) and (3.17). Analyze the effect on the robust dispatch of a storage facility with limited capacity.