The spot and balancing markets for electricity: open- and closed-loop equilibrium models

Abstract

The increasing penetration of inflexible and fluctuating renewable energy generation is often accompanied by a sequential market setup, including a day-ahead spot market that balances forecasted supply and demand with an hourly time resolution and a balancing market in which flexible generation handles unexpected imbalances closer to real-time and with a higher time resolution. Market characteristics such as time resolution, the time of market offering and the information available at this time, price elasticities of demand and the number of market participants, allow producers to exercise market power to different degrees. To capture this, we study oligopolistic spot and balancing markets with Cournot competition, and formulate two stochastic equilibrium models for the sequential markets. The first is an open-loop model which we formulate and solve as a complementarity problem. The second is a closed-loop model that accounts for the sequence of market clearings, but is computationally more demanding. Via optimality conditions, the result is an equilibrium problem with equilibrium constraints which we solve by an iterative procedure. When compared to the closed-loop solution, our results show that the open-loop problem overestimates the ability to exercise market power unless the market allows for speculation. In the presence of a speculator, the open-loop formulation forces spot and balancing market prices to be equal in expectation and indicates substantial profit reductions, whereas speculation has less severe impact in the closed-loop problem. We use the closed-loop model to further analyse market power issues with a higher time resolution and limited access to the balancing market.

Appendices

Appendix A. Nomenclature

Parameters	Units
$d_h^{{da}}$	MWh	Net spot market (forecasted) demand in hour h
$\alpha _{h}^{{da}}$	DKK/MWh$^2$	Slope of the inverse demand curve for hour h
$\beta _{h}^{{da}}$	MWh	Intercept of the inverse demand curve for hour h
$d_{t\omega }^{{ba}}$	MWh	Net balancing market (realized) demand in time interval t and scenario $\omega$
$\alpha _{t}^{{da-bal}}$	DKK/MWh$^2$	Slope of the inverse demand curve for time interval t with respect to forecasted demand
$\alpha _t^{{ba}}$	DKK/MWh$^2$	Slope of the inverse demand curve for time interval t with respect to
		realized demand
$\beta _{t\omega }^{{ba}}$	MWh	Intercept of the inverse demand curve for time interval t and scenario $\omega$
$x_{ih}$	MWh	Spot market offer by producer i in hour h
$x_{it\omega }^+$	MWh	Up-regulation offer by producer i in time interval t and scenario $\omega$
$x_{it\omega }^-$	MWh	Down-regulation offer by producer i in time interval t and scenario $\omega$
$x_{i}^{max}$	MW	Spot market capacity by producer i
$a_i$	DKK/MWh$^2$	Production cost parameter of producer i
$b_i$	DKK/MWh	Production cost parameter of producer i
$\gamma _i^+$	DKK/MWh	Balancing cost parameter of producer i
$\gamma _i^-$	DKK/MWh	Balancing cost parameter of producer i
$\theta _{ih}$	DKK/MWh$^2$	Spot market price response for hour h and producer i
$\theta _{it}^{{da}}$	DKK/MWh$^2$	Balancing price response for time interval t and producer i with respect to forecasted demand
$\theta _{it}^{{ba}}$	DKK/MWh$^2$	Balancing price response for time interval t and producer i with respect to realized demand

Appendix B. Proof of Proposition 1

Proof

The Hessian of the function $f_{it\omega }(\cdot )$ is

$$\begin{aligned} \begin{bmatrix} 2\tau \big (\theta _{ih}-\tau a_i\big )&{}\theta ^{{da}}_{it}-2\tau a_i&{}-\theta ^{{da}}_{it}+2\tau a_i\\ \theta ^{{da}}_{it}-2\tau a_i&{}2\big (\theta _{it}^{{bal}}-a_i\big )&{}-2(\theta ^{{bal}}_{it}-a_i)\\ -\theta ^{{da}}_{it}+2\tau a_i&{}-2(\theta ^{{bal}}_{it}-a_i)&{}2\big (\theta _{it}^{{bal}}-a_i\big )\\ \end{bmatrix} \end{aligned}$$

The third order principal minor is zero, one of the second order principal minors is likewise zero whereas the other two are $4\tau (\theta _{ih}-\tau a_i)(\theta ^{{bal}}_{it}-a_i)-(\theta ^{{da}}_{it}-2\tau a_i)^2$, and one of the first order principal minors is $2\tau \big (\theta _{ih}-\tau a_i\big )$ whereas the other two are $2\big (\theta _{it}^{{bal}}-a_i\big )$.

Since $\theta _{ih}, \theta _{it}^{{bal}}\le 0$ all first order principal minors are non-positive, and by Assumption 1, all second order principal minors are non-negative. As a result, the Hessian is negative semi-definite, and hence, the function is concave. $\square$

Appendix C. Proof of Proposition 2

Proof

If $\underline{\sigma }_{it\omega }^{+}=\underline{\sigma }_{it\omega }^{-}=0$, then (9b), (9c), (9g) and (9i) implies that

$$\begin{aligned}&\pi _\omega \Bigg (\theta _{it}^{{bal}}\big (x_{it\omega }^{+}-x_{it\omega }^{-}\big )+p^{{bal}}_{t\omega }-\frac{\partial c_i}{\partial x_{it\omega }^{+}} \Bigg )=\bar{\sigma }_{it\omega }^{+}\ge 0, \;\\ -&\pi _\omega \Bigg (\theta _{it}^{{bal}}\big (x_{it\omega }^{+}-x_{it\omega }^{-}\big )+p^{{bal}}_{t\omega }+\frac{\partial c_i}{\partial x_{it\omega }^{-}} \Bigg )=\bar{\sigma }_{it\omega }^{-}\ge 0. \end{aligned}$$

By adding the two inequalities,

$$\begin{aligned} -\gamma _i^+-\gamma _i^-=-\frac{\partial c_i}{\partial x_{it\omega }^{+}}-\frac{\partial c_i}{\partial x_{it\omega }^{-}}\ge 0, \end{aligned}$$

which cannot be true. We consider the remaining cases.

Let $\underline{\sigma }_{it\omega }^{+}=0,\underline{\sigma }_{it\omega }^{-}>0$. The conditions (9b), (9c), (9f), (9g), (9h) and (9i) are then

$$\begin{aligned} -&\pi _\omega \Bigg (\theta _{it}^{{bal}}x_{it\omega }^{+}+p^{{bal}}_{t\omega }-\frac{\partial c_i}{\partial x_{it\omega }^{+}} \Bigg )+\bar{\sigma }_{it\omega }^{+}=0, \;\\&\pi _\omega \Bigg (\theta _{it}^{{bal}}x_{it\omega }^{+}+p^{{bal}}_{t\omega }+\frac{\partial c_i}{\partial x_{it\omega }^{-}} \Bigg )+\bar{\sigma }_{it\omega }^{-}-\underline{\sigma }_{it\omega }^{-}=0\\&x_{it\omega }^{+}\ge 0, \; 0\le \tau \big (x_i^{\max }-x_{ih}\big )-x_{it\omega }^{+}\perp \bar{\sigma }_{it\omega }^{+} \ge 0, \;\\&x_{it\omega }^{-}=0, \underline{\sigma }_{it\omega }^{-}>0, \; 0\le \tau x_{ih}\perp \bar{\sigma }_{it\omega }^{-} \ge 0 \end{aligned}$$

If $\bar{\sigma }_{it\omega }^{-}=0$, then this is equivalent to

$$\begin{aligned}&\pi _\omega \Bigg (\theta _{it}^{{bal}}x_{it\omega }^{+}+p^{{bal}}_{t\omega }-2a_i(\tau x_{ih}+x_{it\omega }^{+})-b_i+\gamma _i^-\Bigg )>0, x_{it\omega }^{-}=0\\&\quad 0\le \tau \big (x_i^{\max }-x_{ih}\big )-x_{it\omega }^{+}\perp \pi _\omega \Bigg (\theta _{it}^{{bal}}x_{it\omega }^{+}+p^{{bal}}_{t\omega }-2a_i(\tau x_{ih}+x_{it\omega }^{+})-b_i-\gamma _i^+\Bigg ) \\ {}&\qquad \ge 0, x_{it\omega }^{+}\ge 0. \end{aligned}$$

There are now two cases, that is,

$$\begin{aligned}&\pi _\omega \Bigg (\theta _{it}^{{bal}}\tau \big (x_i^{\max }-x_{ih}\big )+p^{{bal}}_{t\omega }-2a_i\tau x_i^{\max }-b_i+\gamma _i^-\Bigg )>0, x_{it\omega }^{-}=0\\&\pi _\omega \Bigg (\theta _{it}^{{bal}}\tau \big (x_i^{\max }-x_{ih}\big )+p^{{bal}}_{t\omega }-2a_i\tau x_i^{max}-b_i-\gamma _i^+\Bigg )>0, x_{it\omega }^{+}=\tau \big (x_i^{\max }-x_{ih}\big ) \end{aligned}$$

and

$$\begin{aligned}&\pi _\omega \Bigg (\theta _{it}^{{bal}}x_{it\omega }^{+,*}+p^{{bal}}_{t\omega }-2a_i(\tau x_{ih}+x_{it\omega }^{+,*})-b_i+\gamma _i^-\Bigg )>0, x_{it\omega }^{-}=0\\&\quad 0\le x_{it\omega }^{+,*}=\frac{p^{{bal}}_{t\omega }-2a_i\tau x_{ih}-b_i-\gamma _i^+}{2a_i-\theta _{it}^{{bal}}}\le \tau \big (x_i^{\max }-x_{ih}\big ), x_{it\omega }^{+}=x_{it\omega }^{+,*}. \end{aligned}$$

The former is the same as

$$\begin{aligned}&x_{it\omega }^{-}=0, \; x_{it\omega }^{+,*}>\tau \big (x_i^{\max }-x_{ih}\big ), x_{it\omega }^{+}=\tau \big (x_i^{\max }-x_{ih}\big ) \end{aligned}$$

and the latter is the same as

$$\begin{aligned}&x_{it\omega }^{-}=0, \; 0\le x_{it\omega }^{+,*}=\frac{p^{{bal}}_{t\omega }-2a_i\tau x_{ih}-b_i-\gamma _i^+}{2a_i-\theta _{it}^{{bal}}}\le \tau \big (x_i^{\max }-x_{ih}\big ), x_{it\omega }^{+}=x_{it\omega }^{+,*}. \end{aligned}$$

Let $\underline{\sigma }_{it\omega }^{+}>0,\underline{\sigma }_{it\omega }^{-}=0$. By similar reasoning, we obtain the two cases

$$\begin{aligned}&x_{it\omega }^{+}=0, \; x_{it\omega }^{-,*}>\tau x_{ih}, x_{it\omega }^{-}=\tau x_{ih} \end{aligned}$$

and

$$\begin{aligned}&x_{it\omega }^{+}=0, \; 0\le x_{it\omega }^{-,*}=\frac{-p^{{bal}}_{t\omega }+2a_i\tau x_{ih}+b_i-\gamma _i^-}{2a_i-\theta _{it}^{{bal}}}\le \tau x_{ih}, x_{it\omega }^{-}=x_{it\omega }^{-,*}. \end{aligned}$$

Finally, let $\underline{\sigma }_{it\omega }^{+}>0,\underline{\sigma }_{it\omega }^{-}>0$. The KKT-conditions are

$$\begin{aligned}&-\pi _\omega \Bigg (p^{{bal}}_{t\omega }-\frac{\partial c_i}{\partial x_{it\omega }^{+}} \Bigg )+\bar{\sigma }_{it\omega }^{+}-\underline{\sigma }_{it\omega }^{+} =0, \; \pi _\omega \Bigg (p^{{bal}}_{t\omega }+\frac{\partial c_i}{\partial x_{it\omega }^{-}} \Bigg )+\bar{\sigma }_{it\omega }^{-}-\underline{\sigma }_{it\omega }^{-}=0\\&\quad x_{it\omega }^{+}=0, \underline{\sigma }_{it\omega }^{+}> 0, \; 0\le \tau \big (x_i^{\max }-x_{ih}\big )\perp \bar{\sigma }_{it\omega }^{+} \ge 0, \; x_{it\omega }^{-}=0, \underline{\sigma }_{it\omega }^{-} > 0, \;\\&\qquad \qquad \quad 0\le \tau x_{ih}\perp \bar{\sigma }_{it\omega }^{-} \ge 0. \end{aligned}$$

If $\bar{\sigma }_{it\omega }^{+}=\bar{\sigma }_{it\omega }^{-}=0$, then this is equivalent to

$$\begin{aligned} -&\pi _\omega \Bigg (p^{{bal}}_{t\omega }-2a_i\tau x_{ih}-b_i-\gamma _i^+\Bigg )>0, x_{it\omega }^{+}=0, \; \\&\pi _\omega \Bigg (p^{{bal}}_{t\omega }-2a_i\tau x_{ih}-b_i+\gamma _i^-\Bigg )>0, x_{it\omega }^{-}=0. \end{aligned}$$

This is the same as

$$\begin{aligned}&x_{it\omega }^{+,*}<0, x_{it\omega }^{+}=0, \; x_{it\omega }^{-,*}<0, x_{it\omega }^{-}=0. \end{aligned}$$

$\square$

Appendix D. Assumptions on the price response parameters

Let $F:\mathbb {R}^{|I||H|(1+2|T_h||\Omega |)}\rightarrow \mathbb {R}^{|I||H|(1+2|T_h||\Omega |)}$ be the vector-function with entries

$$\begin{aligned} F_{ih}&=-\theta _{ih}x_{ih}-p^{{da}}_h-\sum _{\omega \in \Omega }\pi _\omega \sum _{t\in T_h}\Big (\theta _{it}^{{da}}\big (x_{it\omega }^{+}-x_{it\omega }^{-}\big )-\frac{\partial c_i}{\partial x_{ih}}\Big )\\ F_{it\omega }^+&=-\pi _\omega \Big (\theta _{it}^{{bal}} \big (x_{it\omega }^{+}-x_{it\omega }^{-}\big )+p^{{bal}}_{t\omega }-\frac{\partial c_i}{\partial x_{it\omega }^{+}} \Big )\\ F_{it\omega }^-&=\pi _\omega \Big (\theta _{it}^{{bal}} \big (x_{it\omega }^{+}-x_{it\omega }^{-}\big )+p^{{bal}}_{t\omega }+\frac{\partial c_i}{\partial x_{it\omega }^{-}} \Big ) \end{aligned}$$

for $i\in I, t\in T_h, h\in H, \omega \in \Omega$. Note that F is continuously differentiable.

A necessary condition for the equilibrium problem to have an equivalent optimization problem is the existence of a function $z:\mathbb {R}^{|I||H|(1+2|T_h||\Omega |)}\rightarrow \mathbb {R}$, serving as the objective, with $F=\nabla z$. Such function exists if and only if the Jacobian matrix of F is symmetric (Theorem 4.2 in Gabriel). More specifically, this requires

$$\begin{aligned} \theta _{ih}&=\frac{\partial F_{ih}}{\partial x_{jh}}=\frac{\partial F_{jh}}{\partial x_{ih}}=\theta _{jh}, \\ 0&=\frac{\partial F_{ih}}{\partial x_{jt\omega }^+}=\frac{\partial F_{jt\omega }^+}{\partial x_{ih}}=-\pi _{\omega }\theta _{jt}^{{da}},\\&\quad -\pi _{\omega }\theta _{it}^{{bal}}=\frac{\partial F_{ih}}{\partial x_{jt\omega }^+}=\frac{\partial F_{it\omega }^+}{\partial x_{jh}}=-\pi _{\omega }\theta _{jt}^{{bal}}, \end{aligned}$$

for $i,j\in I$, which are the assumptions on the price response parameters.

Appendix E. Proof of Proposition 3

Proof

If $\theta _{hi}:=\theta _h, \theta _{it}^{{da}}:=\theta _{t}^{{da}}=0, \theta _{it}^{{bal}}:=\theta _{t}^{{bal}}$, the KKT-conditions of the problem (14) are

$$\begin{aligned}&-\theta _{h}x_{ih}-p^{{da}}_h-\sum _{\omega \in \Omega }\pi _\omega \sum _{t\in T_h}\Bigg (\theta _{t}^{{da}}\big (x_{it\omega }^{+}-x_{it\omega }^{-}\big )-\frac{\partial c_i}{\partial x_{ih}}\Bigg )\\&+\bar{\sigma }_{ih}-\underline{\sigma }_{ih}+\tau \sum _{\omega \in \Omega }\sum _{t\in T_h} \big (\bar{\sigma }_{it\omega }^{+}-\bar{\sigma }_{it\omega }^{-}\big )+\lambda _h(\alpha _{h}^{{da}} -\theta _h)=0, \; h\in H, \; i\in I\\&-\pi _\omega \Bigg (\theta _{it}^{{bal}}\big (x_{it\omega }^{+}-x_{it\omega }^{-}\big ) +p^{{bal}}_{t\omega }-\frac{\partial c_i}{\partial x_{it\omega }^{+}} \Bigg )+\bar{\sigma }_{it\omega }^{+} -\underline{\sigma }_{it\omega }^{+} +\lambda _{t\omega }(\alpha _t^{{bal}}-\theta _t^{{bal}})=0, \;\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad t\in T_h,\; h\in H,\; \omega \in \Omega , \; i\in I\\&\quad \pi _\omega \Bigg (\theta _{it}^{{bal}}\big (x_{it\omega }^{+}-x_{it\omega }^{-}\big ) +p^{{bal}}_{t\omega }+\frac{\partial c_i}{\partial x_{it\omega }^{-}} \Bigg )+\bar{\sigma }_{it\omega }^{-} -\underline{\sigma }_{it\omega }^{-}-\lambda _{t\omega }(\alpha _t^{{bal}}-\theta _t^{{bal}})=0, \;\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad t\in T_h,\; h\in H,\; \omega \in \Omega , \; i\in I \end{aligned}$$

(9d)–(9i) for all $i\in I$ and (1), (2), (6), (7), where $\lambda _h,\lambda _{t\omega }, t\in T_h, h\in H, \omega \in \Omega$ are free variables. A solution to the equilibrium problem satisfies the above with $\lambda _h=\lambda _{t\omega }=0, t\in T_h, h\in H, \omega \in \Omega$.

If also $\theta _h=-\alpha _{h}^{{da}}, \theta _{t}^{{da}}=-\alpha _{t}^{{da-bal}}\tau , \theta _{t}^{{bal}}=-\alpha _t^{{bal}}$, the above are the KKT-conditions of (15). $\square$

Appendix F. 30-minute market clearing of the balancing market under perfect competition

Table 13 shows the results under perfect competition.

Table 13 Open-loop perfect competition

Appendix G. 30-minute market clearing of the balancing market under open-loop Cournot

Table 14 shows the open-loop Cournot results. For the base case (i), the solution is the same as with hourly time resolution in the balancing market, cf. Table 5. With intra-hourly variations of inelastic demand, profits increase by 25.63%.

Table 14 Open-loop Cournot

Appendix H. Proof of Lemma 1

Consider the (upper-level) objective function:

$$\begin{aligned} f_{ih}(x_{ih};x_{-ih})=p^{{da}}_hx_{ih}+\sum _{\omega \in \Omega }\pi _{\omega }\sum _{t\in T_h}\left( p^{{bal}}_{t\omega }(x_{it\omega }^{+}-x_{it\omega }^{-})-c_i(x_{ih},x_{it\omega }^{+},x_{it\omega }^{-})\right) , \end{aligned}$$

where $x_{it\omega }^{+}:=x_{it\omega }^{+}(x_{ih};x_{-ih})$ and $x_{it\omega }^{-}:=x_{it\omega }^{-}(x_{ih};x_{-ih})$.

When it exists, define the derivative as

$$\begin{aligned}&\frac{\partial f_{it\omega }}{\partial x_{ih}}(x_{ih};x_{-ih})= \theta _{ih}x_{ih}+p^{{da}}_h+\sum _{\omega \in \Omega }\pi _{\omega }\sum _{t\in T_h}\Bigg (\theta _{it}^{{da}}\big (x_{it\omega }^{+}-x_{it\omega }^{-}\big )-\frac{\partial c_i}{\partial x_{ih}}\Bigg )\\&\quad +\sum _{\omega \in \Omega }\pi _\omega \sum _{t\in T_h}\big (x_{it\omega }^{+}-x_{it\omega }^{-}\big )\sum _{j\ne i}\theta _{jt}^{{bal}}\big (D_{jit\omega }^{+}-D_{jit\omega }^{-}\big )\\&\quad +\sum _{\omega \in \Omega }\pi _{\omega }\sum _{t\in T_h}\Bigg (D_{iit\omega }^{+}\Big (\theta _{it}^{{bal}}\big (x_{it\omega }^{+} -x_{it\omega }^{-}\big )+p^{{bal}}_{t\omega }-\frac{\partial c_i}{\partial x_{it\omega }^+}\Big )\\&\quad -D_{iit\omega }^{-}\Big (\theta _{it}^{{bal}}\big (x_{it\omega }^{+} -x_{it\omega }^{-}\big )+p^{{bal}}_{t\omega }+\frac{\partial c_i}{\partial x_{it\omega }^-}\Big )\Bigg ) \end{aligned}$$

For $\theta _{it}^{{bal}}=0, i=1,2$, this reduces to

$$\begin{aligned}&\theta _{ih}x_{ih}+p^{{da}}_h+\sum _{\omega \in \Omega }\pi _{\omega }\sum _{t\in T_h}\Bigg (\theta _{it}^{{da}}\big (x_{it\omega }^{+}-x_{it\omega }^{-}\big )-\frac{\partial c_i}{\partial x_{ih}}\Bigg )\\&\qquad +\sum _{\omega \in \Omega }\pi _{\omega }\sum _{t\in T_h}\Bigg (D_{iit\omega }^{+}\Big (p^{{bal}}_{t\omega }-\frac{\partial c_i}{\partial x_{it\omega }^+}\Big )-D_{iit\omega }^{-}\Big (p^{{bal}}_{t\omega }+\frac{\partial c_i}{\partial x_{it\omega }^-}\Big )\Bigg )\\&\quad = \theta _{ih}x_{ih}+p^{{da}}_h+\sum _{\omega \in \Omega }\pi _{\omega }\sum _{t\in T_h}\Bigg (\theta _{it}^{{da}}\big (x_{it\omega }^{+}-x_{it\omega }^{-}\big )-\frac{\partial c_i}{\partial x_{ih}}\Bigg )\\&\qquad -\tau \sum _{\omega , t: i\in I_{t\omega }^{+,\ge }}\pi _{\omega }\Big (p^{{bal}}_{t\omega }-\frac{\partial c_i}{\partial x_{it\omega }^+}\Big ) +\sum _{\omega , t: i\in I_{t\omega }^{+,*}}\pi _{\omega }D_{iit\omega }^{+}\Big (p^{{bal}}_{t\omega }-\frac{\partial c_i}{\partial x_{it\omega }^+}\Big )\\&\qquad -\tau \sum _{\omega , t: i\in I_{t\omega }^{-,\ge }}\pi _{\omega }\Big (p^{{bal}}_{t\omega }+\frac{\partial c_i}{\partial x_{it\omega }^-}\Big ) +\sum _{\omega , t: i\in I_{t\omega }^{-,*}}\pi _{\omega }D_{iit\omega }^{-}\Big (p^{{bal}}_{t\omega }+\frac{\partial c_i}{\partial x_{it\omega }^-}\Big )\\&\quad = \theta _{ih}x_{ih}+p^{{da}}_h+\sum _{\omega \in \Omega }\pi _{\omega }\sum _{t\in T_h}\Bigg (\theta _{it}^{{da}}\big (x_{it\omega }^{+}-x_{it\omega }^{-}\big )-\frac{\partial c_i}{\partial x_{ih}}\Bigg )\\&\qquad -\tau \sum _{\omega , t: i\in I_{t\omega }^{+,\ge }}\pi _{\omega }\Big (p^{{bal}}_{t\omega }-2a_i\tau x_i^{max}-b_i\Big )-\tau \sum _{\omega , t: i\in I_{t\omega }^{-,\ge }}\pi _{\omega }\Big (p^{{bal}}_{t\omega }-b_i\Big ) \end{aligned}$$

since $D_{iit\omega }^+=0$ when $i\in I_{t\omega }^{+,\le }$ and $D_{iit\omega }^-=0$ when $i\in I_{t\omega }^{-,\le }$, cf. Proposition 5, $p^{{bal}}_{t\omega }-{\partial c_i}/{\partial x_{it\omega }^+}=0$ for $i\in I_{t\omega }^{+,*}$ and $p^{{bal}}_{t\omega }+{\partial c_i}/{\partial x_{it\omega }^-}=0$ for $i\in I_{t\omega }^{-,*}$, and $x_{it\omega }^-=0$ when $i\in I_{t\omega }^{+,\ge }$ and $x_{it\omega }^+=0$ when $i\in I_{t\omega }^{-,\ge }$, cf. Proposition 4.

When it exists, the derivative of this is

$$\begin{aligned}&2\theta _{ih}+\sum _{\omega \in \Omega }\pi _{\omega }\sum _{t\in T_h}\Bigg ((\theta _{it}^{{da}}-2\tau a_i)\big (D_{iit\omega }^{+}-D_{iit\omega }^{-}\big )-2\tau ^2a_i\Bigg )\\&\quad -\tau \sum _{\omega , t: i\in I_{t\omega }^{+,\ge }}\pi _{\omega }\theta _{it}^{{da}}-\tau \sum _{\omega , t: i\in I_{t\omega }^{-,\ge }}\pi _{\omega }\theta _{it}^{{da}}. \end{aligned}$$

For $a_i=\theta _{it}^{{da}}=0, i=1,2$, the second order derivative is non-positive and constant. Hence, the function $f_{ih}(x_{ih};x_{-ih})$ is piecewise quadratic and concave in $x_{ih}$ (and thus, differentiable on each quadratic and concave segment). Non-concavity and non-differentiability may occur when $x_{jt\omega }^{+,*}(x_{jh},x_{-jh})=0,x_{jt\omega }^{+,*}(x_{jh},x_{-jh})=\tau (x_j^{max}-x_{jh}), x_{jt\omega }^{-,*}(x_{jh},x_{-jh})=0$ or $x_{jt\omega }^{-,*}(x_{jh},x_{-jh})=\tau x_{jh}$ for some $j=1,2$.

Appendix I. Including ramping

Let $H=\{h_1,\dots ,h_H\}$ and $T_h=\{t_{h1},\dots ,t_{hT}\}$. For $i\in I$, ramping restrictions are given by

$$\begin{aligned} -r_ix_{i}^{max}&\le (\tau x_{ih}+x_{it_{h1}\omega }^+-x_{i1\omega }^-)-(\tau x_{ih-1}+x_{it_{h-1T}\omega }^+-x_{it_{h-1T}\omega }^-) \le r_ix_{i}^{max}, \; h\in H,\; \omega \in \Omega \\ -r_ix_{i}^{max}&\le (x_{it\omega }^+-x_{it\omega }^-) -(x_{it-1\omega }^+-x_{it-1\omega }^-)\le r_ix_{i}^{max}, \; t\in T_h\backslash \{t_{h1}\}, \; h\in H,\; \omega \in \Omega \end{aligned}$$

The KKT-conditions of the open-loop equilibrium with ramping restrictions are

$$\begin{aligned}&-\theta _{ih}x_{ih}-p^{{da}}_h-\sum _{\omega \in \Omega }\pi _\omega \sum _{t\in T_h}\Bigg (\theta _{it}^{{da}}\big (x_{it\omega }^{+}-x_{it\omega }^{-}\big )-\frac{\partial c_i}{\partial x_{ih}}\Bigg )\\&+\bar{\sigma }_{ih}-\underline{\sigma }_{ih}+\tau \sum _{\omega \in \Omega }\sum _{t\in T_h}\big (\bar{\sigma }_{it\omega }^{+}-\bar{\sigma }_{it\omega }^{-}\big )\\&+\tau \sum _{\omega \in \Omega }\big (\bar{\mu }_{it_{h1}\omega }-\bar{\mu }_{it_{h+11}\omega } -\underline{\mu }_{it_{h1}\omega }+\underline{\mu }_{it_{h+11}\omega }\big ) =0, \; h\in H\backslash \{h_H\}\\&-\theta _{ih_H}x_{ih_H}-p^{{da}}_{h_H}-\sum _{\omega \in \Omega }\pi _\omega \sum _{t\in T_{h_H}}\Bigg (\theta _{it}^{{da}}\big (x_{it\omega }^{+} -x_{it\omega }^{-}\big )-\frac{\partial c_i}{\partial x_{ih_H}}\Bigg )\\&+\bar{\sigma }_{ih}-\underline{\sigma }_{ih}+\tau _{h}\sum _{\omega \in \Omega } \sum _{t\in T_{h}}\big (\bar{\sigma }_{it\omega }^{+}-\bar{\sigma }_{it\omega }^{-}\big ) +\tau _{h}\sum _{\omega \in \Omega }\big (\bar{\mu }_{it_{h1}\omega }-\underline{\mu }_{it_{h1}\omega }\big ) =0, \; h=h_H\\&-\pi _\omega \Bigg (\theta _{it}^{{bal}}\big (x_{it\omega }^{+}-x_{it\omega }^{-}\big ) +p^{{bal}}_{t\omega }-\frac{\partial c_i}{\partial x_{it\omega }^{+}} \Bigg )+\bar{\sigma }_{it\omega }^{+} -\underline{\sigma }_{it\omega }^{+}\\ {}&\qquad +\bar{\mu }_{it\omega }-\bar{\mu }_{it+1\omega } -\underline{\mu }_{it\omega }+\underline{\mu }_{it+1\omega }=0, \; t\in T\backslash \{t_{TH}\},\; \omega \in \Omega \\&-\pi _\omega \Bigg (\theta _{it}^{{bal}}\big (x_{it\omega }^{+}-x_{it\omega }^{-}\big ) +p^{{bal}}_{t\omega }-\frac{\partial c_i}{\partial x_{it\omega }^{+}} \Bigg )+\bar{\sigma }_{it\omega }^{+} -\underline{\sigma }_{it\omega }^{+} +\bar{\mu }_{it\omega }-\underline{\mu }_{it\omega }=0, \; \\&\qquad t=t_{HT}, \; \omega \in \Omega \\&\quad \pi _\omega \Bigg (\theta _{it}^{{bal}}\big (x_{it\omega }^{+}-x_{it\omega }^{-}\big ) +p^{{bal}}_{t\omega }+\frac{\partial c_i}{\partial x_{it\omega }^{-}} \Bigg )+\bar{\sigma }_{it\omega }^{-} \\&\qquad -\underline{\sigma }_{it\omega }^{-}+\bar{\mu }_{it\omega }-\bar{\mu }_{it+1\omega } -\underline{\mu }_{it\omega }+\underline{\mu }_{it+1\omega }=0, \; t\in T\backslash \{t_{TH}\}, \; \omega \in \Omega \\&\quad \pi _\omega \Bigg (\theta _{it}^{{bal}}\big (x_{it\omega }^{+}-x_{it\omega }^{-}\big ) +p^{{bal}}_{t\omega }+\frac{\partial c_i}{\partial x_{it\omega }^{-}} \Bigg )+\bar{\sigma }_{it\omega }^{-} -\underline{\sigma }_{it\omega }^{-}+\bar{\mu }_{it\omega }-\underline{\mu }_{it\omega }=0, \;\\ {}&t=t_{HT}, \; \omega \in \Omega \\&\quad 0\le x_{ih}\perp \underline{\sigma }_{ih} \ge 0,\; h\in H\\&\quad 0\le x_i^{\max }-x_{ih}\perp \bar{\sigma }_{ih} \ge 0,\; h\in H\\&\quad 0\le x_{it\omega }^{+}\perp \underline{\sigma }_{it\omega }^{+} \ge 0,\; t\in T_h, \; h\in H,\; \omega \in \Omega \\&\quad 0\le \tau \big (x_i^{\max }-x_{ih}\big )-x_{it\omega }^{+}\perp \bar{\sigma }_{it\omega }^{+} \ge 0, \; t\in T_h,\; h\in H,\; \omega \in \Omega \\&\quad 0\le x_{it\omega }^{-}\perp \underline{\sigma }_{it\omega }^{-} \ge 0,\; t\in T_h,\; h\in H,\; \omega \in \Omega \\&\quad 0\le \tau x_{ih}-x_{it\omega }^{-}\perp \bar{\sigma }_{it\omega }^{-} \ge 0,\; t\in T_h,\; h\in H,\; \omega \in \Omega \\&\quad 0\le r_ix_{i}^{max}-(\tau x_{ih}+x_{it_{h1}\omega }^+-x_{i1\omega }^-)\\&\qquad +(\tau x_{ih-1}+x_{it_{h-1T}\omega }^+-x_{it_{h-1T}\omega }^-)\perp \bar{\mu }_{t_{h1}\omega }\ge 0, \; h\in H,\; \omega \in \Omega \\&\quad 0\le r_ix_{i}^{max}-(x_{it\omega }^+-x_{it\omega }^-)+(x_{it-1\omega }^ +-x_{it-1\omega }^-)\perp \bar{\mu }_{it\omega }\ge 0, \; \\&\qquad t\in T_h\backslash \{t_{h1}\},\; h\in H,\; \omega \in \Omega \\&\quad 0\le r_ix_{i}^{max}+(\tau x_{ih}+x_{it_{h1}\omega }^+-x_{i1\omega }^-)\\&\qquad -(\tau x_{ih-1}+x_{it_{h-1T}\omega }^+-x_{it_{h-1T}\omega }^-)\perp \underline{\mu }_{it_{h1}\omega }\ge 0, \; h\in H,\; \omega \in \Omega \\&\quad 0\le r_ix_{i}^{max}+(x_{it\omega }^+-x_{it\omega }^-)-(x_{it-1\omega }^+ -x_{it-1\omega }^-)\perp \underline{\mu }_{it\omega }\ge 0, \; \\&\qquad t\in T_h\backslash \{t_{h1}\},\; h\in H,\; \omega \in \Omega \end{aligned}$$

where $x_{ih-1}, x_{it_{h-1T}\omega }^+, x_{it_{h-1T}\omega }^-$ are given parameters for $h=h_1$. The adaptation of the KKT-conditions of the closed-loop equilibrium with ramping is similar.