The role of demand response in mitigating market power: a quantitative analysis using a stochastic market equilibrium model

Abstract

Market power is a dominant feature of many modern electricity markets with an oligopolistic structure, resulting in increased consumer cost. This work investigates how consumers, through demand response (DR), can mitigate against market power. Within DR, our analysis particularly focusses on the impacts of load shifting and self-generation. A stochastic mixed complementarity problem is presented to model an electricity market characterised by an oligopoly with a competitive fringe. It incorporates both energy and capacity markets, multiple generating firms and different consumer types. The model is applied to a case study based on data for the Irish power system. The results demonstrate how DR can help consumers mitigate against the negative effects of market power and that load shifting and self-generation are competing technologies, whose effectiveness against market power is similar for most consumers. We also find that DR does not necessarily reduce emissions in the presence of market power.

A. Karush-Kuhn-Tucker conditions

This appendix presents the Karush-Kuhn-Tucker (KKT) conditions for optimality for the two types of players modelled in this work. These conditions, along with the market clearing conditions (8), make up the mixed complementarity problem. The “perp” notation $0 \le a \perp b \ge 0$ is equivalent to $a\ge 0$, $b\ge 0$ and $a.b=0$.

1.1 A.1 Firms’ KKT conditions

The firms’ KKT conditions are

$$\begin{aligned} 0& \le \mathrm{gen}_{f, t,p,s} \perp -PR_{s} \bigg ( \gamma _{p,s}+X_{t} +PM_{f,t}\frac{\partial \gamma _{p,s}}{\partial \mathrm{gen}_{f, t,p,s}} \left (\sum _{{\bar{t}} \in T} \mathrm{gen}_{f, {\bar{t}},p,s} \right) \nonumber \\&\quad - \frac{\partial C^{\text {GEN}}_{f,t}}{\partial \mathrm{gen}_{f, t,p,s}} \bigg )+\lambda ^{1}_{f, t,p,s} \ge 0, \>\> \forall f,t,p,s, \end{aligned}$$

(10a)

$$\begin{aligned} 0\le \mathrm{exit}_{f, t} \perp - \mathrm{MTC}^{\text {GEN}}_{t}+\sum _{p,s}\mathrm{NORM}^{\text {G}}_{f,t,p,s}\lambda ^{1}_{f, t,p,s}+\lambda ^{2}_{f, t}\ge 0, \>\> \forall f,t, \end{aligned}$$

(10b)

$$\begin{aligned} 0\le cap_{f,t} \perp \mathrm{DRF}_{t}\times \kappa +\lambda ^{2}_{f, t}\ge 0, \>\> \forall f,t, \end{aligned}$$

(10c)

$$0 \le {\text{ }}\lambda _{{f,t,p,s}}^{1} \bot - {\text{gen}}_{{f,t,p,s}} + (\overline{{{\text{CAP}}_{{f,t}} }} - {\text{exit}}_{{f,t}} ) \times {\text{NORM}}_{{f,t,p,s}}^{G} \ge 0,\quad \forall \;f,t,p,s,{\text{ }}$$

(10d)

$$\begin{aligned} 0\le \lambda ^{2}_{f,t} \perp -cap_{f,t}+\overline{\mathrm{CAP}_{f,t}} - \mathrm{exit}_{f,t} \ge 0, \>\> \forall f,t. \end{aligned}$$

(10e)

If firm f is a price-making firm, then it’s optimal level of generation for technology t, using equation (10a), is

$$\begin{aligned} gen^{*}_{f, t,p,s} =\min \left[ \frac{ \gamma _{p,s}+X_{t}+ \frac{\partial \gamma _{p,s}}{\partial \mathrm{gen}_{f, t,p,s}} \big (\sum _{{\bar{t}} \in T \setminus \{t\}} \mathrm{gen}_{f, {\bar{t}},p,s}\big ) - \frac{\partial C^{\text {GEN}}_{f,t}}{\partial gen_{f, t,p,s}} }{\frac{\partial \gamma _{p,s}}{\partial \mathrm{gen}_{f, t,p,s}} } , \overline{\mathrm{CAP}_{f,t}} \right] . \end{aligned}$$

(11)

1.2 A.2 Consumers’ KKT conditions

The consumers’ KKT conditions are

$$\begin{aligned} 0&\le g^{\text {ls}}_{k,p,s} \perp -PR_{s} \left( \gamma _{p,s}- \frac{\partial C^{\text {LS}}_{k,p}}{\partial g^{\text {ls}}_{k,p,s}} \right) +\mu ^{1}_{k,p,s}\nonumber \\&\quad +\mu ^{8}_{k,p,s} \ge 0, \>\> \forall k,p,s, \end{aligned}$$

(12a)

$$\begin{aligned} 0& \le g^{\text {up}}_{k,p,s} \perp PR_{s} \gamma _{p,s}+\mu ^{2}_{k,p,s}+\sum ^{|H|}_{e=p-{\hat{p}}+1} \big ( \mu ^{6}_{k,{\hat{p}},e,s}\nonumber \\&\quad - \mu ^{7}_{k,{\hat{p}},e,s} \big )-\mu ^{8}_{k,p,s} \ge 0, \>\> \forall k,p,s, \end{aligned}$$

(12b)

$$\begin{aligned} 0 &\le g^{\text {down}}_{k,p,s} \perp -PR_{s} \gamma _{p,s}+\mu ^{3}_{k,p,s}-\sum ^{|H|}_{e=p-{\hat{p}}+1} \big ( \mu ^{6}_{k,{\hat{p}},e,s} \nonumber \\&\quad - \mu ^{7}_{k,{\hat{p}},e,s} \big )+\mu ^{8}_{k,p,s} \ge 0, \forall k,p,s, \end{aligned}$$

(12c)

where

$$\begin{aligned} {\hat{p}}&= \max \{p' | {\hat{p}} \le p \},\nonumber \\ 0\le & {} g^{\text {micro}}_{k,p,s} \perp -PR_{s} \left( \gamma _{p,s}- \frac{\partial C^{\text {MICRO}}_{k,p}}{\partial g^{\text {micro}}_{k,p,s}} \right) \nonumber \\&\quad +\mu ^{4}_{k,p,s}+\mu ^{8}_{k,p,s} \ge 0, \>\> \forall k,p,s, \end{aligned}$$

(12d)

$$\begin{aligned} 0& \le g^{\text {pv}}_{k,p,s} \perp -PR_{s} \left( \gamma _{p,s}+X^{\text {PV}}- C^{\text {PV}}_{k,p} \right) \nonumber \\&\quad +\mu ^{5}_{k,p,s}+\mu ^{8}_{k,p,s} \ge 0, \>\> \forall k,p,s, \end{aligned}$$

(12e)

$$\begin{aligned} 0\le & {} \mu ^{1}_{k,p,s} \perp -g^{\text {ls}}_{k,p,s}+G_{k}^{\text {LS,MAX}} \ge 0 \>\> \forall k,p,s, \end{aligned}$$

(12f)

$$\begin{aligned} 0\le \mu ^{2}_{k,p,s} \perp -g^{\text {up}}_{k,p,s}+FAC_{k}^{\text {STOR}}\times \mathrm{INT}^{\text {STOR}}_{k} \ge 0 \>\> \forall k,p,s, \end{aligned}$$

(12g)

$$\begin{aligned} 0\le \mu ^{3}_{k,p,s} \perp -g^{\text {down}}_{k,p,s}+\mathrm{FAC}_{k}^{\text {STOR}}\times INT^{\text {STOR}}_{k}\ge 0 \>\> \forall k,p,s, \end{aligned}$$

(12h)

$$\begin{aligned} 0\le \mu ^{4}_{k,p,s} \perp -g^{\text {micro}}_{k,p,s}+INT^{\text {MICRO}}_{k} \ge 0 \>\> \forall k,p,s, \end{aligned}$$

(12i)

$$\begin{aligned} 0\le \mu ^{5}_{k,p,s} \perp -g^{\text {pv}}_{k,p,s}+ \mathrm{INT}^{\text {PV}}_{k}\times \mathrm{NORM}^{\text {PV}}_{p,s} \ge 0 \>\> \forall k,p,s, \end{aligned}$$

(12j)

$$\begin{aligned} 0\le \mu ^{6}_{k,p',h,s} \perp -\sum ^{p'+h-1}_{e=p'}\big ( g^{\text {up}}_{k,e,s}-g^{\text {down}}_{k,e,s} \big )\nonumber \\&+\mathrm{INT}^{\text {STOR}}_{k} \ge 0 \>\> \forall k,p',s,h, \end{aligned}$$

(12k)

$$\begin{aligned} 0\le \mu ^{7}_{k,p',h,s} \perp \sum ^{p'+h-1}_{e=p'}\big ( g^{\text {up}}_{k,e,s}-g^{\text {down}}_{k,e,s} \big ) \ge 0 \>\> \forall k,p',s,h, \end{aligned}$$

(12l)

$$\begin{aligned} 0& \le \mu ^{8}_{k,p,s} \perp -g^{\text {ls}}_{k,p,s}-g^{\text {down}}_{k,p,s}-g_{k,p,s}^{\text {micro}}- g^{\text {pv}}_{k,p,s}\nonumber \\&\quad + D^{\text {REF}}_{k,p}+ g^{\text {up}}_{k,p,s}\ge 0 \>\> \forall k,p,s. \end{aligned}$$

(12m)