Application of Supply Function Equilibrium Model to Describe the Interaction of Generation Companies in the Electricity Market

Abstract

The paper studies the trade in the spot electricity market based on submitting bids of energy consumers and producers to the market operator. We investigate supply function equilibrium (SFE) model, in which generation capacities are integrated into large generation companies that have a common purpose of maximizing their profits. For this case we prove the existence and uniqueness of equilibrium for a linear function of aggregate demand and quadratic costs. The mechanism is tested on the basis of the Siberian electric power system, Russia.

Appendix: Proof of the Proposition

Let a generation company have two capacities; $G=\left\{ l,\,\,\, k\right\}$. Then the profit function is:

$$ \pi _{l+k}\left( q_{k},\; q_{l,}\; q_{-(k+l)},\; p\right) =p\cdot q_{k}+p\cdot q_{l}-C_{k}\left( q_{k}\right) -C_{l}\left( q_{l}\right) . $$

Since the company redistributes the output (a residual demand) inside according to the condition of cost optimization it is necessary to equate the marginal revenue (which in this case is the same for any sold unit of commodity) to the marginal costs. For two capacities this is:

$$ \left\{ \begin{array}{c} \frac{\partial \pi _{k+l}}{\partial q_{k}}=\frac{\partial p}{\partial q_{k}}\left( q_{k}+q_{l}\right) -MC_{k}\left( q_{k}\right) =0,\\ \frac{\partial \pi _{k+l}}{\partial q_{l}}=\frac{\partial p}{\partial q_{l}}\left( q_{k}+q_{l}\right) -MC_{l}\left( q_{l}\right) =0. \end{array}\right. $$

Hence, the main condition is: $MC_{k}\left( q_{k}\right) =MC_{l}\left( q_{l}\right) .$ The impact of each unit of output of companies on the market price is equivalent to $\frac{\partial p}{\partial q_{l}}=\frac{\partial p}{\partial q_{k}}$. For the linear supply functions this condition is:

$$ a_{k}+c_{k}\beta _{k}\left( p-\alpha _{k}\right) =a_{l}+c_{l}\cdot \beta _{l}\left( p-\alpha _{l}\right) . $$

Then

$$\begin{aligned} \beta _{k}=\frac{\beta _{l}\cdot c_{l}}{c_{k}},\,\,\,\,\,\,\,\alpha _{k}=\alpha _{l}+\frac{a_{k}-a_{l}}{\beta _{l}\cdot c_{l}}. \end{aligned}$$

(12)

Using (12), the supply function can be written through the costs of the capacities

$$ q_{k}+q_{l}=\beta _{l}\left( 1+\frac{c_{l}}{c_{k}}\right) \cdot p-\beta _{l}\left( 1+\frac{c_{l}}{c_{k}}\right) \alpha _{l}-\frac{a_{k}-a_{l}}{c_{k}}. $$

Hence the linear supply function, submitted by the generation company to operator, is:

$$ q_{k+l}=\beta _{g}\left( p-\alpha _{g}\right) ,\,\,\, $$

where

$$ \alpha _{g}=\alpha _{l}+\frac{a_{k}-a_{l}}{\beta _{l}\cdot \left( c_{k}+c_{l}\right) },\,\,\,\,\,\beta _{g}=\beta _{l}\cdot \left( 1+\frac{c_{l}}{c_{k}}\right) . $$

The total costs of the generation company is the sum of costs of separate capacities:

$$ TC\left( q_{l}+q_{k}\right) =a_{k}q_{k}+0.5\cdot c_{k}q_{k}^{2}+a_{l}q_{l}+0.5\cdot c_{l}q_{l}^{2}. $$

Substitute $q_{i}=\beta _{i}\left( p-\alpha _{i}\right) ,\, i\in \{l,k\}$, and (12):

$$ TC\left( q_{l}+q_{k}\right) = $$

$$ \left( 1+\frac{c_{l}}{c_{k}}\right) \cdot \beta _{l}\left( p-\alpha _{l}\right) -0.5\cdot c_{l}\left( 1+\frac{c_{l}}{c_{k}}\right) \left( \beta _{l}\left( p-\alpha _{l}\right) \right) ^{2}\alpha _{l}-\frac{\left( a_{k}-a_{l}\right) \left( a_{k}+a_{l}\right) }{2c_{k}}. $$

Using the general form of the cost function

$$ TC\left( q_{g}\right) =a_{g}\beta _{g}\left( p-\alpha _{g}\right) +0.5\cdot c_{g}\left( \beta _{g}\left( p-\alpha _{g}\right) \right) ^{2}, $$

we obtain a system of equations from which it is possible to determine the required parameters for the aggregate cost function of the generation company:

$$ \left\{ \begin{array}{l} a_{g}-c_{g}\frac{a_{k}-a_{l}}{c_{k}}=1,\\ c_{g}=\frac{c_{k}\cdot c_{l}}{c_{k}+c_{l}},\\ 2a_{g}-c_{g}\frac{a_{k}-a_{l}}{c_{k}-c_{l}}=a_{k}+a_{l}. \end{array}\right. $$

Thus, we can reduce our problem to the problem with separate capacities. In this case, the competitors consider generating company with the supply function of the form $q_{k+l}=\beta _{g}\left( p-\alpha _{g}\right)$. The company problem is:

$$ \left\{ \begin{array}{l} \pi \left( q_{g},q_{-g},\; p\right) =p\cdot \beta _{g}\left( p-\alpha _{g}\right) -a_{g}\beta _{g}\left( p-\alpha _{G}\right) -0.5\cdot c_{g}\left( \beta _{g}\left( p-\alpha _{g}\right) \right) ^{2}\rightarrow \max \limits _{P};\\ a_{g}=a_{k}+a_{l}-1,\,\,\, c_{g}=\frac{c_{k}\cdot c_{l}}{c_{k}+c_{l}},\\ \alpha _{g}=\alpha _{l}+\frac{a_{k}-a_{l}}{\beta _{l}\cdot \left( c_{k}+c_{l}\right) }=\alpha _{k}+\frac{a_{l}-a_{k}}{\beta _{k}\cdot \left( c_{k}+c_{l}\right) },\\ \beta _{g}=\beta _{l}\cdot \left( 1+\frac{c_{l}}{c_{k}}\right) =\beta _{k}\cdot \left( 1+\frac{c_{k}}{c_{l}}\right) . \end{array}\right. $$

The profit function is concave, therefore the maximum is attained. As in [7], we prove the uniqueness of the positive solution for the coefficients $\beta _{1}\mathrm{,...,}\beta _{\mathrm{g}},...,\beta _{n}$. From the uniqueness of the solution and (7) it follows the uniqueness of a solution for the case with generation companies. The coefficients of the supply function is:

$q_{i}=\beta _{i}\left( P-\alpha _{i}\right) ,\,\,\,$ $\alpha _{i}=a_{i},\,\,\,\beta _{i}=\frac{1}{1+c_{i}\left( \gamma +\sum \limits _{j\ne i}\beta _{j}\right) },\,\,\, i\notin G$

$$ \alpha _{g}=a_{k}+a_{l}-1,\,\,\beta _{g}=\left( \left( 1+\frac{c_{l}\cdot c_{k}}{c_{l}+c_{k}}\left( \gamma +{\displaystyle \sum _{j\notin G}}\beta _{j}\right) \right) \right) ^{-1}. $$

Hence $q_{k}=\beta _{k}\left( p-\alpha _{k}\right) ,\,\,\, q_{l}=\beta _{l}\left( p-\alpha _{l}\right) ,$ where

$$ \beta _{l}=\left( \left( 1+\frac{c_{l}}{c_{k}}+c_{l}\left( \gamma +{\displaystyle \sum _{j\notin G}}\beta _{j}\right) \right) \right) ^{-1},\,\,\, l,\,\, k\in G, $$

$$ \beta _{k}=\left( \left( 1+\frac{c_{k}}{c_{l}}+c_{k}\left( \gamma +{\displaystyle \sum _{j\notin G}}\beta _{j}\right) \right) \right) ^{-1},\,\,\, l,\,\, k\in G, $$

$$ \alpha _{l}=a_{k}+a_{l}-1-\displaystyle \frac{a_{k}-a_{l}}{\beta _{l}\left( c_{l}+c_{k}\right) },\,\,\,\alpha _{k}=a_{k}+a_{l}-1-\displaystyle \frac{a_{l}-a_{k}}{\beta _{k}\left( c_{l}+c_{k}\right) }. $$

The special case considered above can be extended to the general one if the generation company has set G of individual capacities. Then the coefficients for the general supply function of the generation company are:

$$ \alpha _{g}=\sum _{k\in g}a_{k}-1,\,\,\beta _{g}=\displaystyle \left( \left( 1+\displaystyle \frac{1}{{\displaystyle \sum _{k\in G}}\displaystyle \frac{1}{c_{k}}}\left( \gamma +{\displaystyle \sum _{j\notin G}}\beta _{j}\right) \right) \right) ^{-1}. $$

From this we can obtain all coefficients for the supply functions of individual productions.

The Proposition is proved.