The optimal hourly electricity price considering wind electricity uncertainty based on conditional value at risk

ABSTRACT

Wind generation fluctuation results in the electricity supply uncertainty of the company in electricity market. And this uncertainty has an important impact on hourly electricity price. In our paper, we use game models to optimize hourly price for different risk appetites’ companies considering the uncertainty. For the risk-neutral company, we solve the optimal electricity price by maximizing the expected profit. For the risk-averse or risk-preferring company, we consider wind electricity generation as a Weibull distribution and obtain the optimal electricity price based on conditional value at risk. In numerical examples, we find that the risk-preferring company sets the highest electricity price, demonstrating the importance of modeling risk appetites of the company in electricity price. And the results indicate that a risk-averse (risk-preferring) company will set a higher (lower) electricity price for a larger risk confidence level. Besides, we find that the company will lower electricity price with the decrement of users’ price elasticity. Furthermore, our finding shows that the profit of the company increases with the wind penetration due to the decreased cost, which supports the observed and investigated results in electricity market.

KEYWORDS:

• Hourly electricity price
• wind electricity generation
• uncertainty
• game theory
• conditional value at risk

Proof of proposition 2

For the model for a risk-neutral company, the second-order derivative of the objective function with regard to $p_t$ is the same as the deterministic model.

By solving the following first-order derivative,

$\frac{\mathrm{\partial }{\pi }_t}{\mathrm{\partial }{p}_t}=2d_t{\left(\frac{p_t}{{\mu }_t}\right)}^{{\epsilon }_t}\left({\epsilon }_t+1\right)-E\left(g_t^w\right)-g_t^{con}=0$

the interior optimal price is

$p_t={\mu }_t{\left(\frac{E\left(g_t^w\right)+g_t^{con}}{2d_t\left({\epsilon }_t+1\right)}\right)}^{\frac{1}{{\epsilon }_t}}$

When the price cannot meet the constraint, the boundary optimal price can be written as

$p_t={\mu }_t{\left(\frac{E\left(g_t^w\right)+g_t^{con}}{d_t}\right)}^{\frac{1}{{\epsilon }_t}}$

Proof of proposition 3

For the model for a risk-averse company, it still has the same second-order derivative of the objective function with regard to $p_t$ as proposition 1.

The first-order derivative of the objective function with regard to $p_t$ can be written as

$\frac{\mathrm{\partial }CVa{R}_{{\beta }_t}^{{\pi }_t}}{\mathrm{\partial }{p}_t}=2d_t{\left(\frac{p_t}{{\mu }_t}\right)}^{{\epsilon }_t}\left({\epsilon }_t+1\right)-\frac{{\int }_{F^{-1}\left({\beta }_t\right)}^{\mathrm{\infty }}{g}_t^{w}f\left(g_t^w\right)}{1-{\beta }_t}-g_t^{con}=0$

Thus, the interior optimal price can be written as

$p_t={\mu }_t{\left(\frac{g_t^{con}{\beta }_t-g_t^{con}-{\int }_{F^{-1}\left({\beta }_t\right)}^{\mathrm{\infty }}{g}_t^{w}f\left(g_t^w\right)}{2d_t\left({\epsilon }_t+1\right)\left({\beta }_t-1\right)}\right)}^{\frac{1}{{\epsilon }_t}}$

When the price cannot meet the constraint, the boundary optimal price is

$p_t={\mu }_t{\left(\frac{F^{-1}\left(1-{\alpha }_t\right)+g_t^{con}}{d_t}\right)}^{\frac{1}{{\epsilon }_t}}$

Proof of proposition 4

For the model for a risk-preferring company, it also has the same second-order derivative of the objective function with regard to $p_t$ as proposition 1.

The first-order derivative of the objective function with regard to $p_t$ can be written as

The interior optimal price is

$p_t={\mu }_t{\left(\frac{g_t^{con}{\beta }_t+E\left(g_t^w\right)-{\int }_{F^{-1}\left({\beta }_t\right)}^{\mathrm{\infty }}{g}_t^{w}f\left(g_t^w\right)}{2d_t\left({\epsilon }_t+1\right){\beta }_t}\right)}^{\frac{1}{{\epsilon }_t}}$

When the price cannot meet the constraint, the boundary optimal price can be written as

$p_t={\mu }_t{\left(\frac{F^{-1}\left(1-{\alpha }_t\right)+g_t^{con}}{d_t}\right)}^{\frac{1}{{\epsilon }_t}}$

Additional information

Funding

This work was supported by the CityU Strategic Research Grant [No. 7005302]; General Research Fund project from the Research Grants Council of the Hong Kong Special Administrative Region, China [No. CityU 11215418].