Trading Stochastic Production in Electricity Pools

Abstract

Renewable electricity producers must trade in day-ahead electricity markets in the same manner as conventional producers. However, their power production may be highly unpredictable and nondispatchable. This is the case, for example, of wind and solar power producers, which thus need to use the balancing market to mend eventual deviations with respect to their day-ahead schedule. This chapter presents close formulae to determine the optimal offering strategy of stochastic producers in the day-ahead market. The analytical solution to these formulae is available under certain assumptions on the probabilistic structure characterizing power production and market prices. Stochastic programming is then introduced as a powerful mathematical framework to rid the solution to the trading problem for stochastic producers of these simplifying assumptions.

Exercises

7.1

A forecaster predicts that the energy production $\widetilde E$ of a wind farm during a given trading period t is characterized by the following probability density function $p_{\widetilde E}(\cdot)$ :

$$p_{\widetilde E}(E) =\begin{cases} \displaystyle \frac{E - 100}{625}, & 100 \leq E < 125, \\ \displaystyle \frac{150 - E}{625}, & 125 \leq E < 150,\\0, & \text{elsewhere}.\end{cases}$$

(7.56)

• 1. Determine the cumulative distribution function $F_{\widetilde E}(\cdot)$ for wind power production.

• 2. Assume that the imbalance penalties in a two-price market are ${\psi^{\mathrm{UP}}} = \$9$  /MWh and ${\psi^{\mathrm{DW}}} = \$4$ /MWh. Determine the optimal offer and the resulting expectation of the imbalance cost.

• 3. Now determine the optimal offer and the resulting expectation of the imbalance cost assuming that the imbalance penalties are ${\psi^{\mathrm{UP}}} = \$4$ /MWh and ${\psi^{\mathrm{DW}}} = \$9$ /MWh.

7.2

Consider the definition of $\widetilde u$ in Example 7.2. Let us assume that the probability density function for $\widetilde u$ is

$$p_{\widetilde u}(u) =\begin{cases} \displaystyle \frac{u + 20}{625}, & -20 \leq u < 5, \\ \displaystyle \frac{30 - u}{625}, & 5 \leq u < 30,\\0, & \text{elsewhere}.\end{cases}$$

(7.57)

• 1. Determine the expected value of the balancing price in a one-price market. If the forecast for wind power production is the same as in Exercise 7.1, what is the optimal offer?

• 2. Determine the expected values of the imbalance penalties ${\widehat{\psi}^{\mathrm{UP}}}$ and ${\widehat{\psi}^{\mathrm{DW}}}$ in a two-price market. If the forecast for wind power production is the same as in Exercise 7.1, what is the optimal offer?

7.3

In a two-price market, the expectation of the imbalance penalties, conditional on the day-ahead price, is given by the following.

$$\begin{aligned} {\mathbb E} \left\{{\widetilde{\psi}^{\mathrm{UP}}} | {\lambda^{\mathrm D}} \right\} =&10 - \frac{1}{10} {\lambda^{\mathrm D}},\end{aligned}$$

(7.58)

$$\begin{aligned} {\mathbb E} \left\{{\widetilde{\psi}^{\mathrm{DW}}} | {\lambda^{\mathrm D}} \right\} =&\frac{1}{8} {\lambda^{\mathrm D}}.\end{aligned}$$

(7.59)

Let us consider a uniform distribution for the stochastic production with lower and upper bounds equal to 100 MWh and 150 MWh, respectively. Determine the optimal offering curve, assuming that the day-ahead price is nonnegative and can take values up to $100/MWh.

7.4

Consider the distribution of wind power production in Exercise 7.1, and the deterministic penalties ${\psi^{\mathrm{UP}}} = \$9$ /MWh and ${\psi^{\mathrm{DW}}} = \$4$ /MWh. Determine the risk-averse bid with parameter $\alpha = 0.2$. Assuming a day-ahead price ${\lambda^{\mathrm D}} = \$50$ /MWh, determine the expected value of the profit, the value at risk (VaR $_{5\%}$) and the conditional value at risk (CVaR $_{5\%}$) of the profit.

7.5

Suppose that the energy production of a certain wind farm in a given trading period t can be modeled by a uniform distribution between 0 MWh and 50 MWh.

• 1. Construct a set of four equiprobable and uniformly spaced scenarios that approximates this uniform distribution.

• 2. Based on this scenario set and knowing that the imbalance penalties, ${\psi^{\mathrm{UP}}}$ and ${\psi^{\mathrm{DW}}}$, are deterministic and equal to $3/MWh and $6/MWh, respectively, formulate and solve a stochastic programming model to calculate the energy offer in the day-ahead market that minimizes the expected imbalance cost of the wind farm.

7.6

Reformulate the stochastic programming model of the previous exercise to account for the risk aversion of the wind power producer using the Conditional Value-at-Risk of its profit distribution at a confidence level of 99%. Then, obtain the optimal bid of the wind power producer in the day-ahead market as a function of a risk-aversion parameter and draw the resulting efficient frontier.

7.7

Consider a solar power producer that participates in an electricity market with the structure and time framework depicted in Fig. 7.13 of Example 7.9. The solar producer owns a 50-MW photovoltaic power plant whose energy output in time period t is stochastic and can be described as follows:

• The amount of energy ${E}_{t-\Delta t}$ produced by the solar power plant in time period $t-\Delta t$ may be high (45 MWh) with a probability of 0.7 or low (5 MWh) with a probability of 0.3.

• If the solar power production ${E}_{t-\Delta t}$ is high (45 MWh), the amount of energy E produced during the delivery period t may be extremely high (50 MWh) with a probability of 0.35 or relatively high (41 MWh) with a probability of 0.65.

• If the solar power production ${E}_{t-\Delta t}$ is low (5 MWh), the amount of energy E produced during the delivery period t may be extremely low (0 MWh) with a probability of 0.70 or relatively low (10 MWh) with a probability of 0.30.

Furthermore, the day-ahead and adjustment market prices, i.e., ${\lambda^{\mathrm D}}$ and ${\lambda^{\mathrm A}}$, are known to be equal to $45/MWh and $46/MWh, respectively, and the imbalance penalties, ${{\psi}^{\mathrm{UP}}}$ and ${{\psi}^{\mathrm{DW}}}$, equal to $10/MWh and $8/MWh, in that order.

• 1. Construct a scenario tree similar to that in Fig. 7.14 of Example 7.9 to describe the decision-making process faced by the solar power producer.

• 2. Compute the optimal bid of the solar power producer in the day-ahead market if the adjustment market is disregarded.

• 3. Compute the optimal bid of the solar power producer in the day-ahead market if the adjustment market is taken into account and calculate how much the expected profit of the solar power producer increases with respect to the previous case. Determine also how much of this increase is due to the fact that the future energy production of the photovoltaic power plant in period t is known with higher accuracy in the adjustment market than in the day-ahead market.

7.8

Consider a 60-MW solar power plant whose power output in a given trading period t can be either 35 and 60 MW, with probabilities 0.6 and 0.4, respectively. It is known that:

• The price in the day-ahead market for period t can be either $25/MWh or $50/MWh, with probabilities 0.3 and 0.7, in that order.

• If the day-ahead market price is equal to $25/MWh, the imbalance penalties for upward and downward balancing energy are $10 /MWh and $2/MWh, respectively.

• In contrast, if the day-ahead market price is equal to $50/MWh, these imbalance penalties take on the values $5/MWh and $4/MWh instead.

Use the information about offering curves provided in Sect. 8.4 of Chap. 8 (see, in particular, Example 8.9 in this chapter) to formulate and solve a stochastic programming model that computes, as an increasing function of the day-ahead electricity price, the optimal energy offer in the day-ahead market that maximizes the expected profit of the solar power producer.

Based on the comments and explanations in Sect. 7.4.2, justify the obtained solution.

7.9

Repeat Exercise 7.8 for the case that:

• If the day-ahead market price is $25/MWh, the imbalance penalties for upward and downward balancing energy are $5 /MWh and $4/MWh, respectively.

• If the day-ahead market price is equal to $50/MWh, the imbalance penalties are $10/MWh and $2/MWh instead.

7.10

Try to solve Exercises 7.8 and 7.9 analytically. Can both problems be solved this way? If not, motivate why.