4.1.1.1 Electricity Market Participants and Operators

The participants in an electricity market can be divided into three categories

With so many participants in the electricity market, a neutral entity is required to establish sound market rules to guarantee the functioning of the economy and the processing of secure transactions. In the United States, independent system operators (ISOs) or regional transmission organizations (RTOs) are used to fulfill this role. The ISOs or RTOs administer transmission tariffs, maintain system security levels, coordinate and schedule maintenance, and play an important role in coordinating long-term planning. In the electricity market, the ISOs/RTOs play an important role as independent, unaffiliated market operators providing unbiased, open access to all market participants. The ISOs/RTOs also forecast electricity demand, schedule adequate reserve, run market clearing models, and send price signals to all of the market participants.

4.1.1.2 Time Frame of the Short-Term Electricity Market

The time framework for the short-term electricity market is shown in Fig. 4.1. The DA market, as its name implies, refers to the market that operates 24 h prior to the RT market. It is used for scheduling the power production for the next day. The reserve market operates almost at the same time as the DA market. In most US markets, the DA market and reserve market are cooptimized by using a single clearing process, as determined by the ISOs/RTOs, from which the energy and reserve transactions are determined during each hour of the next operating day. Once the markets are cleared, the locational marginal prices (LMP), the reserve clearing prices, and the cleared energy volume of each participant are settled. The RT market is carried out just minutes before the actual power delivery by energy owners to ensure a RT balance between generation and demand. This is done by offsetting the differences between the RT operation and the corresponding energy program settled in the DA market. RT prices are calculated for the RT market based on the operating conditions.

4.1.1.3 Energy Owners in the Electricity Market

In the short-term electricity market, the energy owners submit nondecreasing stepwise offering curves and the customers submit nonincreasing stepwise bidding curves indicating their willingness to sell and buy certain amounts of energy or reserve at certain prices, respectively. An illustration of an energy owner’s offering curve and a customer’s bidding curve is shown in Fig. 4.2. The ISOs/RTOs aggregate all of the bidding and offering curves, and determine the market clearing prices and power quantity committed from each energy owner and customer. As shown in Fig. 4.2, if the market clearing price is λ, the committed power of the energy owner is P. Failing to fulfill their commitments brings penalties to the energy owners. The RT penalty mechanisms are not the same in the electricity markets of different countries. For example, in most European countries, two RT prices are determined separately in two different situations: a positive RT price for positive energy deviation (higher production or lower consumption than scheduled) and a negative RT price for negative energy deviation (lower production or higher consumption than scheduled).

4.2.4.1 Basic Mathematical Formation for the Price-Taker Renewable Energy Owner

The basic mathematical formulation for the price-taker renewable energy owner in DA and RT market are expressed in the following mathematical problem (4.4):

$\begin{array}{c}\underset{W_{t\omega }^D,{\mathrm{\Delta }}_{t\omega }^{+},{\mathrm{\Delta }}_{t\omega }^{-},{\mathrm{\Delta }}_{t\omega },\zeta ,{\eta }_{\omega }}{\max }{\pi }_T=\sum _{\omega =1}^{N_{\mathrm{\Omega }}}{\mathit{pr}}_{\omega }{\sum }_{t=1}^{N_T}\left[{\lambda }_{t\omega }^D{W}_{t\omega }^D+{\lambda }_{t\omega }^r\left({\mathrm{\Delta }}_{t\omega }^{+}-{\mathrm{\Delta }}_{t\omega }^{-}\right)\right]\\ +{\beta }_w(\zeta -\frac{1}{1-\alpha }{\sum }_{\omega =1}^{N_{\mathrm{\Omega }}}{\mathit{pr}}_{\omega }{\eta }_{\omega })\end{array}$ (4.4.1)

(4.4.1)

Subject to

$0\le W_{t\omega }^D\le W^{\mathit{max}},\forall t,\omega$ (4.4.2)

(4.4.2)

${\mathrm{\Delta }}_{t\omega }^{+}-{\mathrm{\Delta }}_{t\omega }^{-}={\mathrm{\Delta }}_{t\omega },\forall t,\omega$ (4.4.3)

(4.4.3)

${\mathrm{\Delta }}_{t\omega }=W_{t\omega }^{\mathit{ac}}-W_{t\omega }^D,\forall t,\omega$ (4.4.4)

(4.4.4)

${0\le \mathrm{\Delta }}_{t\omega }^{+}\le W_{t\omega }^{\mathit{ac}},\forall t,\omega$ (4.4.5)

(4.4.5)

${0\le \mathrm{\Delta }}_{t\omega }^{-}\le W_{t\omega }^D,\forall t,\omega$ (4.4.6)

(4.4.6)

$W_{t\omega }^D=W_{t{\omega }^{\prime }}^D,\forall t,\omega ,{\omega }^{\prime }:{\lambda }_{t\omega }^D={\lambda }_{t{\omega }^{\prime }}^D$ (4.4.7)

(4.4.7)

$\left({\lambda }_{t\omega }^D-{\lambda }_{t{\omega }^{\prime }}^D\right)\left(W_{t\omega }^D-W_{t{\omega }^{\prime }}^D\right)\ge 0,\forall t,\omega ,{\omega }^{\prime }$ (4.4.8)

(4.4.8)

${\eta }_{\omega }\ge 0,\forall \omega$ (4.4.9)

(4.4.9)

$\zeta -{\sum }_{t=1}^{N_T}({\lambda }_{t\omega }^D{W}_{t\omega }^D+{\lambda }_{t\omega }^r{\mathrm{\Delta }}_{t\omega }^{+}-{\lambda }_{t\omega }^r{\mathrm{\Delta }}_{t\omega }^{-})\le {\eta }_{\omega },\forall \omega$ (4.4.10)

(4.4.10)

where $W_{t\omega }^D$, ${\mathrm{\Delta }}_{t\omega }^{+}$, ${\mathrm{\Delta }}_{t\omega }^{-}$,${\mathrm{\Delta }}_{t\omega }$, $\zeta$, and ${\eta }_{\omega }$ are the decision variables. $W_{t\omega }^D$represents the power offered by the renewable energy owner in the DA market at time period t. ${\mathrm{\Delta }}_{t\omega }^{+}$, ${\mathrm{\Delta }}_{t\omega }^{-}$, and ${\mathrm{\Delta }}_{t\omega }$ represent the positive deviation, negative deviation, and total deviation of the renewable energy owner in the RT market at time period t.

The objective function comprises two terms: (1) The expected profit of the renewable energy owner, which equals the revenue from the DA market plus the revenue from positive energy deviations in the RT market minus the cost for negative energy deviations in the RT market; and (2) the CVaR multiplied by a weighting factor β_W, which controls the degree of risk aversion of the renewable energy owner.

In this model, the first-stage decision variable is the hourly bid of the energy volume $W_{t\omega }^D$ of the renewable energy owner, whereas the second-stage variables are the RT energy deviations ${\mathrm{\Delta }}_{t\omega }^{+}$ and ${\mathrm{\Delta }}_{t\omega }^{-}$ of the renewable energy owner. Constraint (4.4.2) limits the amount of wind energy that can be traded in the DA market. Constraints (4.4.3)–(4.4.6) determine the total positive and negative energy deviations incurred by the wind energy owner per period and scenario. Constraint (4.4.7) constitutes the nonanticipativity conditions related to the decisions made in the first-stage. Constraint (4.4.8) enforces a nondecreasing offer curve. Constraints (4.4.9) and (4.4.10) are used to compute CVaR.

4.2.4.2 Examples

In this section, we will present the example to demonstrate the effectiveness of the proposed model in the previous section. The historical data speed and market prices are obtained from PJM website [11]. The maximum wind generating capacity is set to be 300 MW. The ARIMA model is used to generate 5000 scenarios for wind power production, DA, respectively. Then, the scenario reduction method is applied to reduce the number of scenarios for each uncertainty to 5. Table 4.2 shows the information of these scenarios. The risk control parameter is set to be 0.2.

By solving problem (4.4), the bidding strategy capacity into DA market for each scenario is $W_1^D=127.43$, $W_2^D=126.80$,$W_3^D=252.42$,$W_4^D=268.17$,$W_5^D=300$. Using these solved bidding capacities and their corresponding DA LMP, the readers can build the nondecreasing bidding curve shown in Fig. 4.5.

The risk management control parameter can be used to control the risk associated with uncertainties. As we increase β_W, the value of CVaR increases; but the expected profit decreases as shown in Figs. 4.6 and 4.7. This is as expected, the energy owner increases β_W in order to reduce the trading risks, thus, the cost associated with risks increase and the expected profit decreases.

4.2.5.1 Introduction

To mitigate the trading risk associated with renewable energy trading, many approaches have been discussed and applied, such as combining renewable energy units with energy storage system, signing bilateral contract, and others. In this chapter, we will introduce the approach to mitigate the risk for bidding in the electricity market by buying additional amount of power from conventional energy owner and use it as “reserve” to hedge against the RT market penalties.

In Ref. [14], the authors introduced a bilateral reserve settlement mechanism between renewable energy owners and conventional energy owners. The authors proposed several assumptions: (1) The bilateral reserve settlement introduced can be mixed with the system-wide reserve to provide standby power to cover the intermittency and uncertainty of renewable energy sources. The bilateral reserve is provided by conventional energy owners and consumed by renewable energy owners. (2) The bilateral reserve settlement among renewable and conventional energy owners can be seen as a new trading mechanism for a new type of reserve, adding to the existing system-wide reserve. The bilateral reserve settlement price and volumes of specific providers are cleared among the renewable and conventional energy owners who provide this new type of reserve. This new bilateral trading mechanism does not change current implementation of the system-wide reserve, regulation, and other auxiliary services for mitigating other uncertainties, for example, large load uncertainties, in the system.

Both renewable and conventional energy owners try to compete with other energy owners and maximize their profit in the market. The bilateral reserve price and power amount settled between these two types of energy owners can be solved using game theory. The optimal bidding strategy of both energy owners can be modeled using stochastic programming introduced in the previous section.

4.2.5.2 Auction Game and Nash Equilibrium

This section gives an introduction of game theory, Nash Equilibrium, and the application settle the bilateral reserve price and power amount between conventional and renewable energy owners.

A game is a “formal representation of a situation in which a number of individuals interact in a setting of strategic independence [15].” There are four elements in a game: (1) The players, (2) the rules of the game, (3) the outcomes, and (4) the payoff and preferences (utility functions) of the players. A game can be either cooperative, where the players collaborate to achieve a common goal, or noncooperative, where they act on their own.

In this section, different energy owners are the game players. Each player has the historical information of other players’ past actions. A strategy is a rule that tells the players which action(s) they should take. Assuming that the players are noncooperative, know the payoff functions of other players, and try to maximize their payoff functions while considering their rivals’ bidding strategies, the Nash equilibrium [15] occurs when no player has the incentive to change its offering/bidding strategy.

In order to benefit from participating in the bilateral reserve market, renewable energy owners will not buy reserve from the bilateral reserve market if the reserve clearing price exceeds the RT price, whereas the conventional energy owners will not offer reserve in the bilateral reserve market if the RT price is lower than the DA price. This means that the offer price ${\lambda }_t^R$ for reserve at a certain time t can take any value between the DA price ${\lambda }_t^D$ and whichever has the higher value: the RT price ${\lambda }_t^r$ or the energy price cap c^cap specified by the market operator.

${\lambda }_t^D\le {\lambda }_t^R\le \max \mathrm{}({\lambda }_t^r,c^{\mathit{cap}})$ (4.5)

(4.5)

A renewable energy owner can bid reserve energy provided by a conventional energy owner at any value between zero and the predicted power imbalance during the market operation.

$0\le W_t^R\le {\Delta }_t^{-}$ (4.6)

(4.6)

where $W_t^R$ represents the power bought by the wind energy owner in the bilateral reserve market and ${\Delta }_t^{-}$ is the negative deviation of wind power in time period t.

Also, a conventional energy owner can offer reserve energy of its unit g to a wind energy owner at any value between zero and the maximum power output of the unit.

$0\le P_{\mathit{gt}}^R\le P_g^{\mathit{max}}$ (4.7)

(4.7)

where $P_{\mathit{gt}}^R$ represents the power offered by the conventional unit g in the bilateral reserve market at time period t and $P_g^{\mathit{max}}$ is the maximum power output of the conventional unit g.

Let ${\gamma }_i$ denote player i’s strategy and ${\gamma }_{-i}$ denote other players’ strategies. Player i’s total profit is π_i, which includes the revenue from both energy and bilateral reserve markets and can be obtained by solving a two-stage stochastic programming problem.

Let Г_i be the set of continuous strategies of player i. For a continuous game, a strategy tuple ${\{{\gamma }_i^{*}\}}_{i=1\mathrm{to}\mathit{I}}$ is a Nash equilibrium if the following equilibrium condition is satisfied [17] for all continuous strategies ${\gamma }_i$s, where i=1,…, I:

${\pi }_i\left({\gamma }_i^{*},{\gamma }_{-i}^{*}\right)\ge {\pi }_i\left({\gamma }_i,{\gamma }_{-i}^{*}\right),\forall {\gamma }_i\in {\mathrm{\Gamma }}_i,\forall i$ (4.8)

(4.8)

The continuous equilibriums are difficult to obtain because the payoff function π has no explicit formula. To simplify the solution process, the continuous strategy set of player i, Г_i, is appropriately discretized into N_i choices; then the set of the resulting discrete strategies of player i can be written as Ω_i={γ_i,n, n=1,…, N_i}. As player i has N_G generating units and each unit can have N_ig discrete strategies, N_i can be expressed as $N_i={\prod }_{g=1}^{N_G}{N}_{\mathit{ig}}$. Moreover, as there are total I players, a game can then be formed with a total number of $N={\prod }_{i=1}^I{\prod }_{g=1}^{N_G}{N}_{\mathit{ig}}$ strategy tuples. The Nash solution can then be searched among the N strategy tuples. Similar to (3.4), is ${\{{\gamma }_i^{*}\}}_{i=1\mathrm{to}\mathit{I}}$, a Nash equilibrium for the matrix game if the following discrete equilibrium condition is satisfied.

${\pi }_i\left({\gamma }_i^{*},{\gamma }_{-i}^{*}\right)\ge {\pi }_i\left({\gamma }_{i,n},{\gamma }_{-i}^{*}\right),\forall {\gamma }_{i,n}\in {\mathrm{\Omega }}_i,\forall i$ (4.9)

(4.9)

4.2.5.3 Mathematical Model

In this section, we will present the stochastic model to obtain the optimal bidding strategies for both conventional energy owners and renewable energy owners.

1. Conventional energy owner model
The conventional energy owners, such as thermal and hydro power plants, can control their power outputs if no generator failure is considered. The problem (4.10) of obtaining the best bidding strategy for conventional power energy owners is formulated as a two-stage stochastic program to maximize the profit of a thermal energy owner

${\max }_{P_{\mathit{gt}\omega }^D,P_{\mathit{gt}\omega }^r,P_{\mathit{gt}\omega }^{\mathit{ac}},\zeta ,{\eta }_{\omega },u_{\mathit{gt}}}{\pi }_T=\sum _{\omega =1}^{N_{\mathrm{\Omega }}}{\mathit{pr}}_{\omega }{\sum }_{t=1}^{N_T}{\sum }_{g=1}^{N_G}\left[{\lambda }_{t\omega }^D{P}_{\mathit{gt}\omega }^D+{\lambda }_{t\omega }^r{P}_{\mathit{gt}\omega }^r+{\lambda }_t^R{P}_{\mathit{gt}}^R-C_g\left(P_{\mathit{gt}\omega }^D+P_{\mathit{gt}\omega }^r+P_{\mathit{gt}}^R\right)-\max \mathrm{}(0,{\mathit{StUP}}_{\mathit{gt}}(u_{\mathit{gt}}-u_{g(t-1)}\left)\right)\right]+{\beta }_T(\zeta -\frac{1}{1-\alpha }{\sum }_{\omega =1}^{N_{\mathrm{\Omega }}}{\mathit{pr}}_{\omega }{\eta }_{\omega })$ (4.10.1)

(4.10.1)

Subject to:

$u_{\mathit{gt}}{P}_g^{\mathit{min}}\le P_{\mathit{gt}\omega }^D+P_{\mathit{gt}\omega }^r+P_{\mathit{gt}}^R\le u_{\mathit{gt}}{P}_g^{\mathit{max}},\forall t,g,\omega$ (4.10.2)

(4.10.2)

$u_{\mathit{gt}}{P}_g^{\mathit{min}}\le P_{\mathit{gt}\omega }^D\le u_{\mathit{gt}}{P}_g^{\mathit{max}},\forall t,g,\omega$ (4.10.3)

(4.10.3)

$u_{\mathit{gt}}{P}_g^{\mathit{min}}\le P_{\mathit{gt}\omega }^r\le u_{\mathit{gt}}{P}_g^{\mathit{max}},\forall t,g,\omega$ (4.10.4)

(4.10.4)

$P_{\mathit{gt}\omega }^{\mathit{ac}}{P}_{\mathit{gt}\omega }^D+P_{\mathit{gt}\omega }^r+P_{\mathit{gt}}^R\le u_{\mathit{gt}}{P}_g^{\mathit{max}},\forall t,g,\omega$ (4.10.5)

(4.10.5)

$P_{\mathit{gt}\omega }^{\mathit{ac}}-P_{g\left(t-1\right)\omega }^{\mathit{ac}}\le {\mathit{RU}}_g,\forall t,g,\omega$ (4.10.6)

(4.10.6)

$P_{g(t-1)\omega }^{\mathit{ac}}-P_{\mathit{gt}\omega }^{\mathit{ac}}\le {\mathit{RD}}_g,\forall t,g,\omega$ (4.10.7)

(4.10.7)

$C_g\left(P\right)=a_g+b_{g}P+c_g{P}^2$ (4.10.8)

(4.10.8)

$\left({\lambda }_{t\omega }^D-{\lambda }_{t{\omega }^{\prime }}^D\right)\left(P_{\mathit{gt}\omega }^D-P_{\mathit{gt}{\omega }^{\prime }}^D\right)\ge 0,\forall t,g,\omega ,{\omega }^{\prime }$ (4.10.9)

(4.10.9)

$P_{\mathit{gt}\omega }^D=P_{\mathit{gt}{\omega }^{\prime }}^D,\forall t,g,\omega ,{\omega }^{\prime }\mathrm{:}{\lambda }_{t\omega }^D={\lambda }_{t{\omega }^{\prime }}^D$ (4.10.10)

(4.10.10)

${\eta }_{\omega }\ge 0,\forall \omega$ (4.10.11)

(4.10.11)

$\zeta -{\sum }_{t=1}^{N_T}{\sum }_{g=1}^{N_G}{d}_t\left[{\lambda }_{t\omega }^D{P}_{\mathit{gt}\omega }^D+{\lambda }_{t\omega }^r{P}_{\mathit{gt}\omega }^r+{\lambda }_t^R{P}_{\mathit{gt}}^R-C_g\left(P_{\mathit{gt}\omega }^D+P_{\mathit{gt}\omega }^r+P_{\mathit{gt}}^R\right)-\max \left(0,{\mathit{StUP}}_{\mathit{gt}}\left(u_{\mathit{gt}}-u_{g\left(t-1\right)}\right)\right)\right]\le {\eta }_{\omega },\forall \omega$ (4.10.12)

(4.10.12)

where $P_{\mathit{gt}}^D,P_{\mathit{gt}}^r,P_{\mathit{gt}}^{\mathit{ac}},\zeta ,\eta ,u_{\mathit{gt}}$ are the decision variables. $P_{\mathit{gt}}^D$ and $P_{\mathit{gt}}^r$ represent the power offered by the conventional unit g in the DA and RT market at time period t, respectively. $P_{\mathit{gt}}^{\mathit{ac}}$ represents the total actual power output of the conventional unit g at time period t. $u_{\mathit{gt}}$ is the binary variable which represents the state of the conventional unit g in time period t ($\mu =1$means ON and $\mu =0$ means OFF). $\zeta$ and $\eta$ are auxiliary variables used to compute the CVaR.
The random variables include ${\lambda }_t^D$ and ${\lambda }_t^r$ which represent the DA and RT market prices at time period t, respectively. The variables, if augmented with a subscript $\omega$, represent their realization in a scenario $\omega$.
The constants and parameters are ${\mathit{pr}}_{\omega }$, $P_g^{\mathit{min}}$, $P_g^{\mathit{max}}$, ${\mathit{RU}}_g$, ${\mathit{RD}}_g$, $a_g$, $b_g$, $c_g$, $\alpha$, and ${\beta }_T$, where ${\mathit{pr}}_{\omega }$ is the probability of a scenario $\omega$, $P_g^{\mathit{min}}$ and $P_g^{\mathit{max}}$ are the minimum and maximum power outputs of the conventional unit g, respectively, ${\mathit{RU}}_g$ and ${\mathit{RD}}_g$ are the ramp-up and ramp-down rates of the conventional unit g, respectively, $a_g$, $b_g$, and $c_g$ represent the thermal heat rate curve parameters of the conventional unit g, $\alpha$ is the per-unit confidence level, and ${\beta }_T$ is the risk aversion parameter of the conventional power energy owner.
The objective function (4.10.1) comprises two terms: (1) The expected profit, which equals the revenues from the DA, RT, and bilateral reserve markets minus the production and start-up costs; and (2) the CVaR multiplied by a weighting factor β_T, which allows the degree of risk aversion of the conventional power energy owner to be controlled.
In this model, the first-stage decision variable is the hourly bid of the energy volume $P_{\mathit{gt}\omega }^D$ of the thermal units while the second-stage decision variable is the RT output $P_{\mathit{gt}\omega }^r$ of the thermal units. Constraints (4.10.2)–(4.10.4) bind the maximum power capacity of each thermal unit. The actual total power generated by each thermal unit is expressed in constraint (4.10.5). Constraints (4.10.6) and (4.10.7) represent the ramp-up and ramp-down limits of each thermal unit, respectively. The production cost C_g of each thermal unit is expressed as a quadratic constraint (4.10.8). Constraint (4.10.9) enforces a nondecreasing offer curve. Constraint (4.10.10) constitutes the nonanticipativity conditions related to the decisions made in the first-stage. Constraints (4.10.11) and (4.10.12) are used to compute CVaR.

2. Renewable energy owner model
The model (4.11) to maximize the profit of a renewable energy owner is

${\max }_{W_{t\omega }^D,{\mathrm{\Delta }}_{t\omega }^{+},{\mathrm{\Delta }}_{t\omega }^{-},{\mathrm{\Delta }}_{t\omega },\zeta ,{\eta }_{\omega }}{\pi }_T=\sum _{\omega =1}^{N_{\mathrm{\Omega }}}{\mathit{pr}}_{\omega }{\sum }_{t=1}^{N_T}\left[{\lambda }_{t\omega }^D{W}_{t\omega }^D+{\lambda }_{t\omega }^r({\mathrm{\Delta }}_{t\omega }^{+}-{\lambda }_{t\omega }^r{\mathrm{\Delta }}_{t\omega }^{-}-{\lambda }_t^R{W}_t^R\right]+{\beta }_w(\zeta -\frac{1}{1-\alpha }{\sum }_{\omega =1}^{N_{\mathrm{\Omega }}}{\mathit{pr}}_{\omega }{\eta }_{\omega })$ (4.11.1)

(4.11.1)

Subject to

$0\le W_{t\omega }^D\le W^{\mathit{max}},\forall t,\omega$ (4.11.2)

(4.11.2)

${\mathrm{\Delta }}_{t\omega }^{+}-{\mathrm{\Delta }}_{t\omega }^{-}={\mathrm{\Delta }}_{t\omega },\forall t,\omega$ (4.11.3)

(4.11.3)

${\mathrm{\Delta }}_{t\omega }=W_{t\omega }^{\mathit{ac}}+W_t^R-W_{t\omega }^D,\forall t,\omega$ (4.11.4)

(4.11.4)

${0\le \mathrm{\Delta }}_{t\omega }^{+}\le W_{t\omega }^{\mathit{ac}}+W_t^R,\forall t,\omega$ (4.11.5)

(4.11.5)

${0\le \mathrm{\Delta }}_{t\omega }^{-}\le W_{t\omega }^D,\forall t,\omega$ (4.11.6)

(4.11.6)

$W_{t\omega }^D=W_{t{\omega }^{\prime }}^D,\forall t,\omega ,{\omega }^{\prime }:{\lambda }_{t\omega }^D={\lambda }_{t{\omega }^{\prime }}^D$ (4.11.7)

(4.11.7)

$\left({\lambda }_{t\omega }^D-{\lambda }_{t{\omega }^{\prime }}^D\right)\left(W_{t\omega }^D-W_{t{\omega }^{\prime }}^D\right)\ge 0,\forall t,\omega ,{\omega }^{\prime }$ (4.11.8)

(4.11.8)

${\eta }_{\omega }\ge 0,\forall \omega$ (4.11.9)

(4.11.9)

$\zeta -{\sum }_{t=1}^{N_T}({\lambda }_{t\omega }^D{W}_{t\omega }^D+{\lambda }_{t\omega }^r{\mathrm{\Delta }}_{t\omega }^{+}-{\lambda }_{t\omega }^r{\mathrm{\Delta }}_{t\omega }^{-}-{\lambda }_t^R{W}_t^R{d}_t)\le {\eta }_{\omega },\forall \omega$ (4.11.10)

(4.11.10)

where $W_{t\omega }^D$, ${\mathrm{\Delta }}_{t\omega }^{+}$, ${\mathrm{\Delta }}_{t\omega }^{-}$,${\mathrm{\Delta }}_{t\omega }$, $\zeta$, and ${\eta }_{\omega }$ are the decision variables. $W_{t\omega }^D$ represents the power offered by the renewable energy owner in the DA market at time period t. ${\mathrm{\Delta }}_{t\omega }^{+}$, ${\mathrm{\Delta }}_{t\omega }^{-}$, and${\mathrm{\Delta }}_{t\omega }$ represent the positive deviation, negative deviation and total deviation of the renewable energy owner in the RT market at time period t.
The random variables include $W_t^{\mathit{ac}}$, ${\lambda }_t^D$, and ${\lambda }_t^r$, where $W_t^{\mathit{ac}}$ is the actual renewable energy production at time period t. The variables, if augmented with a subscript $\omega$, represent their realization in a scenario $\omega$.
The constants and parameters are ${\mathit{pr}}_{\omega }$,$W^{\mathit{max}}$, $\alpha$, and ${\beta }_w$, where $W^{\mathit{max}}$ is the installed capacity of the wind power energy owner and ${\beta }_w$ is the risk aversion parameter of the conventional power energy owner.
The objective function (4.11.1) comprises two terms: (1) The expected profit, which equals the revenue from the DA market plus the revenue from positive energy deviations in the RT market minus the cost for negative energy deviations in the RT market and the cost in the bilateral reserve market; and (2) the CVaR multiplied by a weighting factor β_W, which controls the degree of risk aversion of the wind energy owner.
Again in this model, the first-stage decision variable is the hourly bid of the energy volume $W_{t\omega }^D$ of the renewable energy owner, whereas the second-stage variables are the RT energy deviations ${\mathrm{\Delta }}_{t\omega }^{+}$ and ${\mathrm{\Delta }}_{t\omega }^{-}$ of the owner. Constraint (4.11.2) limits the amount of renewable energy that can be traded in the DA market. Constraints (4.11.3)–(4.11.6) determine the total positive –and negative energy deviations incurred by the renewable energy owner per period and scenario. Constraint (4.11.7) constitutes the nonanticipativity conditions related to the decisions made in the first-stage. Constraint (4.11.8) enforces a nondecreasing offer curve. Constraints (4.11.9) and (4.11.10) are used to compute CVaR.
The two models of conventional and renewable energy owners are connected through the reserve volumes $P_{\mathit{gt}}^R$ and $W_t^R$ and the reserve clearing price ${\lambda }_t^R$ in the bilateral reserve settlement. The total reserve volume, that is, the sum of reserve volumes $P_{\mathit{gt}}^R$, of conventional energy owner should be equal to $W_t^R$.

3. Solving the matrix game
The discretization of the strategy variables ${\gamma }_i$s may cause loss or artificial creation of a Nash equilibrium [16]. To capture a possibly missing Nash solution, the standard discrete equilibrium condition (4.9) is loosened by $\epsilon$ to yield an approximate Nash equilibrium.

${\pi }_i\left({\gamma }_i^{*},{\gamma }_{-i}^{*}\right)\ge {\pi }_i\left({\gamma }_{i,n},{\gamma }_{-i}^{*}\right)-\epsilon ,\forall {\gamma }_{i,n}\in {\mathrm{\Omega }}_i,\forall i$ (4.12)

(4.12)

The matrix payoffs are suppliers’ profits from both the energy and bilateral reserve markets. These profits are obtained by solving the two-stage constrained stochastic programming problems described in model (4.10) and (4.11) for all strategy tuples in the increasing order of bidding price and energy. After the payoffs for each strategy tuple are obtained, the strategy tuples are examined for the Nash equilibrium condition (4.12). Ref. [16] presented a method to find an appropriate value of ε for determining an approximate Nash equilibrium.
The complete solution process for trading renewable energy in both energy market and the bilateral reserve settlement is depicted in a flow chart in Fig. 4.8, which consists of two parts: obtaining the matrix payoffs by stochastic programming on the left-hand side and solving for the Nash equilibrium on the right-hand side. The strategies contain two variables: the reserve energy volume $P_{\mathit{gt}}^R$ or $W_t^R$ that a conventional generating unit would like to offer or a renewable energy owner would like to buy, respectively, and the reserve clearing price ${\lambda }_t^R$ settled among conventional and renewable energy owners. These variables are discreated separately into a limited number of values within their limits defined by (4.5)–(4.7). A strategy tuple is then created by combining the values of these variables for all of the players in the game. For each strategy tuple, the stochastic optimization models (4.10) and (4.11) are executed for conventional and renewable energy owners, respectively, during which $P_{\mathit{gt}}^R$ (or $W_t^R$) and ${\lambda }_t^R$ are constant values in this strategy tuple. Then, the Nash equilibrium condition is examined for each strategy tuple solved to determine which strategy tuple yields the maximum profits for both conventional and renewable energy owners. Finally, the stochastic programming is executed for the best strategy tuple, that is, the Nash equilibrium, to obtain the optimal bidding curve. The parameter ε_max is relatively small compared with the expected profit obtained from the stochastic models, for example, 1% of the expected profit. If no solution is obtained from the search for the Nash equilibrium, wind energy owners will not bid reserve and conventional power energy owners will not offer reserve in the new bilateral reserve settlement.

4.2.5.4 Case Study

Consider a game with two players: one thermal power plant with three units and one wind power plant. The installed capacities of the wind and thermal power plants are 100 and 140 MW, respectively. As there is only one thermal power energy owner in the market of providing the new type of bilateral reserve, the wind energy owner decides how much reserve it would like to buy from the thermal power energy owner and at what price. The thermal power energy owner decides the price of the reserved power and how much reserve it would like to sell. The thermal units’ operating characteristics are given in Table 4.3. The reserve price bid cap c^cap is $1000/MW h in the reserve market. The ARIMA model is used to generate 5000 scenarios for wind power, DA price, and RT price predictions, respectively. Scenario reduction is then performed to reduce the scenarios of wind power, DA price, and RT price predictions to five each. Therefore, the final reduced scenario tree has 125 scenarios. The wind plant data is obtained from the National Renewable Energy Laboratory website [17]. The energy prices, including both the DA price and RT price, are obtained from the PJM website [11]. The risk aversion parameter is β_T=β_W=0.5. The confidence level is α=0.95. The maximum approximation parameter ε_max (see Figure 4.8) is $10. For each thermal unit, three discrete reserve power volumes and three reserve clearing prices are generated. Assume that the reserve clearing price strategies are identical for the three units. A total of 81 strategy tuples are generated in this case study. The bidding curve for the wind energy owner is generated by solving the two-stage stochastic optimization problem (4.12) where the wind generation, RT price, and DA price are obtained through forecasting and scenario generation and reduction. The reserve price settled between wind and conventional power energy owners is obtained by using game theory. The expected profit is then calculated by applying the bidding curve obtained to the electricity market using real data obtained from the PJM market.

Table 4.3

Thermal Unit Data

Unit Number	1	2	3
$P^{\min }$(MW)	0	5	5
$P^{\max }$(MW)	50	45	45
RU (MW/h)	50	15	15
RD (MW/h)	50	15	15
uIC (h)	0	0	0
StUp ($)	0	88	88
Fuel type	Gas	Gas	Gas
a	0	85.509	82.342
b	80	70.85831	68.23393
c	0	0.18819	0.18122

4.3.2.1 Introduction to MPEC

The strategic bidding model for a price-maker energy owner in the electricity market is different from that for a price-taker energy owner, as the price-maker energy owner has the market power to influence the market clearing prices; and the market clearing price, on the other hand, can influence the clearing power quantity of the price-maker energy owner [23]. To address this dependency, the problem can be formulated as a bilevel model. The upper-level model maximizes the profit of the price-maker energy owner from the electricity market. The lower level model represents the market clearing process.

The general formulation of a bilevel optimization problem (4.14) is the following:

$\underset{x.y}{\min }{f}^U(x,y)$ (4.14.1)

(4.14.1)

Subject to:

$g^U\left(x,y\right)\le 0$ (4.14.2)

(4.14.2)

$h^U\left(x,y\right)=0$ (4.14.2)

(4.14.2)

$y\in {\mathrm{argmin}}_y\mathrm{\{}{f}^L\left(x,y\right)\mathit{subject}\mathit{to}{g}^L\left(x,y\right)\le 0,h^L\left(x,y\right)=0\mathrm{\}}$ (4.14.2)

(4.14.2)

Under the assumption that Karush–Kuhn–Tucker (KKT) conditions are necessary and sufficient for optimality in the lower level problem, we can employ them to replace condition (4.14.2) and transfer the bilevel problem into a singlelevel one (4.15) as follows:

${\min }_{x.y,\lambda ,\mu }{f}^U(x,y)$ (4.15.1)

(4.15.1)

Subject to

$g^U\left(x,y\right)\le 0$ (4.15.2)

(4.15.2)

$h^U\left(x,y\right)=0$ (4.15.3)

(4.15.3)

${\nabla }_y{f}^L\left(x,y\right)+{\lambda }^{\mathrm{T}}{\nabla }_y{f}^L\left(x,y\right)+{\mu }^{\mathrm{T}}{\nabla }_y{g}^L\left(x,y\right)=0$ (4.15.4)

(4.15.4)

$g^L\left(x,y\right)\le 0$ (4.15.5)

(4.15.5)

$h^L\left(x,y\right)=0$ (4.15.6)

(4.15.6)

$\mu \ge 0$ (4.15.7)

(4.15.7)

${\mu }^{\mathrm{T}}{g}^L\left(x,y\right)=0$ (4.15.8)

(4.15.8)

where λ and μ represent the dual variables associated to constraints $h^L\left(x,y\right)=0$ and $g^L\left(x,y\right)\le 0$, respectively, in the lower level problem.

The lower level problem can be replaced by its first-order optimality condition, such as the widely used KKT condition. Thus, the original bilevel optimization problem can be transformed to a singlelevel problem, which is called a mathematical program with equilibrium constraints (MPEC). The MPEC approach has been widely used to obtain optimal bidding strategies for conventional energy owners. Readers who are interested in learning the MPEC approach application can find detailed information in Refs. [19–22].

4.3.2.2 Multistage Stochastic MPEC

The challenge to use the MPEC approach for modeling the bidding strategies of the price-maker renewable energy owners is the uncertainties involving in the modeling. Thus, the stochastic programming and MPEC approach will be combined together to model the bidding strategies of the price-maker renewable energy owner.

1. Uncertainties in the DA market
The main uncertainty in the DA market clearing process comes from the load and renewable energy forecast errors. The load and renewable energy forecast errors are unintentional and depend on the forecasting models. These forecast errors are handled by up and down reserves, which are scheduled as functions of the standard deviations of the forecasted renewable power and the total forecasted demand of the system are as follows:

$R_t^U=R_t^D=\kappa \sqrt{{{(\sigma }_t^D)}^2+\sum _j{{(\sigma }_{\mathit{jt}}^W)}^2}$ (4.16)

(4.16)

where $R_t^U$ and $R_t^D$ are the total up and down reserve scheduled in a time period t, respectively. ${\sigma }_t^D$ and ${\sigma }_{\mathit{jt}}^W$ are the standard deviation of the total forecasted demand and forecasted renewable power of the renewable generating unit j in a time period t, respectively. $\kappa$ is a parameter decided by the system operator and is selected to be 3 in this chapter.
The reserve clearing process is assumed to be independent from the DA energy clearing and is modeled below:

${\min }_{R_{\mathit{it}}^U,R_{\mathit{it}}^D}\sum _i{{\text{λ}}_{\mathit{it}}^{\mathit{RU}}R}_{\mathit{it}}^U+{\text{λ}}_{\mathit{it}}^{\mathit{RD}}{R}_{\mathit{it}}^D$ (4.17.1)

(4.17.1)

Subject to

$0\le R_{\mathit{it}}^U\le R_i^{U\mathit{max}},\forall i,t$ (4.17.2)