(4.17.2)

$0\le R_{\mathit{it}}^D\le R_i^{D\mathit{max}},\forall i,t$ (4.17.3)

(4.17.3)

$\sum _i{R}_{\mathit{it}}^U=R_t^U,\forall t$ (4.17.4)

(4.17.4)

$\sum _i{R}_{\mathit{it}}^D=R_t^D,\forall t$ (4.17.5)

(4.17.5)

where $R_{\mathit{it}}^U$ and $R_{\mathit{it}}^D$ are the decision variables of the optimization problem (4.17). $R_{\mathit{it}}^U$ and $R_{\mathit{it}}^D$ represent the up and down reserve scheduled for the conventional energy owner i in a time period t, respectively. $R_i^{U\mathit{max}}$ and $R_i^{D\mathit{max}}$ are the parameters which represent the maximum up and down reserve that can be provided by the conventional energy owner i, respectively.

2. Uncertainties in the RT market
In the RT market clearing process, the main uncertainties come from the intentional misscheduling of load and wind power production deviation. If the power purchasers predict that the RT price will be lower than the DA price, they may intentionally underschedule load in the DA market and buy extra demand in the RT market at the RT price. Otherwise, if the power purchasers expect that the RT price will be higher than DA price, they may intentionally overschedule load and sell any extra energy procured in the DA market back to the system at the RT price. The behavior of these purchasers will increase the RT price volatility. For a wind power energy owner, the deviation of the actual production from that scheduled in the DA market will be balanced in the RT market. If the actual production is less than the DA scheduled, the wind energy owner will have to buy the deviated power at the RT price. Otherwise, the wind energy owner can bid the extra power into the RT market and get paid for the accepted power at the RT prices.
Overall, the problem to determine the optimal bidding strategy for a strategic wind power energy owner is formulated as a bilevel stochastic optimization model. Fig. 4.18 shows the structure of the stochastic bilevel model, which consists of an upper–lower problem and two lower level problems. The upper-level problem maximizes the total profit obtained by the wind power energy owner in both DA and RT markets to determine its optimal bidding curves in both markets. The maximal profit and the resulting bidding curves depend on the cleared information (LMPs and scheduled energy) in the DA and RT markets, which will be obtained in the lower level problems. One lower level problem collects the bidding information of each energy owner and runs the DA market clearing process to generate the DA LMPs and energy schedule for each energy owner. The other lower level problem is carried out after the DA clearing to deal with the RT energy deviation. This RT market clearing process depends not only on the RT offering curve of each energy owner but also on the DA energy schedule. Both the RT LMPs and RT energy schedule will be announced after the clearing.

1. Upper-level problem
The upper level problem (4.18) maximizing the total profit of the wind power energy owner in the DA and RT market is shown below:

${\min }_{\mathrm{\Xi }}\sum _{\mathit{jt}\omega }{\pi }_{\omega }\left[\sum _b(-{\text{λ}}_{\left(m\mathrm{:}j{\in \mathrm{\Psi }}_m^W\right)t\omega }^{\mathit{DA}}{\text{p}}_{\mathit{bjt}\omega }^{\mathit{WD}})-{\text{λ}}_{\left(m\mathrm{:}j{\in \mathrm{\Psi }}_m^W\right)t\omega }^{\mathit{RT}}{\text{P}}_{\mathit{jt}\omega }^{\mathit{WR}}\right]-\beta (\zeta -\frac{1}{1-\alpha }\sum _{\omega }{\pi }_{\omega }{\eta }_{\omega })$ (4.18.1)

(4.18.1)

Subject to

$0\le {\text{λ}}_{\left(b-1\right)\mathit{jt}\omega }^{\mathit{WD}}\le {\text{λ}}_{\mathit{bjt}\omega }^{\mathit{WD}}\le {\lambda }^{\mathit{CapD}},\forall j,b\ge 2,t,\omega$ (4.18.2)

(4.18.2)

$0\le {\text{λ}}_{\mathit{jt}\omega }^{\mathit{WR}}\le {\lambda }^{\mathit{CapR}},\forall j,t,\omega$ (4.18.3)

(4.18.3)

${\eta }_{\omega }\ge 0,\forall \omega$ (4.18.4)

(4.18.4)

$\zeta -{\eta }_{\omega }\le \sum _{\mathit{jt}}\left[\sum _b(-{\text{λ}}_{\left(m\mathrm{:}j{\in \mathrm{\Psi }}_m^W\right)t\omega }^{\mathit{DA}}{\text{p}}_{\mathit{bjt}\omega }^{\mathit{WD}})+{\text{λ}}_{\left(m\mathrm{:}j{\in \mathrm{\Psi }}_m^W\right)t\omega }^{\mathit{RT}}{\text{P}}_{\mathit{jt}\omega }^{\mathit{WR}}\right],\forall \omega$ (4.18.5)

(4.18.5)

where $\mathit{m:w}{\in \mathrm{\Psi }}_m^W$ denotes the bus where the wind power generating unit w is located; and Ξ={${\text{λ}}_{\mathit{bjt}\omega }^{\mathit{WD}}$,${\text{λ}}_{\mathit{jt}\omega }^{\mathit{WR}}$, $\zeta$, ${\eta }_{\omega }$, Ξ^D, Ξ^R } are the set of all decision variables of the problem (4.18), where Ξ^D and Ξ^R are the sets of all the decision variables in the lower level problems (4.19) and (4.20), respectively, and will be defined later. It should be noted that the total formulation for (4.18) consists of (4.18.1)–(4.18.5) and in addition the total formulations of (4.19) and (4.20). ${\text{λ}}_{\mathit{bjt}}^{\mathit{WD}}$ represents the offer price of block b of the wind generating unit j in a time period t in the DA market. ${\text{λ}}_{\mathit{jt}}^{\mathit{WR}}$ represents the offer price of the wind generating unit j in a time period t in the RT market.$\zeta$ and $\eta$ are auxiliary variables used to compute the CVaR. These variables, if augmented with a subscript $\omega$, represent their realization in a scenario $\omega$. ${\lambda }^{\mathit{CapD}}$ and ${\lambda }^{\mathit{CapR}}$ are constants which represent the cap bidding price in the DA and RT markets, respectively.
The objective function (4.18.1) minimizes the sum of two terms: (1) The negative expected profit of the wind power energy owner, which is the negative revenue obtained in the DA market plus the negative revenue from the RT market, and (2) the negative CVaR_α multiplied by the weighting factor β. In (4.18.1), ${\text{λ}}_{(\mathit{m:j}{\in \mathrm{\Psi }}_m^W)t\omega }^{\mathit{DA}}$ and ${\text{p}}_{\mathit{bjt}\omega }^{\mathit{WD}}$ are the variables determined in the lower level problem (4.19); while ${\text{λ}}_{(\mathit{m:j}{\in \mathrm{\Psi }}_m^W)t\omega }^{\mathit{RT}}$ and ${\text{P}}_{\mathit{jt}\omega }^{\mathit{WR}}$ are the variables determined in the lower level problem (4.20). The constraints (4.18.2) and (4.18.3) enforce the acceptable DA and RT bidding prices and nondecreasing DA bidding curves of the wind power energy owner. The constraints (4.18.4) and (4.18.5) are used to compute the CVaR, which is used in this paper as a measurement for risk. The CVaR_α represents the expected profit associated with the (1−α)×100% worst scenarios. The weighting parameter β is set by the wind power energy owner to model the tradeoff between profit and risk. If the wind power energy owner is willing to gain less profit in order to bear less risk, it will select a higher value for β.

2. Lower level problem—DA market clearing
The lower level problem (4.19) is formulated as follows to represent the DA market clearing process.

${\min }_{{\Xi }^D}\sum _{\mathit{bi}}{\text{λ}}_{\mathit{bit}\omega }^{\mathit{CD}}{\text{p}}_{\mathit{bit}\omega }^{\mathit{CD}}+\sum _{\mathit{bj}}{\text{λ}}_{\mathit{bjt}\omega }^{\mathit{WD}}{\text{p}}_{\mathit{bwt}\omega }^{\mathit{WD}}-\sum _{\mathit{ld}}{\text{λ}}_{\mathit{ldt}}^{\mathit{LD}}{\text{p}}_{\mathit{ldt}\omega }^{\mathit{LD}}$ (4.19.1)

(4.19.1)

Subject to

$\sum _{l\left(d\in {\Psi }_m^D\right)}{\text{p}}_{\mathit{ldt}\omega }^{\mathit{LD}}-\sum _{b\left(i\in {\Psi }_m^I\right)}{\text{p}}_{\mathit{bit}\omega }^{\mathit{CD}}-\sum _{b\left(j\in {\Psi }_m^W\right)}{\text{p}}_{\mathit{bjt}\omega }^{\mathit{WD}}+\sum _{n\in {\Phi }_m}{B}_{\mathit{mn}}\left({\delta }_{\mathit{mt}\omega }^D-{\delta }_{\mathit{nt}\omega }^D\right)=0\mathrm{:}{\text{λ}}_{\mathit{mt}\omega }^{\mathit{DA}},\forall m,t,\omega$ (4.19.2)

(4.19.2)

$0\le {\text{p}}_{b\mathit{it}\omega }^{\mathit{CD}}\le {\text{p}}_{\mathit{bit}}^{\mathit{CD}\mathit{max}}\mathrm{:}{\mu }_{\mathit{bit}\omega }^{C\mathit{max}},{\mu }_{\mathit{bit}\omega }^{C\mathit{min}},\forall b,i,t,\omega$ (4.19.3)

(4.19.3)

$0\le {\text{p}}_{\mathit{bjt}\omega }^{\mathit{WD}}\le {\text{P}}_{\mathit{jtb}}^{\mathit{Wf}}\mathrm{:}{\mu }_{\mathit{bjt}\omega }^{W\mathit{max}},{\mu }_{\mathit{bjt}\omega }^{W\mathit{min}},b=1,\forall j,t,\omega$ (4.19.4)

(4.19.4)

$0\le {\text{p}}_{\mathit{bjt}\omega }^{\mathit{WD}}\le {\text{P}}_{\mathit{jtb}}^{\mathit{Wf}}-{\text{P}}_{\mathit{jt}\left(b-1\right)}^{\mathit{Wf}}\mathrm{:}{\mu }_{\mathit{bjt}\omega }^{W\mathit{max}},{\mu }_{\mathit{bjt}\omega }^{W\mathit{min}},\forall b\ge 2,j,t,\omega$ (4.19.5)

(4.19.5)

$0\le {\text{p}}_{\mathit{ldt}\omega }^{\mathit{LD}}\le {\text{p}}_{\mathit{ldt}}^{\mathit{LD}\mathit{max}}\mathrm{:}{\mu }_{\mathit{ldt}\omega }^{L\mathit{max}},{\mu }_{\mathit{ldt}\omega }^{L\mathit{min}},\forall l,d,t,\omega$ (4.19.6)

(4.19.6)

${\mathrm{|}B}_{\mathit{mn}}\left({\delta }_{\mathit{mt}\omega }^D-{\delta }_{\mathit{nt}\omega }^D\right)\mathrm{|}\le C_{\mathit{mn}}^{\mathit{max}}\mathrm{:}{\beta }_{\mathit{mnt}\omega }^{D\mathit{max}},{\beta }_{\mathit{mnt}\omega }^{D\mathit{min}},\forall m,n\in {\Phi }_m,t,\omega$ (4.19.7)

(4.19.7)

${{\delta }_m^{\mathit{min}}\le \delta }_{\mathit{mt}\omega }^D\le {\delta }_m^{\mathit{max}}\mathrm{:}{\theta }_{\mathit{mt}\omega }^{D\mathit{max}},{\theta }_{\mathit{mt}\omega }^{D\mathit{min}},\forall m,t,\omega$ (4.19.8)

(4.19.8)

${\delta }_{\mathit{mt}\omega }^D=0\mathrm{:}{\theta }_{t\omega }^{D1},m=1,\forall t,\omega$ (4.19.9)

(4.19.9)

where ${\text{P}}_{\mathit{jtb}}^{\mathit{Wf}}$ is the bth element in the set WP(j,t); and Ξ^D={${\text{p}}_{\mathit{bit}\omega }^{\mathit{CD}}$,${\text{p}}_{\mathit{bjt}\omega }^{\mathit{WD}}$,${\text{p}}_{\mathit{ldt}\omega }^{\mathit{LD}}$,${\delta }_{\mathit{mt}\omega }^D$, ${\text{λ}}_{\mathit{mt}\omega }^{\mathit{DA}}$,${\mu }_{\mathit{bit}\omega }^{C\mathit{max}}$,${\mu }_{\mathit{bit}\omega }^{C\mathit{min}}$,${\mu }_{\mathit{bjt}\omega }^{W\mathit{max}}$,${\mu }_{\mathit{bjt}\omega }^{W\mathit{min}}$,${\mu }_{\mathit{ldt}\omega }^{L\mathit{max}}$,${\mu }_{\mathit{ldt}\omega }^{L\mathit{min}}$,${\beta }_{\mathit{mnt}\omega }^{D\mathit{max}}$,${\beta }_{\mathit{mnt}\omega }^{D\mathit{min}}$,${\theta }_{\mathit{mt}\omega }^{D\mathit{max}}$,${\theta }_{\mathit{mt}\omega }^{D\mathit{min}}$,${\theta }_{t\omega }^{D1}$} is the set of all decision variables of the problem (4.19). The decision variable set contains both primal and dual variables, where the dual variables are defined following the colon in each constraint. ${\text{p}}_{\mathit{bit}}^{\mathit{CD}}$ and ${\text{p}}_{\mathit{bjt}}^{\mathit{WD}}$ represent the power produced in block b by the conventional power energy owner i and the wind power energy owner j in a time period t in the DA market, respectively. ${\text{p}}_{\mathit{ldt}}^{\mathit{LD}}$ is the power bought by demand d in block l in a time period t in the DA market. ${\delta }_{\mathit{mt}}^D$ is voltage angle of bus m in a time period t in the DA market. ${\text{λ}}_{\mathit{mt}}^{\mathit{DA}}$ is the DA LMP at bus m in a time period t.
${\text{λ}}_{\mathit{bit}\omega }^{\mathit{CD}}$ is the random variable which represents the offer price of block b of the conventional power energy owner i in a time period t in the DA market. Other variables include ${\text{p}}_{\mathit{bit}}^{\mathit{CD}\mathit{max}}$ and ${\text{p}}_{\mathit{ldt}}^{\mathit{LD}\mathit{max}}$, where ${\text{p}}_{\mathit{bit}}^{\mathit{CD}\mathit{max}}$ is the capacity of block b of the conventional power energy owner i in a time period t in the DA market and ${\text{p}}_{\mathit{ldt}}^{\mathit{LD}\mathit{max}}$ is the capacity of block l of the demand d in a time period t in the DA market. The constants are $B_{\mathit{mn}}$, $C_{\mathit{mn}}^{\mathit{max}}$, ${\delta }_m^{\mathit{min}}$, and ${\delta }_m^{\mathit{max}}$, where $B_{\mathit{mn}}$ is the imaginary part of the admittance of line m–n, $C_{\mathit{mn}}^{\mathit{max}}$ is the transmission capacity of lines m–n ${\delta }_m^{\mathit{max}}$ and ${\delta }_m^{\mathit{min}}$ are the upper and lower limits of voltage magnitude of bus m, respectively.
The objective function (4.19.1) minimizes the total cost of energy offered by both conventional and wind power energy owners minus the revenue from supplying demand in each period and each scenario. Eq. (4.19.2) enforces the DA power balance at each bus. The constraints (4.19.3)–(4.19.5) represent the limits of the power offered by the conventional and wind power units in each block. The constraint (4.19.6) represents the bounds of the demand in each block. The constraint (4.19.7) imposes the transmission capacity limits of each power line. The voltage angle limits of each bus are expressed in the constraint (4.19.8). The reference bus is selected in (4.19.9). Note that if the uncertainty of the bidding prices of the conventional power energy owner i is not considered, then ${\text{λ}}_{\mathit{bit}\omega }^{\mathit{CD}}={\text{λ}}_{\mathit{bit}{\omega }^{\prime }}^{\mathit{CD}}$, $\forall \omega ,{\omega }^{\prime }$.

3. Lower level problem—RT market clearing
The lower level problem (4.20) modeling the RT market clearing process is formulated below:

${\min }_{{\mathrm{\Xi }}^R}\sum _i({\text{λ}}_{\mathit{it}}^{\mathit{CR}+}{\text{P}}_{\mathit{it}\omega }^{\mathit{CR}+}-{\text{λ}}_{\mathit{it}}^{\mathit{CR}-}{\text{P}}_{\mathit{it}\omega }^{\mathit{CR}-})+\sum _j{\text{λ}}_{\mathit{jt}\omega }^{\mathit{WR}}{\text{P}}_{\mathit{jt}\omega }^{\mathit{WR}}+\sum _i({\text{λ}}_{\mathit{it}}^{\mathit{RU}}{\text{r}}_{\mathit{it}\omega }^U+{\text{λ}}_{\mathit{it}}^{\mathit{RU}}{\text{r}}_{\mathit{it}\omega }^U-\sum _d{\text{λ}}_{\mathit{dt}\omega }^{\mathit{LR}}{\text{P}}_{\mathit{dt}\omega }^{\mathit{LR}}$ (4.20.1)

(4.20.1)

Subject to

$\sum _{d\in {\Psi }_m^D}({\text{P}}_{\mathit{dt}\omega }^{\mathit{LR}}+\sum _l{\text{p}}_{\mathit{ldt}\omega }^{\mathit{LD}})-\sum _{b\left(i\in {\Psi }_m^I\right)}{\text{p}}_{\mathit{bit}\omega }^{\mathit{CD}}-\sum _{i\in {\Psi }_m^I}({\text{P}}_{\mathit{it}\omega }^{\mathit{CR}+}-{\text{P}}_{\mathit{it}\omega }^{\mathit{CR}-}+{\text{r}}_{\mathit{it}\omega }^U-{\text{r}}_{\mathit{it}\omega }^D)-\sum _{b\left(j\in {\Psi }_m^W\right)}{\text{p}}_{\mathit{bjt}\omega }^{\mathit{WD}}-\sum _{j\in {\Psi }_m^W}{\text{P}}_{\mathit{jt}\omega }^{\mathit{WR}}+\sum _{n\in {\Phi }_m}{B}_{\mathit{mn}}\left({\delta }_{\mathit{mt}\omega }^R-{\delta }_{\mathit{nt}\omega }^R\right)=0\mathrm{:}{\text{λ}}_{\mathit{mt}\omega }^{\mathit{RT}},\forall m,t,\omega$ (4.20.2)

(4.20.2)

$0\le {\text{P}}_{\mathit{it}\omega }^{\mathit{CR}+}\le {\text{P}}_i^{\mathit{max}+}\mathrm{:}{\mu }_{\mathit{it}\omega }^{\mathit{max}+},{\mu }_{\mathit{it}\omega }^{\mathit{min}+},\forall i,t,\omega$ (4.20.3)

(4.20.3)

$0\le {\text{P}}_{\mathit{it}\omega }^{\mathit{CR}-}\le {\text{P}}_i^{\mathit{max}-}\mathrm{:}{\mu }_{\mathit{it}\omega }^{\mathit{max}-},{\mu }_{\mathit{it}\omega }^{\mathit{min}-},\forall i,t,\omega$ (4.20.4)

(4.20.4)

${\text{P}}_{\mathit{it}\omega }^{\mathit{CR}+}\le {\text{P}}_i^{\mathit{max}}-\sum _b{\text{p}}_{\mathit{bit}}^{\mathit{CD}}\mathrm{:}{\mu }_{\mathit{it}\omega }^{\mathit{CR}+},\forall i,t,\omega$ (4.20.5)

(4.20.5)

${\text{P}}_{\mathit{it}\omega }^{\mathit{CR}-}\le \sum _b{\text{p}}_{\mathit{bit}}^{\mathit{CD}}\mathrm{:}{\mu }_{\mathit{it}\omega }^{\mathit{CR}-},\forall i,t,\omega$ (4.20.6)

(4.20.6)

$-\sum _b{\text{p}}_{\mathit{bjt}}^{\mathit{WD}}\le {\text{P}}_{\mathit{jt}\omega }^{\mathit{WR}}\le P_{\mathit{jt}\omega }^{\mathit{Wf}}-\sum _b{\text{p}}_{\mathit{bjt}}^{\mathit{WD}}\mathrm{:}{\mu }_{\mathit{jt}\omega }^{\mathit{WR}\mathit{max}},{\mu }_{\mathit{jt}\omega }^{\mathit{WR}\mathit{min}},\forall j,t,\omega$ (4.20.7)

(4.20.7)

$\{\begin{array}{c}0\le P_{\mathit{dt}\omega }^{\mathit{LR}}\le P_{\mathit{dt}\omega }^{\mathit{LRf}},:{\mu }_{\mathit{dt}\omega }^{\mathit{LR}\mathit{max}},{\mu }_{\mathit{dt}\omega }^{\mathit{LR}\mathit{min}},\forall d,t,\omega ,\mathit{if}{P}_{\mathit{dt}\omega }^{\mathit{LRf}}\ge 0\\ P_{\mathit{dt}\omega }^{\mathit{LRf}}\le P_{\mathit{dt}\omega }^{\mathit{LR}}\le 0,:{\mu }_{\mathit{dt}\omega }^{\mathit{LR}\mathit{max}},{\mu }_{\mathit{dt}\omega }^{\mathit{LR}\mathit{min}},\forall d,t,\omega ,\mathit{if}{P}_{\mathit{dt}\omega }^{\mathit{LRf}}<0\end{array}$ (4.20.8)

(4.20.8)

$0\le {\text{r}}_{\mathit{it}\omega }^U\le {\text{R}}_{\mathit{it}}^U\mathrm{:}{\phi }_{\mathit{it}\omega }^{U\mathit{max}},{\phi }_{\mathit{it}\omega }^{U\mathit{min}},\forall i,t,\omega$ (4.20.9)

(4.20.9)

$0\le {\text{r}}_{\mathit{it}\omega }^D\le {\text{R}}_{\mathit{it}}^D\mathrm{:}{\phi }_{\mathit{it}\omega }^{D\mathit{max}},{\phi }_{\mathit{it}\omega }^{D\mathit{min}},\forall i,t,\omega$ (4.20.10)

(4.20.10)

${\mathrm{|}B}_{\mathit{mn}}\left({\delta }_{\mathit{mt}\omega }^R-{\delta }_{\mathit{nt}\omega }^R\right)\mathrm{|}\le C_{\mathit{mn}}^{\mathit{max}}\mathrm{:}{\beta }_{\mathit{mnt}\omega }^{R\mathit{max}},{\beta }_{\mathit{mnt}\omega }^{R\mathit{min}},\forall m,n\in {\Phi }_m,t,\omega$ (4.20.11)

(4.20.11)

${{\delta }_m^{\mathit{min}}\le \delta }_{\mathit{mt}\omega }^R\le {\delta }_{\mathit{mt}}^{\mathit{max}}\mathrm{:}{\theta }_{\mathit{mt}\omega }^{R\mathit{max}},{\theta }_{\mathit{mt}\omega }^{R\mathit{min}},\forall m,t,\omega$ (4.20.12)

(4.20.12)

${\delta }_{\mathit{mt}\omega }^R=0\mathrm{:}{\theta }_{t\omega }^{R1},m=1,\forall t,\omega$ (4.20.13)

(4.20.13)

where Ξ^R is the set of all primal and dual decision variables in the problem (4.20.1), where Ξ^R={${\text{P}}_{\mathit{it}\omega }^{\mathit{CR}+}$, ${\text{P}}_{\mathit{it}\omega }^{\mathit{CR}-}$, ${\text{P}}_{\mathit{jt}\omega }^{\mathit{WR}}$, ${\text{P}}_{\mathit{dt}\omega }^{\mathit{LR}}$, ${\text{r}}_{\mathit{it}\omega }^U$, ${\text{r}}_{\mathit{it}\omega }^D$, ${\delta }_{\mathit{mt}\omega }^R$,${\text{λ}}_{\mathit{mt}\omega }^{\mathit{RT}}$, ${\mu }_{\mathit{it}\omega }^{\mathit{max}+}$, ${\mu }_{\mathit{it}\omega }^{\mathit{min}+}$, ${\mu }_{\mathit{it}\omega }^{\mathit{max}-}$, ${\mu }_{\mathit{it}\omega }^{\mathit{min}-}$, ${\mu }_{\mathit{it}\omega }^{\mathit{CR}+}$, ${\mu }_{\mathit{it}\omega }^{\mathit{CR}-}$, ${\mu }_{\mathit{jt}\omega }^{\mathit{WR}\mathit{max}}$, ${\mu }_{\mathit{jt}\omega }^{\mathit{WR}\mathit{min}}$, ${\mu }_{\mathit{dt}\omega }^{\mathit{LR}\mathit{max}}$, ${\mu }_{\mathit{dt}\omega }^{\mathit{LR}\mathit{min}}$, ${\phi }_{\mathit{it}\omega }^{U\mathit{max}}$, ${\phi }_{\mathit{it}\omega }^{U\mathit{min}}$, ${\phi }_{\mathit{it}\omega }^{D\mathit{max}}$, ${\phi }_{\mathit{it}\omega }^{D\mathit{min}}$, ${\beta }_{\mathit{mnt}\omega }^{R\mathit{max}}$, ${\beta }_{\mathit{mnt}\omega }^{R\mathit{min}}$, ${\theta }_{\mathit{mt}\omega }^{R\mathit{max}}$, ${\theta }_{\mathit{mt}\omega }^{R\mathit{min}}$, ${\theta }_{t\omega }^{R1}$}. ${\text{P}}_{\mathit{it}}^{\mathit{CR}+}$ and ${\text{P}}_{\mathit{it}}^{\mathit{CR}-}$ represent the increased and decreased power of the conventional power energy owner i in a time period t in the RT market, respectively. ${\text{P}}_{\mathit{jt}}^{\mathit{WR}}$ is the rescheduled power of the wind generating unit j in a time period t in the RT market. ${\text{P}}_{\mathit{dt}}^{\mathit{LR}}$ is accepted deviation of the demand d in a time period t in the RT market. ${\text{r}}_{\mathit{it}}^U$ and ${\text{r}}_{\mathit{it}}^D$ represent up and down reserve deployed by the conventional power energy owner i in a time period t in the RT market, respectively. ${\delta }_{\mathit{mt}}^R$ is voltage angle of bus m in a time period t in the RT market. ${\text{λ}}_{\mathit{mt}}^{\mathit{RT}}$ is the RT LMP at bus m in a time period t.
The random variables include $P_{\mathit{jt}}^{\mathit{Wf}}$ and $P_{\mathit{dt}}^{\mathit{LRf}}$, where $P_{\mathit{jt}}^{\mathit{Wf}}$ is the forecasted wind power production of the wind generating unit j in a time period t and $P_{\mathit{dt}}^{\mathit{LRf}}$ is the forecasted deviation of demand d in a time period t.
The constants include ${\text{P}}_i^{\mathit{max}+}$,${\text{P}}_i^{\mathit{max}-}$, and ${\text{P}}_i^{\mathit{max}}$, where ${\text{P}}_i^{\mathit{max}+}$ and ${\text{P}}_i^{\mathit{max}-}$ represent the maximum increased and decreased power that can be provided by the conventional power energy owner i, respectively, and ${\text{P}}_i^{\mathit{max}}$ is the maximum power output of the conventional power energy owner i.
The objective function (4.20.1) minimizes the total cost of redispatching energy and deploying reserve minus the revenue from the deviation demand in each period and each scenario. Eq. (4.20.2) enforces the RT power balance at each bus. The constraints (4.20.3)–(4.20.6) limit the power of each conventional generator that can be sold into the RT market. The amount of wind power that can be sold in the RT market in each scenario is constrained by the forecasted wind power in the scenario, as described by (4.20.7). The bounds of the deviated demand in each scenario are represented by the constraint (4.20.8). The up and down reserves deployed are bounded by the scheduled up and down reserves in the DA market, respectively, as expressed in (4.20.9) and (4.20.10), respectively. The constraint (4.20.11) imposes the transmission capacity limits of each power line. The voltage angle limits of each bus are expressed in the constraint (4.20.12). The reference bus is selected in (4.20.13).
The wind power trading in the RT market is different from the conventional power energy owners due to the uncertainty in the wind power production. The wind power energy owner may fail to fulfill the productions settled in the DA market and is forced to correct its negative deviation in the RT market. Meanwhile, the wind power energy owner can bid the extra power into the RT market if the deviation is positive. The rescheduled wind power is constrained by (4.20.8) for both cases. As the DA market is cleared prior to the RT market, the decision variables ${\text{R}}_{\mathit{it}}^U$ and ${\text{R}}_{\mathit{it}}^D$ of the problem (4.18) and ${\text{p}}_{\mathit{bit}\omega }^{\mathit{CD}}$, ${\text{p}}_{\mathit{bjt}\omega }^{\mathit{WD}}$, and ${\text{p}}_{\mathit{ldt}\omega }^{\mathit{LD}}$ of the problem (4.19) are considered to be parameters in the problem (4.20).

1. The $\sum _{\mathit{jb}}{\text{λ}}_{\left(m\mathrm{:}j{\in \Psi }_m^W\right)t\omega }^{\mathit{DA}}{\text{p}}_{\mathit{bjt}\omega }^{\mathit{WD}}$ term in (4.19.1).
According to (4.21.2), (4.21.3), (4.21.9), (4.21.10) and the strong duality theorem, this term can be linearized to (4.23).

$\begin{array}{c}\sum _{\mathit{jb}}{\text{λ}}_{\left(m\mathrm{:}j{\in \Psi }_m^W\right)t\omega }^{\mathit{DA}}{\text{p}}_{\mathit{bjt}\omega }^{\mathit{WD}}=\sum _{\mathit{jb}}({\text{λ}}_{\mathit{bjt}\omega }^{\mathit{WD}}{+\mu }_{\mathit{bjt}\omega }^{W\mathit{max}}-{\mu }_{\mathit{bjt}\omega }^{W\mathit{min}}){\text{p}}_{\mathit{bjt}\omega }^{\mathit{WD}}\\ =-\sum _{\mathit{bi}}{\text{λ}}_{\mathit{bit}\omega }^{\mathit{CD}}{\text{p}}_{\mathit{bit}\omega }^{\mathit{CD}}-\sum _{\mathit{bi}}{\text{p}}_{\mathit{bit}}^{\mathit{CD}\mathit{max}}{\mu }_{\mathit{bit}\omega }^{C\mathit{max}}+\sum _{\mathit{ld}}{\text{λ}}_{\mathit{ldt}}^{\mathit{LD}}{\text{p}}_{\mathit{ldt}\omega }^{\mathit{LD}}-\sum _{\mathit{ld}}{\text{p}}_{\mathit{ldt}}^{\mathit{LD}\mathit{max}}{\mu }_{\mathit{ldt}\omega }^{L\mathit{max}}-\\ \sum _{m\left(n\in {\Phi }_m\right)}{C}_{\mathit{mn}}^{\mathit{max}}\left({\beta }_{\mathit{mnt}\omega }^{D\mathit{max}}+{\beta }_{\mathit{mnt}\omega }^{D\mathit{min}}\right)-\sum _m\left({\delta }_m^{\mathit{max}}{\theta }_{\mathit{mt}\omega }^{D\mathit{max}}-{\delta }_m^{\mathit{min}}{\theta }_{\mathit{mt}\omega }^{D\mathit{min}}\right)\end{array}$ (4.23)