## 3 Mathematical Formulation
### Integrated Bidding Strategies for Coordinated Wind-DR Power Plant  
When a WPP and a DR aggregator become a coalition, the WPP can provide low-priced energy to the DR aggregator to reduce customers' purchase costs and obtain DR capacities to minimize its imbalance penalty. Based on the appropriate cooperation, revenues of WPPs and DR aggregators can be improved. The detailed trading strategy can be described as follows:  
1. In the coalition, we assume that the WPP provides low-priced energy to the DR aggregator. The benefit received by the WPP equals the purchase cost of DR aggregators. Therefore, within the coalition, the energy provided by a WPP to a DR aggregator is free. In return, DR aggregators can offer DR capacity for mitigating wind output imbalances caused by forecasting errors. Similarly, within the coalition, the DR capacity provided by the DR aggregator to the WPP is free.
2. As described in Section 2, wind power and DR resources are complementary when providing frequency regulation. Therefore, through the coordinated bidding strategy, the coalition can optimize the operation states of wind generation and DR resources to respond to regulation signals more economically.  
The purpose of the joint bidding model is to optimize the coalition's decision to maximize its participation in the markets. The objective function and corresponding constraints are formulated as follows:  
\[\text{max}\ \ PF_{\text{W}R}^{s}+\beta\left(\theta_{w}-\frac{1}{1-\alpha} \underset{s}{\overset{N_{s}}{\sum}}\pi_{s}\eta_{w}^{s}\right) \tag{25}\]  
\[\ PF_{\text{W}R}^{s}=\underset{s\in N_{s}}{\sum}\underset{t}{\sum}\underset {s}{\sum}\underset{t}{\sum}\underset{s}{\sum}\underset{t}{\pi_{s}}(P_{t}^{ DA,s}\lambda_{t}^{DA}+P_{DR,t}^{s}\lambda_{DR,t}+\lambda_{reg,t}^{exp}C_{WR,t}^{ reg,s}S_{t}+\lambda_{reg,t}^{perf}C_{WR,t}^{reg,s}R_{t}^{mill}S_{t}+ \lambda_{t}^{RT}\Delta P_{t}^{+\beta}-\lambda_{t}^{RT}\Delta P_{t}^{-\beta}) \tag{26}\]  
The market benefit of the coalition \(P_{\text{W}R}^{s}\) includes four parts as presented in Equation (26):  
1. the revenue from participating in the day-ahead energy market, (2) the revenue from participating in the DR programs, (3) the revenue from bidding in the regulation market, (4) the imbalance revenue resulting from the real-time market. \[C_{WR,t}^{reg,s}=C_{WR,t}^{up,s}+C_{WR,t}^{dn,s}\] (27) \[C_{WR,t}^{up,s}=C_{DR,t}^{reg,s}\] (28) \[C_{WR,t}^{dn,s}=C_{w,t}^{reg,s}\] (29) \[0\leq C_{WR,t}^{up,s}\leq C_{DR,t}^{s}\] (30) \[0\leq C_{WR,t}^{dn,s}\leq P_{t}^{\text{max}}\] (31) \[\Delta P_{t}^{s}=P_{w,t}^{ac,s}-P_{w,t}^{s}-P_{DR,t}^{s}\] (32) \[\Delta P_{t}^{s}=\Delta P_{w,t}^{+,s}-\Delta P_{w,t}^{-,s}-P_{DR,t}^ {s}\] (33) \[0\leq\Delta P_{t}^{+,s}\leq P_{w,t}^{ac,s}\] (34) \[0\leq\Delta P_{t}^{-,s}\leq W\] (35) \[PF_{\text{W}R}+\eta_{w}-\theta_{w}\geq 0\] (36)Equations (28) and (29) indicate the up- and down-regulation capacities (\(C_{WR,t}^{up,s}\) and \(C_{WR,t}^{dn,s}\)) that are individually supplied by different operators. \(C_{WR,t}^{reg,s}\) represents total regulation capacity bids. The up-regulation capacity is offered from DR resources, whereas the down-regulation capacity is provided by wind generation. Equations (30) and (31) express the limits for submitted regulation capacity. Equations (32)-(35) show the constraints for system imbalance \(\Delta P_{t}^{s}\). In Equation (36), the CVaR constraint is modeled.