## 3 Upper-Level Multiple Resources Coordinated Optimization Modeling and Solving
### Mathematical Model for Multiple Resources Coordinated Optimization
#### 3.2.2 Resource Control Cost  
The three control resources are implemented in different ways, and their control cost factors are significantly different. In the actual power grid, the control cost factor of DC modulation is the lowest. Since VEPR does not cause a large area blackout, its control cost is usually less than that of the ELS. For HVDC emergency power support, its cost factor can be assumed to be constant. In actual operation, the ELS is to cut off the line without informing the customers. The larger the load-shedding amount, the greater the resulting loss due to power outages and the impact on the normal production and life, so the cost increases with the increase of load-shedding amount. In this work, the first-order function is used to describe the characteristics of \(c_{\text{tl}}\) increasing with the increase of the shedding proportion. The cost factor of ELS can be expressed as  
\[c_{\text{tl},j}=b_{\text{tl},0,j}+b_{\text{tl},j}P_{\text{tl},j}/P_{\text{L},j} \tag{5}\]  
where \(P_{\text{L},j}\) is the total load active power before disturbance of the bus where the \(j\)th ELS station is located, and \(b_{\text{tl},j}\) is the linear growth coefficient of the cost factor. The control cost factor of VPP is not only related to the response proportion but is also affected by the VEPR evaluation error. Since the VEPR of the VPP is refreshed periodically, the evaluation results will lag behind the actual DER operating state at off-cycle moments. On the other hand, since the VEPR capacity is the maximum power change that VPP can provide, when the response amount of VPP is close to the VEPR capacity, there will be more DER devices approaching their operation status limit, resulting in user discomfort. Therefore, when the response amount is close to VEPR capacity, the control cost of VPP will increase. The modeling idea of the VPP control cost factor is as follows: when the VPP response proportion is small, the VPP control cost factor will be significantly smaller than the ELS cost factor. As the VPP response proportion increases, the control cost factor first increases slowly, and when the VPP response proportion approaches 100%, the control cost factor increases rapidly and is comparable to the ELS. The control cost factor of VPP can be expressed as  
\[c_{\mathrm{VPP},k}=b_{\mathrm{VPP0},k}+b_{\mathrm{VPP},k}P_{\mathrm{VPP},k}/P_ {\mathrm{VPP},k}^{\mathrm{max}}+\max(0,K_{\mathrm{VPP},k}^{\mathrm{c}}(P_{ \mathrm{VPP},k}/P_{\mathrm{VPP},k}^{\mathrm{max}}-k_{\mathrm{VPP},k}^{\mathrm{ c}})^{\theta}) \tag{6}\]  
where the first part of the expression, \(b_{\mathrm{VPP},k}\), is the constant term of the cost factor; the second part is the linear growth part with the VPP response proportion (\(P_{\mathrm{VPP},k}/P_{\mathrm{VPP},k}^{\mathrm{max}}\)), \(b_{\mathrm{tl},j}\) is the growth coefficient; the third part is an additional term that takes into account VERR evaluation error, \(k_{\mathrm{VPP},k}^{\mathrm{c}}\) is a constant between 0 and 1, \(\theta\) is an odd number, and \(K_{\mathrm{VPP},k}^{\mathrm{c}}\) is a constant. \(k_{\mathrm{VPP},k}^{\mathrm{c}}\), \(\theta\), and \(K_{\mathrm{VPP},k}^{\mathrm{c}}\) of the third part of the cost factor should be determined by the VPP control center according to the actual VERR evaluation error and ELS cost factor. For example, when \(b_{\mathrm{VPP0},k}\), \(b_{\mathrm{VPP},k}\), and \(b_{\mathrm{tl0},j}\) are 3, 0.5, and 5, respectively, Figure 6 shows the VPP cost factor curves when \([K_{\mathrm{VPP},k}^{\mathrm{c}},k_{\mathrm{VPP},k}^{\mathrm{c}}]\) are [40, 0.7], [70, 0.7], and [70, 0.8], respectively. As Figure 6 shows, when the other parameters are given, the larger the \(k_{\mathrm{VPP},k}^{\mathrm{c}}\), the larger is the proportion at which the cost factor starts to grow rapidly. The larger \(K_{\mathrm{VPP},k}^{\mathrm{c}}\) is, the larger the maximum value of \(c_{\mathrm{VPP},k}\) is.