## 3 DR Resource Allocation Strategy Based on Multiple Response Modes  
### Scenario Generation and Reduction Models Considering Wind Uncertainty  
The uncertainty of wind speed causes wind power forecast errors, affecting the stable operation of power systems. Wind power forecast errors result from multiple factors such as season, climate, geography, and the distribution of wind farms. If the wind power \begin{table}
\begin{tabular}{c c c} \hline \hline
**Constraint Type** & **DR Resources** & **Thermal Units** \\ \hline \multirow{3}{*}{Output constraint} & Minimum response capacity & Maximum output \\  & Maximum response capacity & Minimum output \\  & Minimum response interval time & Minimum offline hours \\ Time-dependent constraint & Maximum response amount & Startup/shutdown power \\  & Maximum response duration & constraint \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of regulation constraints between thermal units and DR resources. forecast value is used as the only reference data for day-ahead dispatching, it will inevitably make the dispatch results deviate from the real value. Therefore, it is necessary to use probabilistic analysis of wind power forecast errors to improve the accuracy of the wind power forecast so that it can be combined with DR to achieve cooperative effects [23]. Wind power forecast errors are generally considered to obey the Gaussian distribution. They are assumed to be with the variance of \(\sigma_{w}^{2}\) and a mean value of 0 at time \(t\) before the day [24], which can be expressed as:  
\[\left\{ \begin{array}{l}\varepsilon_{w}^{t}=W_{a}^{t}-W_{f}^{t}\\ \varepsilon_{w}^{t}:N(0,\sigma_{w}^{2})\end{array} \right. \tag{10}\]  
where \(W_{f}^{t}\) and \(W_{a}^{t}\) are the predicted value and the actual output of wind power in each period, respectively. \(\sigma_{w}^{t}\) represents the standard deviation, set to 0.1 \(W_{f}^{t}\) here. \(\varepsilon_{w}^{t}\) represents the wind power forecast error. (1) Initial scenario generation  
To address the uncertainty of the wind power forecast, the Monte Carlo simulation (MCS) method is used to generate the initial scenarios [25]. Here, we set the predicted wind power output at each time of the day as \(p_{w}(t),t\in(1,2,\ldots,T)\). A scenario \(s\) is a complete scheduling time cycle. The different scenarios form a set of scenarios \(\Omega_{w}^{s}\)[26], \(\Omega_{w}^{s}=\left\{p_{w,1}^{s},p_{w,2}^{s}\cdots p_{w,T}^{s}\right\},s\in (1,2,\ldots S)\). (2) Scenario reduction  
The above set of scenarios tends to enlarge the set scale because of the need to ensure the sampling accuracy, which leads to a low efficiency of problem solving. To balance the computation efficiency and accuracy, the backward reduction (BR) method is applied to the scenario reduction process [27]. The probability distance of the set of predicted scenarios \(\Omega_{w}^{s}\) for wind farms for two different scenarios is defined as: if scenario \(\Omega_{w}^{i}\) and scenario \(\Omega_{w}^{j}\) occur with probability \(\lambda_{i}\) and \(\lambda_{j}\), respectively, and the sum of the probability of all the scenarios is 1, then the probability distance between scenario \(\Omega_{w}^{i}\) and scenario \(\Omega_{w}^{j}\) can be expressed as:  
\[d(\Omega_{i}\Omega_{j})=\sqrt{\sum\limits_{t=0}^{T}{(p_{w,t}^{i}-p_{w,t}^{j}) }^{2}} \tag{11}\]  
The process of the set of the wind power forecast scenario includes the following three steps:  
* Step 1: Set a set of scenarios \(J=J_{0}\) as the initial set and make \(J_{0}\) as an empty set. Set the initial iteration number \(k=0\). * Step 2: Calculate \(\sum\limits_{i\in J}{\lambda_{i}*\min_{i\
otin J}{d(\Omega_{i},\Omega_{j})}}\). Each iteration needs to determine the deleted scenario, for example, the \(k\)th iteration needs to delete the scenario \(a_{k}\).