## 3 Generation-Load Uncertainty Model
### Load Uncertainty Model Considering Demand-Side Management
#### 3.2.1 Uncertainty Model of Day-Ahead Price Demand Response Virtual Unit  
The electricity consumption behavior of users in the electricity price demand response is fixed, and the response time is long, but the adjustable range is limited. In the day-ahead dispatching stage, the electricity price mechanism is used to guide users to choose more economical electricity consumption mode, so that load can be flexibly adjusted. Under the background of the energy Internet of things and the strong smart grid system, the number of users participating in demand response increases. Therefore, it is necessary that electricity price demand response in a specific area should be assembled into a virtual unit for scheduling, named a price demand response virtual unit. The decision variable of this virtual unit is the electricity price, i.e., the change of the price mechanism affects the output of the virtual unit.  
The demand response based on the price elasticity coefficient is studied from the economic point of view. When the electricity price is high, less electricity is used. On the contrary, more electricity is used. The power sector adjusts the user's electricity consumption through the electricity price. The influence of the price change rate on the load change rate is represented by the self-elastic coefficient, which is defined as follows:  
\[\phi_{\Delta\text{L},t}=e_{tt}\times\phi_{\Delta\rho,t} \tag{4}\]  
where \(\phi_{\Delta\text{L},t}\) is the load response rate of price response virtual unit at time \(t\), \(\phi_{\Delta\rho,t}\) is the price change rate at time \(t\), and \(e_{tt}\) is the self-elasticity coefficient at time \(t\).  
Users participate in the demand response according to the voluntary principle. The actual response quantity of load is random and cannot be completely determined. In this paper, the triangular membership function is used to describe the uncertainty of the PDR load response rate:  
\[\begin{split}&\widetilde{\phi}_{\Delta t,t}=(\phi_{\Delta\text{L },1,t},\phi_{\Delta\text{L},2,t},\phi_{\Delta\text{L},3,t})\\ &\left\{\begin{array}{l}\phi_{\Delta\text{L},2,t}=e_{tt}\phi_{ \Delta\rho,t}\\ \phi_{\Delta\text{L},1,t}=\phi_{\Delta\text{L},2,t}-\Delta\delta_{t}\\ \phi_{\Delta\text{L},3,t}=\phi_{\Delta\text{L},2,t}+\Delta\delta_{t}\end{array}\right. \end{split} \tag{5}\]  
where \(\widetilde{\phi}_{\Delta t,t}\) is the fuzzy expression of load response rate \(\phi_{\Delta\text{L},t}\) in time period \(t\). \(\phi_{\Delta\text{L},1,t}\), \(\phi_{\Delta\text{L},2,t}\) and \(\phi_{\Delta\text{L},3,t}\) are membership parameters. \(e_{tt}\) is the self-elastic coefficient of the price elasticity matrix in time period \(t\). \(\Delta\delta_{t}\geq 0\), is the maximum error value of load response rate prediction at time \(t\), which is related to the price change rate and the detailed calculation process is shown in [20].  
The triangular numbers in the fuzzy expression of load demand response rate are converted into a certain variable, then the expected value of them can represent the users' actual response quantity due to price changes [21], as follows:  
\[P_{\text{PDR},t,\text{act}}=\frac{1}{4}P_{\text{PDR},t}(\phi_{\Delta\text{L},t}+2\phi_{\Delta\text{L},1,t}+\phi_{\Delta\text{L},3,t}) \tag{6}\]  
where \(P_{\text{PDR},t,\text{act}}\) is the PDR users' actual response power in day-ahead stage at time \(t\), \(P_{\text{PDR},t}\) is the system load when the users do not participate in the price demand response at time \(t\).