# The Optimal Dispatch of a Power System Containing Virtual Power Plants under Fog and Haze Weather
## 4 The Mathematical Model of Dispatch under F-H Weather
### Objective Functions  
(1) Minimizing generation cost and start-up (shut-down) cost of TPPs: \(f_{1}\). \[ \begin{gathered}\min f_{1}=\sum\limits_{i=1}^{M}\sum\limits_{t=1} ^{T}\left[A_{i}(P_{Gi,t})\cdot I_{Gi,t}+SU_{i,t}+SD_{i,t}\right]\\ A_{i}(P_{Gi,t})=a_{Qi}\cdot(P_{Gi,t})^{2}+b_{Qi}\cdot P_{Gi,t}+c_{Qi },SU_{i,t}=C_{SUi}\cdot y_{Gi,t},SD_{i,t}=C_{SDi}\cdot z_{Gi,t-1}\end{gathered}  \tag{14}\]  
where \(M\) is the number of thermal power units; \(T\) is the number of time intervals during the dispatch period; \(P_{Gi,t}\) and \(A_{i}(P_{Gi,t})\) are the output and generation cost of unit \(i\) in period \(t\); \(I_{Gi,t}\) is the running state of unit \(i\) in period \(t\), with unit \(i\) on/off in period \(t\) when \(I_{Gi,t}=1/0\); \(a_{Qi}\), \(b_{Qi}\) and \(c_{Qi}\) are the cost coefficients (constants) of unit \(i\); \(SU_{i,t}\) and \(SD_{i,t}\) are the start-up and shut-down costs of unit \(i\) in period \(t\); \(C_{SUi}\) and \(C_{SDi}\) are the cost coefficients of the start-up and shut-down of unit \(i\); and \(y_{Gi,t}=1\) (\(z_{Gi,t}=1\)) means that unit \(i\) is in the process of start-up (shut-down) in period \(t\); otherwise, \(y_{Gi,t}=0\) (\(z_{Gi,t}=0\)). Figure 8: Schematic diagram of VPPs participating in the dispatch. TPP, thermal power plant; HPP, hydraulic power plant; GPP, geothermal power plant; NPP, nuclear power plant. The relations among \(y_{Gi,t}\), \(z_{Gi,t}\) and \(I_{Gi,t}\) (constraints \(C_{0}\)) are shown as Formula (15), which are additional equality constraints (\(C_{01}\)) and inequality constraints (\(C_{02}\)), because of the added variables (\(y_{Gi,t}\) and \(z_{Gi,t}\)). \[ \begin{array}{c}y_{Gi,t}-z_{Gi,t}=I_{Gi,t}-I_{Gi,t-1},y_{Gi,t}-z_{Gi,t}\leqslant 1 \\ i=1,2,...M,t=1,2,...T\end{array}  \tag{15}\]  
where \(I_{Gi,0}=1/0\) (\(t=1\)) means that unit \(I\) is on/off before the dispatch period. (2) Minimizing pollution emission of TPPs (considering the penalty factor of AQI): \(f_{2}\). Gaseous pollutants, such as SO\({}_{2}\), NO\({}_{\times}\), PM10 and PM2.5, are the main emissions considered in this paper. The concept of nominal environmental compensation cost is introduced to transform the pollutant emission of the thermal power unit into emission cost. \[ \begin{array}{c}\min f_{2}=\sum_{i=1}^{M}\left[C_{Fi}\cdot\sum_{t=1}^{T}B_{i }(P_{Gi,t})\cdot I_{Gi,t}\right],B_{i}(P_{Gi,t})=a_{Wi}\cdot(P_{Gi,t})^{2}+b_ {Wi}\cdot P_{Gi,t}+c_{Wi}\end{array}  \tag{16}\]  
where \(B_{i}(P_{Gi,t})\) is the emission amount of unit \(i\) in period \(t\); \(a_{Wi}\), \(b_{Wi}\) and \(c_{Wi}\) are the pollutant discharge coefficients of unit \(i\); and \(C_{Fi}\) is the cost coefficient of pollutant emissions of unit \(i\). In addition, the AQI level (\(L_{\text{AQI}}\)), as a penalty term, is added to the calculation of \(C_{Fi}\) to consider the effect of F-H in the dispatch. The economic punishment varies with \(L_{\text{AQI}}\) and the emission characteristic of the unit; namely, a unit with heavier pollution would receive more serious punishment when \(L_{\text{AQI}}\) is higher. \[C_{Fi}=C_{Bi}\cdot\varphi(L_{AQI}),L_{AQI}=1,2,...,6 \tag{17}\]  
where \(\varphi(L_{\text{AQI}})\) is the growth factor function of emission cost and \(C_{Bi}\) is the cost coefficient of the pollutant emission of unit \(i\) when \(L_{\text{AQI}}=1\) (the air condition is excellent); namely, \(\varphi(1)=1\) and \(C_{Fi}=C_{Bi}\). Generally, the worse the emission characteristics of a unit is, the larger the \(C_{Bi}\) value, which causes the unit to stay in an inferior position in the dispatch considering the environmental benefits. Therefore, it could be deduced according to the above analysis that \(\varphi(L_{\text{AQI}})\) might be an increasing function, which would be supposed to be a linear one since there are no studies on this expression.