# Power Flow Calculation Methods for Power Systems with Novel Structure UPFC
## 3 Three Power Flow Calculation Methods
### Handling Conventional Structure UPFC  
For the conventional structure UPFC, according to Figure 3, the power mismatch equations of UPFC buses (bus \(l\) and bus \(m\)) are:  
\[\left\{\begin{array}{l}\Delta P_{l}=P_{ls}-(P_{lm}+P_{ne})-u_{l}\underset{i \in l}{\sum}u_{i}(G_{li}\cos\theta_{li}+B_{li}\sin\theta_{li})\\ \Delta Q_{l}=Q_{ls}-(Q_{lm}+Q_{ne})-u_{l}\underset{i\in l}{\sum}u_{i}(G_{li} \sin\theta_{li}-B_{li}\cos\theta_{li})\\ \end{array}\right. \tag{17}\] \[\left\{\begin{array}{l}\Delta P_{m}=P_{ms}-P_{ml}-u_{m}\underset{i \in m}{\sum}u_{i}(G_{mi}\cos\theta_{mi}+B_{mi}\sin\theta_{mi})\\ \Delta Q_{m}=Q_{ms}-Q_{ml}-u_{m}\underset{i\in m}{\sum}u_{i}(G_{mi}\sin\theta_{ mi}-B_{mi}\cos\theta_{mi})\end{array}\right.\]  
The Jacobian matrix of the system still can be represented by Equation (14). If we assume:  
\[\left\{\begin{array}{l}\Delta\mathbf{P_{2}}=\left[\begin{array}{cc}\Delta P_{ l}&\Delta P_{m}\\ \Delta Q_{2}=\left[\begin{array}{cc}\Delta Q_{l}&\Delta Q_{m}\\ \end{array}\right]^{\mathrm{T}}\\ \Delta\mathbf{\theta_{2}}=\left[\begin{array}{cc}\Delta\theta_{l}&\Delta\theta_{m} \\ \end{array}\right]^{\mathrm{T}}\\ \Delta\mathbf{U_{2}}/\mathbf{U_{2}}=\left[\begin{array}{cc}\Delta u_{l}/u_{l}& \Delta u_{m}/u_{m}\\ \end{array}\right]^{\mathrm{T}}\end{array}\right. \tag{18}\]  
Then, the modifications of the Jacobian matrix are:  
\[\begin{array}{l}\mathbf{H_{U}}=-\left[\begin{array}{cc}\frac{\partial(P_{lm}+P _{ne})}{\partial\theta_{l}}&\frac{\partial(P_{lm}+P_{ne})}{\partial\theta_{m}} \\ \frac{\partial P_{ml}}{\partial\theta_{l}}&\frac{\partial P_{ml}}{\partial\theta_{m}} \\ \end{array}\right]\mathbf{J_{U}}=-\left[\begin{array}{cc}u_{l}\frac{\partial(P_{lm}+P _{ne})}{\partial u_{l}}&u_{m}\frac{\partial(P_{lm}+P_{ne})}{\partial u_{m}} \\ u_{l}\frac{\partial P_{ml}}{\partial\theta_{l}}&u_{m}\frac{\partial P_{ml}}{ \partial u_{m}}\\ \end{array}\right]\mathbf{L_{U}}=-\left[\begin{array}{cc}u_{l}\frac{\partial(Q_{ lm}+Q_{ne})}{\partial u_{l}}&u_{m}\frac{\partial(Q_{lm}+Q_{ne})}{\partial u_{m}} \\ u_{l}\frac{\partial Q_{ml}}{\partial\theta_{l}}&u_{m}\frac{\partial(Q_{lm}+Q_{ne})}{ \partial u_{m}}\\ u_{l}\frac{\partial Q_{ml}}{\partial u_{l}}&u_{m}\frac{\partial(Q_{ml}+Q_{ne})}{ \partial u_{m}}\\ \end{array}\right]\end{array}\right.\end{array} \tag{19}\]  
Compared with Equation (16), it can be observed that because the number of UPFC buses decreases, the items in the modifications decrease. Especially, when the UPFC series converter controls \(P_{ml}+jQ_{ml}\)and the UPFC shunt converter controls the voltage magnitude of bus \(l\), only the partial derivatives of \(P_{lm}+P_{ne}\) need to be calculated in Equation (19). At the same time, according to Equations (1), (2) and (8)-(11), \(P_{lm}+P_{ne}\) can be calculated as:  
\[P_{lm}+P_{ne}=P_{bm}=-P_{c}+\frac{\left(S_{1}{}^{2}+S_{2}{}^{2}\right)\!\!r_{lm} }{u_{m}{}^{2}} \tag{20}\]  
where:  
\[\left\{\begin{array}{l}S_{1}=Q_{c}\times\cos(\theta_{l}-\theta_{m})+P_{c} \times\sin(\theta_{l}-\theta_{m})+u_{m}{}^{2}\times\cos(\theta_{l}-\theta_{m}) b_{lm}\\ S_{2}=Q_{c}\times\sin(\theta_{l}-\theta_{m})-P_{c}\times\cos(\theta_{l}-\theta_{m}) +u_{m}{}^{2}\times\sin(\theta_{l}-\theta_{m})b_{lm}\end{array}\right. \tag{21}\]  
According to Equation (20), only the partial derivative of \(P_{lm}+P_{ne}\) to \(u_{m}\) is not zero. Therefore, at this time, only one item in the Jacobian matrix needs to be modified with the partial derivative of \(P_{lm}+P_{ne}\) to \(u_{m}\) in \(\mathbf{N}\mathbf{u}\).