# Power Flow Calculation Methods for Power Systems with Novel Structure UPFC
## 3 Three Power Flow Calculation Methods
### Three Power Flow Calculation Methods  
Using the power mismatch equations and the Jacobian matrix given above, the power flow of the power systems with novel structure UPFC can be solved by the Newton-Raphson algorithm. However, when the above method (which is referred to as the \"original method\") is used for power flow calculation, the power mismatch equations need to be recalculated according to Equation (13) (or Equation (17)), and the Jacobian matrix needs to be modified according to Equation (14). Notably, because the Jacobian matrix needs to be modified in each step of iterations it is difficult to implement this method in existing power flow calculation programs. Therefore, in practice, the Jacobian matrix is usually not modified, but the equivalent power injections are updated during iterations. The power mismatch equations of this method (which is referred to as the \"simplified method\") are the same as those of the original method, so the same calculation accuracy as the original method can be achieved. However, due to the simplification of the Jacobian matrix, the convergence characteristics of the power flow calculation will deteriorate. Under the same condition, the simplified method may require more iterations or even may not converge. To solve this problem, under the condition that the Jacobian matrix is not changed during the iterations, the impact of the Jacobian matrix modifications can be considered in the UPFC equivalent power injections. According to Equation (14), we have:  
\[\left[ \begin{array}{c}\Delta\mathbf{P_{1}}\\ \Delta\mathbf{P_{2}}\\ \Delta\mathbf{Q_{1}}\\ \Delta\mathbf{Q_{2}}\end{array} \right]-\left[ \begin{array}{cccc}\mathbf{0}&\mathbf{0}& \mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{H_{U}}&\mathbf{0}&\mathbf{N_{U}}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{J_{U}}&\mathbf{0}&\mathbf{L_{U}}\end{array} \right]\left[ \begin{array}{c} \Delta\mathbf{\theta_{1}}\\ \Delta\mathbf{\theta_{2}}\\ \Delta\mathbf{U_{1}}/\mathbf{U_{1}}\\ \Delta\mathbf{U_{2}}/\mathbf{U_{2}}\end{array} \right]=\left[ \begin{array}{c}H_{\mathbf{ 11}}&H_{\mathbf{12}}&N_{\mathbf{11}}&N_{\mathbf{12}}\\ H_{\mathbf{21}}&H_{\mathbf{22}}&N_{\mathbf{21}}&N_{\mathbf{22}}\\ J_{\mathbf{11}}&J_{\mathbf{12}}&L_{\mathbf{11}}&L_{\mathbf{12}}\\ J_{\mathbf{21}}&J_{\mathbf{22}}&L_{\mathbf{21}}&L_{\mathbf{22}}\end{array} \right] \tag{22}\]  
The right side of Equation (22) is the Jacobian matrix used in the simplified method. Therefore, if the power mismatch equations of the UPFC buses are modified so that the system power deviations have the form of the left side of Equation (22), the power flow calculation process without the Jacobian matrix modifications can have the same convergence characteristics as the original method. To achieve this goal, the second items of the left side of Equation (22) can be added to the equivalent power injections of the UPFC. Then, the new equivalent power injections of the UPFC become:  
\[\left[ \begin{array}{c}P_{lm}{}^{\prime}\\ P_{ml}{}^{\prime}\\ P_{me}{}^{\prime}\\ Q_{lm}{}^{\prime}\\ Q_{ml}{}^{\prime}\\ Q_{me}{}^{\prime}\end{array} \right]=\left[ \begin{array}{c}P_{lm}\\ P_{ml}\\ P_{ne}\\ Q_{lm}\\ Q_{me}\end{array} \right]+\left[ \begin{array}{cc}H_{\mathbf{U}}&N_{\mathbf{U}}\\ J_{\mathbf{U}}&L_{\mathbf{U}}\end{array} \right]\left[ \begin{array}{c}\Delta\mathbf{ \theta_{2}}\\ \Delta\mathbf{U_{2}}/\mathbf{U_{2}}\end{array} \right] \tag{23}\]And the power mismatch equations of UPFC buses become:  
\[\left\{