# Power Flow Calculation Methods for Power Systems with Novel Structure UPFC
## 2 The General Structure UPFC and Its Equivalent Model
### Power Flow of the UPFC Series Side  
There are multiple control strategies for the UPFC series converter, mainly including: power flow regulation, series reactive compensation, phase angle shifting and terminal voltage regulation [24; 25]. When these control strategies are adopted, the power flows of the UPFC series side can be calculated by the following method:  
(1) Power flow regulation  
Under this strategy, the active power flow and the reactive power flow of the transmission line are controlled. If we assume \(P_{ml}+jQ_{ml}\) is controlled to \(P_{c}+jQ_{c}\) (\(P_{c}\) and \(Q_{c}\) are the targets of power flow regulation), then according to Figure 2c we have:  
\[\dot{\underline{i}}_{2}=\left(\frac{P_{c}+jQ_{c}}{\underline{u}_{m}}\right)^{ \star}-\underline{u}_{m}\times jb_{lm} \tag{1}\]  
\[\underline{u}_{k}=\underline{u}_{m}-(r_{lm}+jx_{lm})\times\dot{\underline{i}} _{2} \tag{2}\]  
where \({}^{\star}\) denotes the conjugate of a complex number.  
(2) Series reactive compensation  
Under this strategy, the series side of the UPFC is controlled to be an equivalent series impedance. If we assume the equivalent impedance is \(\underline{Z}_{c}\), then according to Figure 2c we have:  
\[\underline{u}_{k}=\frac{\underline{Z}_{c}\times\underline{u}_{l}+\underline{ Z}_{l}\times\underline{u}_{m}}{\underline{Z}_{c}+\overline{Z}_{l}+jb_{lm} \underline{Z}_{l}\underline{Z}_{c}} \tag{3}\]  
where:  
\[\underline{Z}_{l}=r_{lm}+jx_{lm} \tag{4}\]  
(3) Phase angle shifting  
Figure 2: The general structure of UPFC and its equivalent model: (**a**) the general structure of UPFC; (**b**) the equivalent model of the general structure UPFC; (**c**) the equivalent circuit of the general structure UPFC.  
Under this strategy, the voltage magnitude of bus \(k\) is the same as that of bus \(l\). If we assume the desired phase angle difference between bus \(k\) and bus \(l\) is \(\theta\), then we have:  
\[\underline{u}_{k}=u_{k}\angle\theta_{k}=u_{l}\angle(\theta_{l}+\theta_{c}) \tag{5}\]  
where:  
\[\underline{u}_{l}=u_{l}\angle\theta_{l} \tag{6}\]  
(4) Terminal voltage regulation  
Under this strategy, the phase angle of bus \(k\) is the same as that of bus \(l\). If we assume the desired voltage magnitude of bus \(k\) is \(u_{c}\), then we have:  
\[\underline{u}_{k}=u_{k}\angle\theta_{k}=u_{c}\angle\theta_{l} \tag{7}\]  
According to Equations (2), (3), (5), and (7), it can be found that \(\underline{u}_{k}\) can always be represented by \(\underline{u}\), \(\underline{u}_{m}\) and the control targets of UPFC, no matter what control strategy is adopted. Then, according to Figure 2c, the power flows of the UPFC series side can be calculated by:  
\[P_{lm}+jQ_{lm}=\underline{u}_{l}\times\left(\underline{u}_{k}\times jb_{lm}+ \frac{\underline{u}_{k}-\underline{u}_{m}}{r_{lm}+jx_{lm}}\right)^{*} \tag{8}\]  
\[P_{ml}+jQ_{ml}=\underline{u}_{m}\times\left(\underline{u}_{m}\times jb_{lm}- \frac{\underline{u}_{k}-\underline{u}_{m}}{r_{lm}+jx_{lm}}\right)^{*} \tag{9}\]  
\[P_{km}+jQ_{km}=\underline{u}_{k}\times\left(\underline{u}_{k}\times jb_{lm}+ \frac{\underline{u}_{k}-\underline{u}_{m}}{r_{lm}+jx_{lm}}\right)^{*} \tag{10}\]  
When the UPFC has multiple series converters connected in series with multiple lines, the power flows of each line can be calculated according to Equations (8)-(10).