A GOUPFC can be represented by three controllable voltage source converters, and two-phase shifting transformers connected in two transmission lines is shown in Fig. 2.

The series converter controllable voltages and PST voltages are given by,

$$\left. {\begin{array}{*{20}l} {\overline{{V_{{{\rm{se}}_{1} }} }} = r_{1} \overline{{V_{i} }} e^{{j\gamma_{1} }} ;} \hfill & {\overline{{V_{{{\rm{se}}_{2} }} }} = r_{2} \overline{{V_{i} }} e^{{j\gamma_{2} }} } \hfill \\ {\overline{{V_{{\sigma_{1} }} }} = k_{1} \overline{{V_{i} }} e^{{j\sigma_{1} }} ;} \hfill & {\overline{{V_{{\sigma_{2} }} }} = k_{2} \overline{{V_{i} }} e^{{j\sigma_{2} }} } \hfill \\ \end{array} } \right\}$$

(1)

where *r*_{1}, *r*_{2} and *k*_{1}, *k*_{2} are the per unit voltage magnitudes of two series converters and two PSTs, respectively;\(\gamma_{1}\) and \(\gamma_{2}\) are the phase angles of series converters; \(\sigma_{1}\) and \(\sigma_{2}\) are the PST phase angles; and *V*_{i} is the bus-*i* voltage. The voltages and phase angles are operating in the limits specified as follows.

$$\begin{array}{*{20}l} {r_{\min } \le r \le r_{\max } ,} \hfill & {k_{\min } \le k \le k_{\max } } \hfill \\ {\gamma_{\min } \le \gamma \le \gamma_{\max } ,} \hfill & {\sigma_{\min } \le \sigma \le \sigma_{\max } } \hfill \\ \end{array}$$

GOUPFC having three voltage source converters among those one is shunt converter, and the remaining two are series converters. Shunt converter is placed at bus-*I,* and the two series converters are placed in the lines *i*–*j* and *i*–*k*, respectively. In addition to the voltage source converters, GOUPFC employing two separate PSTs which are incorporated in the two separate lines where the two series converters are placed. The total voltages injected in the two lines are the phasor sum of the voltages obtained from the series converters and PSTs, and these are given by,

$$\overline{{V_{{{\rm{inj}}_{1} }} }} = \overline{{V_{{{\rm{se}}_{1} }} }} + \overline{{V_{{\sigma_{1} }} }} ; \quad \overline{{V_{{{\rm{inj}}_{2} }} }} = \overline{{V_{{{\rm{se}}_{2} }} }} + \overline{{V_{{\sigma_{2} }} }}$$

The GOUPFC is modeled in series-connected and shunt-connected voltage source models. The voltages in the two transmission lines behind the line reactance can be written mathematically as,

$$\overline{{V^{\prime }_{ij} }} = \overline{{V_{{{\rm{inj}}_{1} }} }} + \overline{{V_{i} }} ;\;\;\overline{{V^{\prime }_{ik} }} = \overline{{V_{{{\rm{inj}}_{2} }} }} + \overline{{V_{i} }}$$

### Series-connected voltage source model

The series-connected voltage source model of GOUPFC with three buses *i*, *j*, and *k* is shown in Fig. 2a, and the corresponding Norton’s equivalent current source model is shown in Fig. 2b. Let the currents flowing in the two lines *i*–*j* and *i*–*k* are \(I_{{{\rm{se}}_{1} }}\) and \(I_{{{\rm{se}}_{2} }}\), respectively, are given by

$$\left. \begin{gathered} \overline{{I_{{{\rm{se}}_{1} }} }} = \frac{{\overline{{V_{{{\rm{inj}}_{1} }} }} }}{{jX_{{{\rm{se}}_{1} }} }} = - jB_{{{\rm{se}}_{1} }} \overline{{V_{{{\rm{inj}}_{1} }} }} = - B_{{{\rm{se}}_{1} }} V_{i} \left( {r_{1} e^{{j\left( {90 + \gamma_{1} + \theta_{i} } \right)}} + k_{1} e^{{j\left( {90 + \sigma_{1} + \theta_{i} } \right)}} } \right) \hfill \\ \overline{{I_{{{\rm{se}}_{2} }} }} = \frac{{\overline{{V_{{{\rm{inj}}_{2} }} }} }}{{jX_{{{\rm{se}}_{2} }} }} = - jB_{{{\rm{se}}_{2} }} \overline{{V_{{{\rm{inj}}_{2} }} }} = - B_{{{\rm{se}}_{2} }} V_{i} \left( {r_{2} e^{{j\left( {90 + \gamma_{2} + \theta_{i} } \right)}} + k_{2} e^{{j\left( {90 + \sigma_{2} + \theta_{i} } \right)}} } \right) \hfill \\ \end{gathered} \right\}$$

(2)

where \(X_{{{\text{se}}_{{1}} }}\) and \(X_{{{\text{se}}_{{2}} }}\) are the line reactances of the two lines and their corresponding susceptances, respectively, given by, \(B_{{{\text{se}}_{{1}} }} = 1{/}X_{{{\text{se}}_{{1}} }}\) and \(B_{{{\text{se}}_{{2}} }} = 1{/}X_{{{\text{se}}_{{2}} }}\).

The independent complex power injections of GOUPFC at buses *i*, *j* and *k* are given as,

$$\overline{{S_{i{\rm{se}}} }} = - \overline{{V_{i} }} \left( {\overline{{I_{{{\rm{se}}_{1} }} }} } \right)^{*} - \overline{{V_{i} }} \left( {\overline{{I_{{{\rm{se}}_{1} }} }} } \right)^{*}$$

$$\overline{{S_{i{\rm{se}}} }} = - V_{i}^{2} B_{{{\rm{se}}_{1} }} \left( {r_{1} e^{{ - j\left( {90 + \gamma_{1} } \right)}} + k_{1} e^{{ - j\left( {90 + \sigma_{1} } \right)}} } \right) - V_{i}^{2} B_{{{\rm{se}}_{2} }} \left( {r_{2} e^{{ - j\left( {90 + \gamma_{2} } \right)}} + k_{2} e^{{ - j\left( {90 + \sigma_{2} } \right)}} } \right)$$

(3)

$$\overline{{S_{j{\rm{se}}} }} = \overline{{V_{j} }} \left( {\overline{{I_{{{\rm{se}}_{1} }} }} } \right)^{*} = - V_{i} V_{j} B_{{{\rm{se}}_{1} }} \left( {r_{1} e^{{ - j\left( {90 + \gamma_{1} + \theta_{i} - \theta_{j} } \right)}} + k_{1} e^{{ - j\left( {90 + \sigma_{1} + \theta_{i} - \theta_{j} } \right)}} } \right)$$

(4)

$$\overline{{S_{k{\rm{se}}} }} = \overline{{V_{k} }} \left( {\overline{{I_{{{\rm{se}}_{2} }} }} } \right)^{*} = - V_{i} V_{k} B_{{{\rm{se}}_{2} }} \left( {r_{2} e^{{ - j\left( {90 + \gamma_{2} + \theta_{i} - \theta_{k} } \right)}} + k_{2} e^{{ - j\left( {90 + \sigma_{2} + \theta_{i} - \theta_{k} } \right)}} } \right)$$

(5)

Let \(\theta_{ij} = \theta_{i} - \theta_{j}\); \(\theta_{ik} = \theta_{i} - \theta_{k}\). By using Euler’s and trigonometric identities, the real and reactive power injections at the buses *i*, *j* and *k* are calculated as follows.

$$P_{{i_{\rm{se}} }} = - V_{i}^{2} B_{{{\rm{se}}_{1} }} \left( {r_{1} \sin \left( {\gamma_{1} } \right) + k_{1} \sin \left( {\sigma_{1} } \right)} \right) - V_{i}^{2} B_{{{\rm{se}}_{2} }} \left( {r_{2} \sin \left( {\gamma_{2} } \right) + k_{2} \sin \left( {\sigma_{2} } \right)} \right)$$

(6)

$$Q_{{i_{\rm{se}} }} = - V_{i}^{2} B_{{{\rm{se}}_{1} }} \left( {r_{1} \cos \left( {\gamma_{1} } \right) + k_{1} \cos \left( {\sigma_{1} } \right)} \right) - V_{i}^{2} B_{{{\rm{se}}_{2} }} \left( {r_{2} \cos \left( {\gamma_{2} } \right) + k_{2} \cos \left( {\sigma_{2} } \right)} \right)$$

(7)

$$P_{{j_{\rm{se}} }} = V_{i} V_{j} B_{{{\rm{se}}_{1} }} (r_{1} \sin (\gamma_{1} + \theta_{ij} ) + k_{1} \sin (\sigma_{1} + \theta_{ij} ))$$

(8)

$$Q_{{j_{\rm{se}} }} = V_{i} V_{j} B_{{{\rm{se}}_{1} }} (r_{1} \cos (\gamma_{1} + \theta_{ij} ) + k_{1} \cos (\sigma_{1} + \theta_{ij} ))$$

(9)

$$P_{{k_{\rm{se}} }} = V_{i} V_{k} B_{{{\rm{se}}_{2} }} (r_{2} \sin (\gamma_{2} + \theta_{ik} ) + k_{2} \sin (\sigma_{2} + \theta_{ik} ))$$

(10)

$$Q_{{k_{\rm{se}} }} = V_{i} V_{k} B_{{{\rm{se}}_{2} }} (r_{2} \cos (\gamma_{2} + \theta_{ik} ) + k_{2} \cos (\sigma_{2} + \theta_{ik} ))$$

(11)

The equivalent power injection modeling for the series-connected voltage source model is shown in Fig. 3a. The amount of complex power supplied by the combination of series converter and PST in the individual lines is derived as follows

$$\begin{aligned} \overline{{S_{{{\rm{se}}_{1} }} }} & = P_{{{\rm{se}}_{1} }} + jQ_{{{\rm{se}}_{1} }} = \overline{{V_{{{\rm{inj}}_{1} }} }} \left( {\overline{{I_{ij} }} } \right)^{*} \\ \overline{{S_{{{\rm{se}}_{1} }} }} & = jB_{{{\rm{se}}_{1} }} V_{i} \left( {r_{1} e^{{j\left( {\gamma_{1} + \theta_{i} } \right)}} + k_{1} e^{{j\left( {\sigma_{1} + \theta_{i} } \right)}} } \right)\left( {\overline{{V^{^{\prime}}_{ij} }} - \overline{{V_{j} }} } \right)^{*} \\ \end{aligned}$$

(12)

$$\begin{aligned} \overline{{S_{{{\rm{se}}_{2} }} }} & = P_{{{\rm{se}}_{2} }} + jQ_{{{\rm{se}}_{2} }} = \overline{{V_{{{\rm{inj}}_{2} }} }} \left( {\overline{{I_{ik} }} } \right)^{*} \\ \overline{{S_{{{\rm{se}}_{2} }} }} & = jB_{{{\rm{se}}_{2} }} V_{i} \left( {r_{2} e^{{j(\gamma_{2} + \theta_{i} )}} + k_{2} e^{{j(\sigma_{2} + \theta_{i} )}} } \right)\left( {\overline{{V^{\prime }_{ik} }} - \overline{{V_{k} }} } \right)^{*} \\ \end{aligned}$$

(13)

The real and reactive powers supplied by the two converters in the two transmission lines are given by,

$$\begin{aligned} P_{{{\rm{se}}_{1} }} & = - r_{1} B_{{{\rm{se}}_{1} }} V_{i}^{2} \sin \left( {\gamma_{1} } \right) - k_{1} B_{{{\rm{se}}_{1} }} V_{i}^{2} \sin \left( {\sigma_{1} } \right) \\ & \quad + r_{1} B_{{{\rm{se}}_{1} }} V_{i} V_{j} \sin \left( {\gamma_{1} + \theta_{ij} } \right) + k_{1} B_{{{\rm{se}}_{1} }} V_{i} V_{j} \sin \left( {\sigma_{1} + \theta_{ij} } \right) \\ \end{aligned}$$

(14)

$$\begin{aligned} Q_{{{\rm{se}}_{1} }} & = B_{{{\rm{se}}_{1} }} V_{i}^{2} \left( {r_{1}^{2} + k_{1}^{2} } \right) + 2r_{1} k_{1} B_{{{\rm{se}}_{1} }} V_{i}^{2} \cos \left( {\sigma_{1} - \gamma_{1} } \right) \\ & \quad + r_{1} B_{{{\rm{se}}_{1} }} V_{i}^{2} \cos \left( {\gamma_{1} } \right) + k_{1} B_{{{\rm{se}}_{1} }} V_{i}^{2} \cos \left( {\sigma_{1} } \right) \\ & & \quad - r_{1} B_{{{\rm{se}}_{1} }} V_{i} V_{j} \cos \left( {\gamma_{1} + \theta_{ij} } \right) - k_{1} B_{{{\rm{se}}_{1} }} V_{i} V_{j} \cos \left( {\sigma_{1} + \theta_{ij} } \right) \\ \end{aligned}$$

(15)

$$\begin{aligned} P_{{{\rm{se}}_{2} }} & = - r_{2} B_{{{\rm{se}}_{2} }} V_{i}^{2} \sin \left( {\gamma_{2} } \right) - k_{2} B_{{{\rm{se}}_{2} }} V_{i}^{2} \sin \left( {\sigma_{2} } \right) \\ & \quad + r_{2} B_{{{\rm{se}}_{2} }} V_{i} V_{k} sin\left( {\gamma_{2} + \theta_{ik} } \right) + k_{2} B_{{{\rm{se}}_{2} }} V_{i} V_{k} sin\left( {\sigma_{2} + \theta_{ik} } \right) \\ \end{aligned}$$

(16)

$$\begin{aligned} Q_{{{\rm{se}}_{2} }} & = B_{{{\rm{se}}_{2} }} V_{i}^{2} \left( {r_{2}^{2} + k_{2}^{2} } \right) + 2r_{2} k_{2} B_{{{\rm{se}}_{2} }} V_{i}^{2} \cos \left( {\sigma_{2} - \gamma_{2} } \right) \\ & \quad + r_{2} B_{{{\rm{se}}_{2} }} V_{i}^{2} \cos \left( {\gamma_{2} } \right) + k_{1} B_{{{\rm{se}}_{1} }} V_{i}^{2} \cos \left( {\sigma_{2} } \right) \\ & \quad - r_{2} B_{{{\rm{se}}_{2} }} V_{i} V_{k} \cos \left( {\gamma_{2} + \theta_{ik} } \right) - k_{2} B_{{{\rm{se}}_{2} }} V_{i} V_{k} \cos \left( {\sigma_{2} + \theta_{ik} } \right) \\ \end{aligned}$$

(17)

### Shunt-connected voltage source model

The equivalent circuit for the shunt-connected voltage source model is shown in Fig. 3b. This model provides the equivalent power injections at the GOUPFC shunt bus to the two series branches through the converter and PST combination. The voltage at the sending end is controlled by the reactive power injection at the GOUPFC shunt converter. The amount of real power supplied by the shunt converter is equal to the real power consumed by the two series converters. Therefore, the real power inserted at the shunt converter is given by:

$$P_{\rm{sh}} = - \left( {P_{{{\rm{se}}_{1} }} + P_{{{\rm{se}}_{2} }} } \right)$$

$$\begin{aligned} P_{\rm{sh}} & = r_{1} B_{{{\rm{se}}_{1} }} V_{i}^{2} \sin \left( {\gamma_{1} } \right) + k_{1} B_{{{\rm{se}}_{1} }} V_{i}^{2} \sin \left( {\sigma_{1} } \right) + r_{2} B_{{{\rm{se}}_{2} }} V_{i}^{2} \sin \left( {\gamma_{2} } \right) + k_{2} B_{{{\rm{se}}_{2} }} V_{i}^{2} \sin \left( {\sigma_{2} } \right) \\ & \quad - r_{1} B_{{{\rm{se}}_{1} }} V_{i} V_{j} \sin \left( {\gamma_{1} + \theta_{ij} } \right) - k_{1} B_{{{\rm{se}}_{1} }} V_{i} V_{j} \sin \left( {\sigma_{1} + \theta_{ij} } \right) \\ & \quad - r_{2} B_{{{\rm{se}}_{2} }} V_{i} V_{k} \sin \left( {\gamma_{2} + \theta_{ik} } \right) - k_{2} B_{{{\rm{se}}_{2} }} V_{i} V_{k} \sin \left( {\sigma_{2} + \theta_{ik} } \right) \\ \end{aligned}$$

(18)

Let assume a constant reactive power injection *Q*_{sh} at bus-*i* and the apparent power injection at the shunt bus-*i* is given as

$$S_{{{\text{sh}}}} = P_{{{\text{sh}}}} + jQ_{{{\text{sh}}}}$$

### Final GOUPFC modeling

The final PIM of GOUPFC is achieved by summing up the equations obtained in series- and shunt-connected models. The corresponding equivalent circuit representing GOUPFC power injection is shown in Fig. 4. The resultant real and reactive power equations at the GOUPFC buses are given as

$$P_{{i_{{{\text{goupfc}}}} }} = P_{{i_{{{\text{se}}}} }} + P_{{{\text{sh}}}}$$

$$\begin{aligned} P_{{i_{{{\text{goupfc}}}} }} & = - r_{1} B_{{{\rm{se}}_{1} }} V_{i} V_{j} \sin \left( {\gamma_{1} + \theta_{ij} } \right) - k_{1} B_{{{\rm{se}}_{1} }} V_{i} V_{j} \sin \left( {\sigma_{1} + \theta_{ij} } \right) \\ & \quad - r_{2} B_{{{\rm{se}}_{2} }} V_{i} V_{k} \sin \left( {\gamma_{2} + \theta_{ik} } \right) - k_{2} B_{{{\rm{se}}_{2} }} V_{i} V_{k} \sin \left( {\sigma_{2} + \theta_{ik} } \right) \\ \end{aligned}$$

(19)

$$Q_{{i_{{{\text{goupfc}}}} }} = Q_{{i_{{{\text{se}}}} }} + Q_{{{\text{sh}}}}$$

$$\begin{aligned} Q_{{i_{{{\text{goupfc}}}} }} & = - V_{i}^{2} B_{{{\rm{se}}_{1} }} \left( {r_{1} \cos \left( {\gamma_{1} } \right) + k_{1} \cos \left( {\sigma_{1} } \right)} \right) \\ & \quad - V_{i}^{2} B_{{{\rm{se}}_{2} }} \left( {r_{2} \cos \left( {\gamma_{2} } \right) + k_{2} \cos \left( {\sigma_{2} } \right)} \right) + Q_{\rm{sh}} \\ \end{aligned}$$

(20)

$$P_{{j_{{{\text{goupfc}}}} }} = P_{{j_{\rm{se}} }} = V_{i} V_{j} B_{{{\rm{se}}_{1} }} \left( {r_{1} \sin \left( {\gamma_{1} + \theta_{ij} } \right) + k_{1} \sin \left( {\sigma_{1} + \theta_{ij} } \right)} \right)$$

(21)

$$Q_{{j_{{{\text{goupfc}}}} }} = Q_{{j_{\rm{se}} }} = V_{i} V_{j} B_{{{\rm{se}}_{1} }} \left( {r_{1} \cos \left( {\gamma_{1} + \theta_{ij} } \right) + k_{1} \cos \left( {\sigma_{1} + \theta_{ij} } \right)} \right)$$

(22)

$$P_{{k_{{{\text{goupfc}}}} }} = P_{{k_{\rm{se}} }} = V_{i} V_{k} B_{{{\rm{se}}_{2} }} \left( {r_{2} \sin \left( {\gamma_{2} + \theta_{ik} } \right) + k_{2} \sin \left( {\sigma_{2} + \theta_{ik} } \right)} \right)$$

(23)

$$Q_{{k_{{{\text{goupfc}}}} }} = Q_{{k_{\rm{se}} }} = V_{i} V_{k} B_{{{\rm{se}}_{2} }} \left( {r_{2} \cos \left( {\gamma_{2} + \theta_{ik} } \right) + k_{2} \cos \left( {\sigma_{2} + \theta_{ik} } \right)} \right)$$

(24)