## 2 Model of VPSSH  
According to topological differences, VPSSH can be divided into doubly-fed types and full-size converter types, both of which have the ability to quickly adjust active power and reactive power. VPSSH has five steady-state operation conditions: generating, pumping, generating phase modulation, pumping phase modulation and shutdown. Consideringthese five operation conditions, the active power and reactive power constraints of VPSH can be expressed as follows:  
\[P_{t,i}^{\text{VPSSH}}=P_{t,i,g}^{\text{VPSSH}}-P_{t,i,p}^{\text{VPSSH}}+P_{t,i, gp}^{\text{VPSSH}}+P_{t,i,pp}^{\text{VPSSH}} \tag{1}\]  
\[Q_{t,j}^{\text{VPSSH}}=Q_{t,j,g}^{\text{VPSSH}}+Q_{t,j,p}^{\text{VPSSH}}+Q_{t,j, gp}^{\text{VPSSH}}+Q_{t,j,pp}^{\text{VPSSH}} \tag{2}\]  
\[\sqrt{\left(P_{t,i}^{\text{VPSSH}}\right)^{2}+\left(Q_{t,i}^{\text{VPSSH}} \right)^{2}}\leq S_{i}^{\text{VPSSH}} \tag{3}\]  
where \(t\) and \(i\) represent the \(t\)th time and \(i\)th VPSSH, \(P_{t,i}^{\text{PSH}}\) and \(Q_{t,i}^{\text{PSH}}\) are the active and reactive power outputs of the VPSSH, respectively; \(P_{t,i,g}^{\text{VPSSH}}\), \(P_{t,i,p}^{\text{VPSSH}}\), \(P_{t,i,pp}^{\text{VPSSH}}\) are the active power output of VPSSH under generating, pumping, generating phase modulation, and pumping phase modulation conditions, respectively; \(Q_{t,i,g}^{\text{VPSSH}}\), \(Q_{t,i,p}^{\text{VPSSH}}\), \(Q_{t,i,pp}^{\text{VPSSH}}\) are the reactive power outputs of VPSSH under generating, pumping, generating phase modulation, modulation, and pumping phase modulation conditions, respectively; \(Q_{t,i,g}^{\text{VPSSH}}\), \(Q_{t,i,p}^{\text{VPSSH}}\), \(Q_{t,i,pp}^{\text{VPSSH}}\) are the reactive power outputs of VPSSH under generating, pumping, generating phase modulation, and pumping phase modulation conditions, respectively; and \(S_{i}^{\text{VPSSH}}\) is the rated capacity of the VPSSH converter. The operation constraints of VSPHS under multi-operation conditions are as follows [18]:  
\[\mu_{t,i,p}^{\text{VPSSH}}+\mu_{t,i,g}^{\text{VPSSH}}+\mu_{t,i,gp}^{\text{VPSSH }}+\mu_{t,i,pp}^{\text{VPSSH}}\leq 1 \tag{4}\]  
\[\mu_{t,i,g}^{\text{VPSSH}}P_{t,i,\min}^{\text{VPSSH}}\leq P_{t,i,g}^{\text{VPSSH }}\leq\mu_{t,i,g}^{\text{VPSSH}}P_{t,i,\max}^{\text{VPSSH}} \tag{5}\]  
\[\mu_{t,i,p}^{\text{VPSSH}}P_{t,i,\min}^{\text{VPSSH}}\leq P_{t,i,g}^{\text{VPSSH }}\leq\mu_{t,i,p}^{\text{VPSSH}}P_{t,i,\max}^{\text{VPSSH}}\]  
\[ \begin{array}{c}-\sqrt{1-\lambda^{2}}P_{t,i,g}^{\text{VPSSH}}/\lambda\leq Q_ {t,i,g}^{\text{VPSSH}}\leq\sqrt{1-\lambda^{2}}P_{t,i,g}^{\text{VPSSH}}/\lambda \\ -\sqrt{1-\lambda^{2}}P_{t,i,p}^{\text{VPSSH}}/\lambda\leq Q_{t,i,p}^{\text{VSH }}\leq\sqrt{1-\lambda^{2}}P_{t,i,p}^{\text{VPSSH}}/\lambda\\ \mu_{t,i,g}^{\text{VPSSH}}Q_{t,i,\min}^{\text{VPSSH}}\leq Q_{t,i,g}^{\text{VPSSH }}\leq\mu_{t,i,gp}^{\text{VPSSH}}P_{t,\max}^{\text{VPSSH}}\\ \mu_{t,ipp}^{\text{VPSSH}}Q_{t,i,\min}^{\text{VPSSH}}\leq Q_{t,i,pp}^{\text{VPSSH }}\leq\mu_{t,i,pp}^{\text{VPSSH}}Q_{t,\max}^{\text{VPSSH}}\end{array}  \tag{6}\]  
where \(\mu_{t,i,p}^{\text{VPSSH}}\), \(\mu_{t,i,g}^{\text{VPSSH}}\), \(\mu_{t,i,gp}^{\text{VPSSH}}\), \(\mu_{t,i,pp}^{\text{VPSSH}}\) are 0-1 variables, and represent whether VPSSH is in generating, pumping, generating phase modulation, and pumping phase modulation conditions, respectively; \(P_{t,i,\min}^{\text{VPSSH}}\), \(P_{t,i,\max}^{\text{VPSSH}}\) are the minimum and maximum active power output of VPSSH, respectively; \(Q_{t,i,\min}^{\text{VPSSH}}\), \(Q_{t,\max}^{\text{VPSSH}}\) are the minimum and maximum reactive power output of VPSSH, respectively; \(P_{t,i,\min}^{\text{VPSSH}}\) is the minimum active power required to maintain VPSSH idling operation at rated speed; and \(\lambda\) is the minimum power factor of the VPSSH converter. Equation (4) is the constraint that the VPSSH can only be under one operation condition at any time. Equation (5) is the active power constraint of VPSSH under each operation condition. Equation (6) is the reactive power constraint of VPSSH under each operation condition.