## 2 Description of the Proposed Method
### Optimazation Problem to Determine Operating Points
#### 2.1.1 Model Description  
To formulate the constraints used in the optimization problem, power balance equations and converter models are required. In the case of the VSC, which can synthesize sine-wave AC voltage regardless of the residual network, the active and reactive power output--\(P_{VSC}\) and \(Q_{VSC}\)respectively--are regulated independently [28]. Therefore, a VSC can be considered similar to a \(PQ\) load for the optimization, and the only consideration is the rated output power, as follows:  
\[\sqrt{P_{VSC}^{2}(n)+Q_{VSC}^{2}(n)}<S_{VSC,rate}(n), \tag{1}\]  
where \(S_{VSC,rate}(n)\) is the rated apparent power of the VSC HVDC connected to the \(n\)-th bus. Note that conversion loss of a VSC HVDC is ignored, because the input variables of controller are active and reactive power output at the inverter side. However, for future work, the conversion loss of VSC should be considered for solving overall optimization problems, including a DC system. The reactive power output of the LCC, \(Q_{LCC}\), is different from the VSC, and cannot be regulated independent of the active power, \(P_{LCC}\)[29]. As conversion loss of LCC is less than 1% in general, the loss can be ignored [30]. Thus, \(Q_{LCC}\) can be presented as:  
\[Q_{LCC}(n)=P_{LCC}(n)\tan\Phi(n) \tag{2}\]  
where \(\Phi(n)\) is the power factor angle of LCC HVDC connected to the \(n\)-th bus. The power factor angle can be expressed as follows [31]:  
\[\tan\Phi(n)=\frac{2\mu(n)+\sin(2\alpha(n))-\sin(2\alpha(n)+2\mu(n))}{\cos(2 \alpha(n))-\cos(2\alpha(n)+2\mu(n))} \tag{3}\]  
where  
\[\alpha(n)=\cos^{-1}\Bigg{[}\frac{\pi}{3\sqrt{2}B(n)T(n)V(n)}\bigg{(}V_{DC,rate }(n)+\frac{3X_{C}(n)P_{LCC}(n)}{\pi V_{DC,rate}(n)}\bigg{)}\Bigg{]} \tag{4}\]  
\[\mu(n)=\cos^{-1}\Bigg{[}\cos(\alpha(n))-\frac{\sqrt{2}X_{C}(n)P_{LCC}(n)}{B( n)T(n)V(n)V_{DC,rate}}\Bigg{]}-\alpha \tag{5}\]  
where \(V(n)\) represents voltage magnitude at \(n\)-th bus, and \(V_{DC,rate}(n)\) is the rated DC voltage of the LCC HVDC connected to the \(n\)-th bus. The transformer turn ratio and the number of six-pulse bridges at the \(n\)-th bus are represented by \(T(n)\) and \(B(n)\), respectively. The reactance of the converter transformer at the \(n\)-th bus is described by \(X_{C}(n)\). An inequality constraint of an LCC HVDC can be represented as  
\[P_{LCC}(n)<P_{LCC,rate}(n) \tag{6}\]  
where \(P_{LCC,rate}(n)\) is the rated active power of the LCC HVDC connected to \(n\)-th bus. Note that the index of the bus number (\(n\)) is utilized to express HVDC system connected to the \(n\)-th bus, which means multiple HVDCs can be considered in the proposed models, because each converter is connected to a different bus. For every bus, the injected active (\(P\)) and reactive (\(Q\)) power, including generator output (\(P_{GEN}\) and \(Q_{GEN}\)), VSC output (\(P_{VSC}\) and \(Q_{VSC}\)), LCC output (\(P_{LCC}\) and \(Q_{LCC}\)), and connected load (\(P_{LOAD}\) and \(Q_{LOAD}\)) satisfy the following constraints [32]:  
\[P_{GEN}(n)+P_{VSC}(n)+P_{LCC}(n)-P_{LOAD}(n)-\sum\limits_{k=1}^{N}V(n)V(k) \{G(n,k)\cos(\theta(n)-\theta(k))+B(n,k)\sin(\theta(n)-\theta(k))\}=0 \tag{7}\]  
\[Q_{GEN}(n)+Q_{VSC}(n)+Q_{LCC}(n)-Q_{LOAD}(n)-\sum\limits_{k=1}^{N}V(n)V(k) \{G(n,k)\sin(\theta(n)-\theta(k))-B(n,k)\cos(\theta(n)-\theta(k))\}=0 \tag{8}\]  
where the phase angle of \(n\)-th bus is represented by \(\theta(n)\). An injected active and reactive power of the \(n\)-th bus can be represented by \(P(n)\) and \(Q(n)\), respectively. The real and imaginary parts of the admittance matrix between the \(n\)-th bus and \(k\)-th bus are represented as \(G(n\),\(k)\) and \(B(n\),\(k)\), respectively. If the generator, VSC, LCC, or load is not connected to bus \(n\), the value of each symbol is zero. Note that \(P_{GEN}\) and \(Q_{GEN}\) are pre-determined by the system operator, and \(P_{LOAD}\) and \(Q_{LOAD}\)are previously forecasted values, which are considered as constant in the optimization problem. The security constraint of the AC voltage magnitude for per unit (p.u.)