# Minimum Frequency and Voltage Stability Constrained Unit Commitment for AC/DC Transmission Systems
## 2 HVDC Transmission System Model
### LCC-HVDC Model  
Figure 1 presents a typical LCC-HVDC system which is connected to AC bus \(m\) and \(n\) through coupling transformers. In order to model the HVDC which is linked to the AC system, seven converter variables, i.e., \(V_{\text{dR}}\), \(V_{\text{dI}}\), \(I_{\text{d}}\), \(k_{\text{R}}\), \(K_{\text{I}}\), \(\alpha\), and \(\gamma\) are considered in Figure 1. These variables determine the HVDC operating state.  
The whole HVDC is modeled by Equations (1)-(7). The converter Equations (1) and (2) express \(V_{\text{dR}}\) (\(V_{\text{dI}}\)) in terms of \(V_{\text{m}}\) (\(V_{\text{n}}\)), \(k_{\text{R}}\) (\(k_{\text{I}}\)), \(\alpha\) (\(\gamma\)), and other converter variables while the coupling transformers are assumed to be lossless. Equation (3) represents the DC voltage-current relationship which depends on the DC transmission system configuration (i.e., \(V_{\text{dR}}\), \(V_{\text{dI}}\), and \(I_{\text{d}}\)). After getting the active power of both rectifier and inverter by Equations (4) and (5), the exchange reactive power between AC and DC system could be calculated through Equations (6) and (7) by considering reactive compensation capacity \(Q_{\text{CI}}\) and \(Q_{\text{CR}}\).  
\[V_{dR}=\frac{3\sqrt{2}}{\pi}n_{R}k_{\text{R}}V_{m}\cos\alpha-\frac{3}{\pi}n_{R }X_{c,R}I_{d} \tag{1}\]  
\[V_{dl}=\frac{3\sqrt{2}}{\pi}n_{l}k_{\text{I}}V_{n}\cos\gamma-\frac{3}{\pi}n_{l }X_{c}I_{d} \tag{2}\]  
\[V_{dR}=V_{dl}+R_{d}I_{d} \tag{3}\]  
\[P_{dR}=V_{dR}I_{d} \tag{4}\]  
\[P_{dl}=V_{dl}I_{d} \tag{5}\]  
\[Q_{dR}=Q_{CR}-\sqrt{\left(V_{dR}+\frac{3}{\pi}n_{R}X_{cR}I_{d}\right)^{2} \frac{I_{d}^{2}}{\cos^{2}\alpha}-P_{dR}^{2}} \tag{6}\]  
\[Q_{dl}=Q_{CI}-\sqrt{\left(V_{dl}+\frac{3}{\pi}n_{l}X_{cI}I_{d}\right)^{2} \frac{I_{d}^{2}}{\cos^{2}\gamma}-P_{dl}^{2}} \tag{7}\]  
Each LCC-HVDC (rectifier or inverter) is regulated by two out of five control modes. The modes are as follows.  
(1) Constant current (CC).  
\[I_{\text{d}}=\text{const}\] (2) Constant voltage (CV).  
\[V_{\text{dI}}=\text{const}\]  
Figure 1: Diagram of HVDC system.  
(3) Constant power (CP).  
\[P_{\text{dR}}=V_{\text{dR}}I_{\text{d}}=\text{const}\] (4) Constant firing angle (CFA).  
\[\alpha=\text{const}\] (5) Constant extinction angle (CEA).  
\[\gamma=\text{const}\]  
Usually, two control modes are selected to maximize the economic benefits of HVDC transmission systems while keeping all variables within their limits. In this paper, CP is used in rectifier, meanwhile CV is preferred to the inverter.  
At the situation of the per-unit system, nodal power balance equations at the AC bus \(m\) that is linked to converter \(l\) are showed as Equations (8) and (9). The sign of \(P^{\text{t}}_{\text{d,l}}\) depends on rectifier or inverter. \(P^{\text{t}}_{\text{d,l}}\) is decided by the state of converter and reactive power compensation.  
\[0=P^{t}_{L,m}\pm\sum_{l\in m}P^{t}_{d,l}+V^{t}_{m}\sum_{n\in m}V^{t}_{n}(G_{ mn}\cos\theta^{t}_{mn}+B_{mn}\sin\theta^{t}_{mn}) \tag{8}\]  
\[0=Q^{t}_{L,m}+\sum_{l\in m}Q^{t}_{d,l}+V^{t}_{m}\sum_{n\in m}V^{t}_{n}(G_{mn} \sin\theta^{t}_{mn}-B_{mn}\cos\theta^{t}_{mn}) \tag{9}\]  
\[m\in N_{B},\,t\in T\]