# Decision Optimization for Power Grid Operating Conditions with High- and Low-Voltage Parallel Loops
## 3 Generation of Parallel Loop Opening Schemes
### Weighted Multi-Infeed Short Circuit Ratio  
With the development of HVDC transmission systems, parallel loops might emerge in typical multi-infeed HVDC configurations. MISCR can represent the interactions between AC and DC systems and among different DC links in AC/DC interconnected power systems [PERSON] (2003). A practical definition of MISCR based on impedance matrix can be derived as [PERSON] (2003):  
\[MISCR_{i}=\frac{U_{\text{aci}i}^{2}/\left|Z_{\text{eq}ii}\right|}{P_{\text{di }}+\sum\limits_{j=1,j\
eq i}^{n}\left|Z_{\text{eq}ij}/Z_{\text{eq}ii}\right| P_{\text{d}j}}, \tag{8}\]where \(U_{aci}\) is the inverter bus voltage of the \(i\)-th DC; and \(Z_{eqij}\) is the element of the equivalent impedance matrix \(Z_{eq}\), which can be obtained by multi-port Thevenin equivalent method. It should be noted that the value of \(Z_{qij}\) is equal to the voltage of node \(i\) when unit current is only being injected to node \(j\).  
It is obvious that opening parallel loops would change the elements of the equivalent impedance matrix \(Z_{eq}\). Thus, it can be deduced from (8) that opening parallel loops would influence the MISCR of each DC link, which indicates system stability would be influenced by any parallel loops opening scheme. In general, system stability should not descend a lot after opening parallel loops. In order to evaluate the influence of opening schemes on system stability in multi-infeed HVDC configurations, the evaluation index must be defined based on MISCR.  
Although the stability of multi-infeed HVDC systems is determined by all of the infeed DC links, the contribution of each DC link is different. Based on this evidence, the system stability evaluation index can be defined as the weighted MISCR, which is derived as:  
\[M^{\prime}=\sum_{i=1}^{n}\omega_{i}MISCR_{i}, \tag{9}\]  
where \(n\) is the number of DCs; \(MISCR_{i}\) is the multi-infeed short-circuit ratio of the \(i\)-th DC; and \(\omega_{i}\) is the weighted factor of the \(i\)-th DC, which represents the importance of the \(i\)-th DC. A larger value of \(\omega_{i}\) indicates greater importance of \(i\)-th DC and greater influence on other DCs.  
Obviously, the weighted factor \(\omega_{i}\) is quite essential to define the weighted MISCR, which must meet three requirements:  
1. The weighted factor should not be assigned a value manually;
2. The weighted factor should represent the interaction between different DC links; and
3. The weighted factor should represent the contribution to system stability of the specific DC link.  
To define the weighted factor \(\omega_{i}\), the definition of MISCR should be analyzed firstly. Equation (8) can be transformed as:  
\[MISCR_{i}=\frac{U_{aci}^{2}/\left|Z_{eqii}\right|}{P_{di}+\sum_{j=1,j\
eq i}^{ n}\left|Z_{eqij}/Z_{eqii}\right|P_{dj}}=\frac{S_{aci}}{P_{di}(1+\sum_{j=1,j\
eq i}^{ n}\left|Z_{eqii}\right|P_{dj})}=\mu_{i}\cdot\frac{S_{aci}}{P_{di}}, \tag{10}\]  
where \(\mu_{i}\) is a penalty factor. It should be noted that \(S_{aci}/P_{di}\) is the definition of single-infeed short circuit ratio (SISCR) without considering the interaction between different DC links. Based on the physical meaning of SISCR, a small \(\mu_{i}\) indicates the \(i\)-th DC can be influenced by other DCs easily, which means a smaller contribution to system stability. Thus, this penalty factor can represent the importance of the \(i\)-th DC in multi-infeed DC systems.  
Thus, the weighted factor of the \(i\)-th DC can be defined as:  
\[\omega_{i}=\frac{1}{\mu_{i}}=1+\sum_{j=1,j\
eq i}^{n}\left|\frac{Z_{eqij}}{Z _{eqii}}\right|\frac{P_{dj}}{P_{di}}, \tag{11}\]  
where \(Z_{eqij}\) is the equivalent mutual-impedance between the \(i\)-th DC and the \(j\)-th DC; \(Z_{eqii}\) is the equivalent self-impedance of the \(i\)-th DC; and \(P_{di}\) and \(P_{dj}\) are the rated transmission powers of the \(i\)-th DC and the \(j\)-th DC, respectively. The definition of (11) considers not only the electrical distance, but also the transmission capacity of a DC link, which fulfills the three requirements above.