# Stochastic Approach to Hosting Limit of Transmission System and Improving Method Utilizing HVDC
## 2 Background Knowledge
### Sensitivity Approach on HVDC Allocation and Operation  
In general, HVDC operates to deliver quantities determined by market operations. However, it is necessary to ensure flexible operation, such as increasing the utilization rate of the surrounding lines by fully utilizing the power flow control ability compared to the alternating current (AC) line of the HVDC line. Moreover, various attempts have been made to reduce the adverse effects of the fixed operating point of the HVDC. The improvement in the stability is discussed through the control that emulates the AC line by focusing on the transient stability [25]. To determine the operating point to increase the limitations determined by the above approaches, we proposed a methodology using a sensitivity-based approach called power transfer distribution factors (PTDFs) for transmission expansion and operation using HVDC. This sensitivity analysis was conducted based on the DC power flow, topology, and some assumptions as follows; more details can be found in [26]. * Transmission line conductance \(G\) is negligible (reactance \(X\) is larger than resistance \(R\)) \[G_{l,k}=0\ (X>>R)\] (2)
* The system operated ideally in the bus \(i\) voltage \(V_{i}\) aspect \[|V_{i}|=1\ p.u.\] (3)
* Voltage angle differences between adjacent buses \(l\) and \(k\) are small (linearization) \[\sin(\delta_{l}-\delta_{k})\ \approx\ \delta_{l}-\delta_{k}\] (4) \[\cos(\delta_{l}-\delta_{k})\ \approx\ 1\] (5) Based on the above assumptions, the sensitivity analysis can be conducted as follows: \[f_{l,k}=(\delta_{l}-\delta_{k})\ B_{l,k}\] (6) \[\Delta\delta=B^{-1}\Delta P\] (7) \[\psi_{l,k}^{i}=B_{l,k}\ (\Delta\delta_{l}-\ \Delta\delta_{k})\] (8) \[\Delta f_{l,k}=\psi_{l,k}^{i}\ \Delta P^{i}\] (9)  
where \(\delta_{l}\) is the angle of bus \(l\), \(f_{l,k}\) is the branch flow from buses \(l\) to \(k\), \(B_{l,k}\) is the susceptance of the transmission line from buses \(l\) to \(k\), and \(\psi_{l,k}^{i}\) is the PTDF when power injected into the bus \(i\). These DC PTDFs can be calculated from the system impedance matrix (i.e., based on the system topology). Figure 2 shows the sensitivity analysis results for each injection case in the test system used in this study. However, this has the disadvantage of accuracy owing to the above DC power flow assumptions. For simplicity and fast calculation in this study, this sensitivity analysis was adopted instead of AC PTDF through a Jacobian matrix with the additional assumption that voltage and reactive power problems are solved by themselves within each area. By using these sensitivity factors and determining the target integration and buses for new DER integration points, it is possible to evaluate the effect of these new generations on the transmission system. In this situation, an allocation candidate was selected by solving the optimization problem by comprehensively considering the target integration amount, connection points, capacity of the planned HVDC, average loading statistics of the current system, and impact on the power generation scheme. The results of the sensitivity analysis of the test system and a feasibility study result for this sensitivity are also included in the Appendix A. In addition, the optimization method utilizes a mixed-integer non-linear programming (MINLP) method for determining location and mixed-integer linear programming (MILP) for finding operating point based on the linearity of the sensitivity. It is also possible to use a fast optimization method, such as the internal point method, for the real-time operation to dispatch the HVDC operating points. The process of solving this optimization problem is illustrated in Figure 3. The input data include all the information needed for this proposed method.