## 2 Optimal Reactive Power Dispatch Problem
### A Set of Constraints  
The set of constraints imposed on the ORPD is described as follows:  
Lower bound and upper bound constraints of generators: As explained in Section 2.2, active power output of generators are known and always within lower bound and upper bound. Thus, the control of each generator is only to monitor and adjust reactive power and voltage magnitude within an allowed range. Generators can be working stably if the following rules are always obeyed:  
\[Q_{Gen,\text{min}}\leq Q_{Gen,i}\leq Q_{Gen,\text{max}};\ i=1,\ldots,NG \tag{4}\]  
\[V_{Gen,\text{min}}\leq V_{Gen,i}\leq V_{Gen,\text{max}};\ i=1,\ldots,NG \tag{5}\]  
Lower bound and upper bound constraints of shunt capacitor banks: At some buses in transmission power network, shunt capacitor banks are additionally installed with intent to supply reactive power to loads and reduce voltage drop in transmission lines. The reactive power output of shunt capacitor banks must follow the rule below:  
\[Q_{cap,i,\text{min}}\leq Q_{cap,i}\leq Q_{cap,i,\text{max}};\ i=1,\ldots,NC \tag{6}\]  
In the study, the value of reactive power output of shunt capacitors is classified into two different cases. It is a continuous variable in the first case as long as it is within lower and upper bounds as shown in Equation (6), but it is discrete variable with a step of 100 KVAR in the second case.  
Lower bound and upper bound constraints of tap changer of transformers: During the operation of transformers, voltage of secondary side is always controlled by setting tap changer, while voltage of primary side is dependent on current flowing through transmission lines, voltage of generators, and reactive power of shunt capacitor banks. The setting of tap changer of transformer at the bus must obey the following inequality:  
\[TT_{\text{min}}\ \ \leq TT_{i}\leq TT_{\text{max}};\ i=1,\ldots,NT \tag{7}\]  
Similar to shunt capacitors, tap changer of transformers is also considered to be a continuous variable and discrete variable for two study cases. In the second case, it is constrained by each change step of 0.01 pu.  
Capacity constraint of transmission lines: On the contrary to other devices, transmission lines are conductors, which are not imposed on lower bound, but upper bound constraint is strictly supervised. Apparent power flowing through transmission lines should not be higher than its capacity. If the following inequality can be accurately responded, the transmission line is working stably.  
\[S_{branch,ij}\leq S_{branch,ij,\text{max}};\ i=1,\ldots,\ NB;\ j=1,\ldots,NB\ \&\ i\
eq j \tag{8}\]  
Lower bound and upper bound constraints of voltage at load buses: In addition to two objectives regarding voltage such as total voltage deviation and \(L\) index, voltage at each load bus is also imposed on lower and upper bounds for the purpose of stable operation of load. The requirement of load bus voltage is as follows:  
\[V_{load,\text{min}}\leq V_{load,i}\leq V_{load,\text{max}};\ i=1,\ldots,NL \tag{9}\]  
Power balance constraint at each bus: In addition to constraints from electric components in transmission lines, the ORPD problem also considers active and reactive power balance at each bus as a serious concern due to issues regarding frequency and voltage. Power generated by generator and capacitor, power consumed by load, and power of connected transmission line have the following relationship:  
\[P_{Gen,i}-P_{load,i}-V_{i}\overset{NB}{\underset{j=1}{\sum}}V_{j}\Big{[}G_{ branch,ij}\cos\big{(}\alpha_{i}-\alpha_{j}\big{)}+B_{branch,ij}\sin\big{(} \alpha_{i}-\alpha_{j}\big{)}\Big{]}=0;\ i=1,\ldots,NB \tag{10}\]  
\[Q_{Gen,i}+Q_{cap,j}-Q_{load,i}-V_{i}\overset{NB}{\underset{j=1}{\sum}}V_{j} \Big{[}G_{branch,ij}\cos\big{(}\alpha_{i}-\alpha_{j}\big{)}-B_{branch,ij}\sin \big{(}\alpha_{i}-\alpha_{j}\big{)}\Big{]}=0;\ i=1,\ldots,NB \tag{11}\]