# Development of an Intelligent Voltage Control System for Bulk Power Systems
## 2 Intelligent Voltage Control System
### Sensitivity Matrices for Reactive Power Devices  
The expert system uses sensitivity matrices to predict voltage fluctuations for the application of various reactive power units. It is clear that experienced human operators would be aware of the voltage impact of each reactive power device at the site.  
The basic formula for identifying the relationship between the control amount and voltage violation is as follows. The reactive power in the \(i\)-th bus is expressed by the following power equations:  
\[Q_{i}=-Q_{Gi}+Q_{Li}+Q_{Ti}=0\ (i=1,2,\cdots,n) \tag{1}\]  
\[Q_{Ti}=\sum_{j=1}^{n}V_{i}V_{j}y_{ij}sin(\delta_{i}-\delta_{j}-\gamma_{ij})\ (i=1,2, \cdots,n) \tag{2}\]  
where \(Q_{Gi}\), \(Q_{Li}\), and \(Q_{Ti}\) are the reactive power of the generator, load, and transmission respectively, in the \(i\)-th bus.  
The vector function and state variable for the bus voltage and control equipment are respectively defined as follows:  
\[\mathbf{X}=[V_{1},V_{2},V_{3},\cdots V_{n}]^{T},\mathbf{U}=[U_{1},U_{2},U_{3},\cdots U_{n}]^{T} \tag{3}\]  
\[H_{i}=-Q_{Gi}+Q_{Li}+\sum_{j=1}^{n}V_{i}V_{j}y_{ij}\sin(\delta_{i}-\delta_{j }-\gamma_{ij}),i=1,2,\ldots,\ n \tag{4}\]  
Then, at a normal steady state, \(\mathbf{X}\ =\mathbf{X}_{0}\), \(\mathbf{U}\ =\ \mathbf{U}_{0}\),  
\[\mathbf{H}(\mathbf{X}_{0}\,\ \mathbf{U}_{0})=0 \tag{5}\]  
If some violation occurs, then the perturbation equation is:  
\[\mathbf{H}(\mathbf{X}_{0}\ +\Delta\mathbf{X},\ \mathbf{U}_{0}+\Delta\mathbf{U}) =0 \tag{6}\]  
Through a Taylor series expansion, the linearized equation is obtained:  
\[\mathbf{H}(\mathbf{X}_{0}\,\ \mathbf{U}_{0})+\mathbf{H}_{\mathbf{X}}(\mathbf{X}_ {0}\,\ \mathbf{U}_{0})\Delta\mathbf{X}+\mathbf{H}_{\mathbf{U}}(\mathbf{X}_{0}\,\ \mathbf{U}_{0})\Delta \mathbf{U}=0 \tag{7}\]  
Therefore,  
\[\Delta\mathbf{X}=\mathbf{S}\cdot\Delta\mathbf{U},\ \text{where}\ \mathbf{S}=- \mathbf{H}_{\mathbf{X}}^{-1}\cdot\mathbf{H}_{\mathbf{U}} \tag{8}\]  
Here, \(\mathbf{S}\) is the sensitivity matrix for the control variable \(\mathbf{U}\).  
In the proposed system, we considered the generator terminal voltage, shunt capacitor, shunt reactor, and tap of the on-load tap changer (OLTC) as the control devices. Therefore, the sensitivity matrices for each control device are represented as follows:  
\[\Delta\mathbf{X}=\mathbf{S}_{\mathbf{Vg}}\cdot\Delta\mathbf{U}_{\text{Vg}} \tag{9}\]  
\[\Delta\mathbf{X}=\mathbf{S}_{\mathbf{T}}\cdot\Delta\mathbf{U}_{\text{T}} \tag{10}\]  
\[\Delta\mathbf{X}=\mathbf{S}_{\mathbf{sh}}\cdot\Delta\mathbf{U}_{\text{sh}} \tag{11}\]  
where \(\mathbf{S}_{\mathbf{Vg}}\,\ \mathbf{S}_{\mathbf{T}}\,\) and \(\mathbf{S}_{\mathbf{sh}}\) are the sensitivity matrices for the generator terminal voltages, OLTC tap, and shunt capacitors or reactors respectively, and \(\Delta\mathbf{U}_{\mathbf{Vg}}\,\ \Delta\mathbf{U}_{\text{T}}\), and \(\Delta\mathbf{U}_{\text{sh}}\) are the control input of the generator terminal voltages, OLTC tap, and shunt capacitors or reactors, respectively.  
Since the induction process of the sensitivity matrix is rather long and complex, detailed derivation processes and final expressions will be presented in a subsequent paper [18].