Comparative Study of the Voltage Stability of an Hight Voltage Power Grid: Case of the Power Grid of the Electric Community of Benin

The voltage profile at the buses of an higth voltage power grid makes it possible to predict the voltage stability of said power grid in order to guarantee production-consumption adequacy. The study presented in this paper is initially restricted to the variation of the active power demand of a consumption bus (PQ) in order to obtain the voltage profile of the entire electricity transmission network. Then, it makes it possible to predict the limit of the maximum increase in power demand at a PQ bus with the corresponding voltage level of all the other buses in order to anticipate the instability of the voltages liable to cause the collapse of the network. Finally, to correct the voltage levels linked to the observed instability, the study proposes the comparison by voltage sensitivity factors of two types of compensation such as shunt compensation and compensation by adding reactance to the line. transmission. The Newton-Raphson method coupled with Predictor-Corrector methods was used for the Continuation Power Flow (CPF) on the electricity transmission network of the Benin Electric Community (CEB). The results from the bus voltage profile curves for the case of the CEB power grid have shown that the two types of compensation make it possible to recover the lost voltage stability. However, shunt compensation is best due to its lower voltage stability sensitivity factor. This study has the advantage of allowing the power grid operator to anticipate the instability of the tensions in the power grid in order to avoid its collapse. It thus helps the manager to properly plan the voltage stability of his power grid.


Introduction
High Voltage (HV) power grid serve as a link between production centers and large areas of electrical energy consumption; these networks are said to be electrical energy transport or even interconnection [1]. They are characterized by buses with which are associated production groups, loads (consumption), lines and electrical energy transformers. This leads to the electrical system consisting of the production, transport and consumption (loads) of electrical energy, the main objectives of which are the search for quality and the reduction of operating costs while respecting the security constraints of said system. The major safety constraints in an HV power grid are the stability limit and the voltage profile [2].
Stability is linked to the power transit capacity of the transmission lines [3,4]. The profile of the voltages at the power grid buses makes it possible to evaluate the variation in the voltage of each bus of the network in order to control the stability of the said power grid within its limits to guarantee a production -consumption adequacy [5]. Several studies have been done on voltage stability based on repeated analyzes of the power flow (PF) with the Newton-Raphson method [6]. The major difficulty of this method is the Jacobian matrix of Newton-Raphson which becomes singular at the critical point of the limit of stability in tension; which leads to an error and a divergence in the solution of the power flow at the critical point, [6,7,8]. To remedy these shortcomings, we reformulate the PF equations by applying a local technique of continuous parameterization called Continuation Power Flow (CPF) [6,9]. The CPF method makes it easy to study the voltage stability of electrical systems as in several studies [6,10,12]. The stability of the power grid using the CPF after insertion of a wind power plant in said grid has been studied in [11]. CPF was usesd in [12] to simulate the near-stable state of an electrical system. In [13], it developed the simulation of dynamic load restoration using CPF. (ZHAO & al. 2015) contributed in [10] from CPF to the study of voltage stability according to the evolution of the load at the border between the transmission network and the energy distribution network electric. Sudden drops in voltage at buses in an power grid have been investigated in [14] using the CPF. These studies are aimed at preventing voltage instability due to increased demand for electrical energy, which could lead to the collapse of power grid. Avoiding the collapse of power grid implies finding a solution for restoring voltage stability, hence the compensation measures or at worst the shedding of certain loads. There are several compensation modes [9] among which we can cite Generator AVRs, Under-Load Tap Changers, Shunt Capacitors, Series Capacitors, Shunt Reactors, Synchronous Condensers. In this paper, the objective is to predict not only the limit of the maximum increase in power demand at a PQ bus but also the voltage instability in order to provide an acceptable compensation. Two compensation modes such as Shunt Capacitors and Series Capacitors are presented and compared for the case of the CEB's HV power grid. For the choice of the appropriate compensation mode, it was first carried out to vary the active power demand of a PQ consumption bus in order to obtain the voltage profile of the entire electricity transmission network. Then, the limit of the maximum increase in power demand at the PQ bus with the corresponding voltage level of all other buses was predicted in order to anticipate the instability of the voltages that could cause the grid to collapse. Finally, the correction of the voltage levels linked to the observed instability, is obtained through compensations such as shunt compensation and compensation by adding reactance to the transmission line with the determination of their voltage sensitivity factors. The study presented in this paper is based on the Newton-Raphson method coupled with the Predictor-Corrector methods used for Continuation Power Flow (CPF). So this paper is structured as follows. After the introduction, we first explain the CPF method used, followed by the presentation of the CEB's HV power grid. Then the data processing with the results is carried out and finally the analyzes and discussions are presented followed by the conclusion.

The Continuation Power Flow Method: CPF
The CPF method begins with the basic conditions using conventional solutions of load distribution (Load Flow: LF) from the Newton-Raphson algorithm to calculate the basic parameter denoted . To do this, we reformulate the Load Flow equation to introduce a load parameter . We rewrite the Load Flow equation in a matrix form known as the Jacobian matrix J.
where: Pi and Qi are respectively the active and reactive powers at node i; Pgi and Qgi the active and reactive powers generated at node i; P Li and Q L i the active and reactive power consumed at the bus i; ∠ and ∠ the voltages at buses i and k; ∠ the admittance (i, k) element of the Ybus admittance matrix; n is the total number of buses in the power grid.
By asking = P gi -P Li and Q = Q gi -Q Li , we rewrite relation (1) in relation (4).
where: , V #$% , V are the vectors of the active and reactive power of the power grid; , V variable vector composed of the angle and the magnitude of the voltage of each bus of the power grid; P inj and Q inj are the vectors of the active and reactive power injected from each bus of the network.
In order to know the state of the electrical system for different load factors, we must add a state variable λ to equation (4); we can obtain the plot of δ "and V" by varying λ [7]. The system of equation (4) then becomes that of relation (5).
In the Newton-Raphson method (method not developed in this paper) for Load Flow, the reformulated Jacobian matrix of the system of equations (5) is written at relation (6), [7].
This Jacobian matrix makes it possible to find the point of bifurcation of the system by systematic gradual increase of the load factor λ of the system thanks to the CPF. CPF techniques are a very robust tool for the calculation of trajectories of one or more parameters [7]. The considered system is summarized by relation (5).
Continuation Power Flow (CPF) is an iterative process which from an initial solution defined by , V , , consist to calculate a new situation ,-, V ,-, ,with > ,-. This process thus converges towards )*+ .
The CPF is then carried out in three steps, namely parametrization, prediction and correction.

Parametrization
Parametrization is mathematically a means of identifying each solution so that the next or previous solution can be evaluated. In this paper, the natural parametrization which takes directly as a parameter has been used according to relation (7).
where: is the initial parameter; the new parameter and 0 continuous measurement of the step.

Prediction
Prediction is the process of producing an estimate ?
where: : = H% dV d I is the tangent vector; : is the normalized tangent vector; C = E / @ j-1 is the appropriate and dimensioned row vector such that all its elements are zero except the k ième element that is worth ±1 he sign of the variation; k is the index of the maximum component of the tangent vector respecting the relation (11), [14].

Correction
The correction step provides the new solution ,-, V ,-, ,by correcting the predicted solution 1 ,-, ^ ,-, ^ ,-3. The Newton-Raphson method is used to find this new solution by solving the equation system of relation (12).
The principle of Continuation power flow is shown in Figure 1.

Materials
The CEB's HV power grid shown in Figure 2 and made up of 13 buses and 14 power transmission lines is used in this paper. Matpower 6.0 [15] software in a Matlab R2016a environment was used.

Methodology
The adopted compensation comparison methodology is presented in Figure 3. In this methodology, we define at each load bus i PQ the sensitivity factor S vi of the voltage stability by equation (13).
where dV i is the variation of the voltage in per unit (pu) at the bus i and dP Total The lower this sensitivity factor to a bus i compared to another bus j, the more stable the bus i. The voltages of the buses of the power grid are subjected to a constraint of magnitude margin V according to equation 14 to guarantee the stability of the network.

Results
By varying the active power demand on bus 6 of the CEB's power grid via the load factor lambda λ, the voltage magnitude of the nine (09) PQ buses is obtained through the network voltage profile. Figure 4 shows the voltage profile.
The critical voltage at bus 6 is 0.5616 pu for a maximum load factor λ max equal to 0.2705. This led to making the two types of compensation, namely the shunt compensation to bus 6 and the compensation by reactance of the transmission line on line (2)(3)(4)(5)(6) of the CEB's power grid. Figures 5 and 6 show the voltage profile curves of the two types of compensation at bus 6.
While respecting the constraints related to the minimum and maximum magnitudes on bus 6, the results retained for the two types of compensation are summarized in Table 3.
Once the two types of compensation have been able to bring the maximum voltage of bus 6 to values greater than 0.9 pu, then we add a shunt of 0.6 pu to bus 6 and we bring back the reactance X 2-6 of the line (2-6) to 0.1 X 2-6 . For each type of compensation with the optimal values in table 3, continuous power flow is carried out to obtain the profile of the voltage amplitudes of all the PQ buses in the network. Figures 7 and 8 show these profiles.      Tables 4 and 5 contain the maximum value of the load parameter λ max with the amplitudes of the corresponding critical voltages V critical and of the sensitivity factor of the voltage stability for each PQ bus of the power grid respectively after the line transmission reactance compensation and the shunt compensation.   Figure 9 illustrates the comparison of the consequences of the line transmission reactance compensation of line X 2-6 and of the shunt compensation of bus 6 on all PQ buses compared to the constraints related to the voltage of each PQ bus of the CEB's power grid.

Discussion
The increase in the active power demand on bus 6 of the CEB power grid to the critical point reduced the voltage of this bus to 0.5616 pu for a maximum load factor λ max equal to 0.2705 as shown in Figure 4.
The constraint linked to the voltage of each bus being 0.9 ≤ V ≤ 1.3 and presented in table 1 to make the line transmission reactance compensation on line X 2-6 ( Figure 5) and the shunt compensation on bus 6 ( Figure 6) in order to bring the bus voltage back within the above mentioned range. The results from these two types of compensation (Table 3) sufficiently show the advantage of shunt compensation on the line transmission reactance by their voltage variation interval which is respectively 0.79 ≤V≤0.95 and 0.95 ≤V≤ 1.25. The voltage stability sensitivity factor S v6 confirms this advantage at its value for the shunt compensation which is 0.010 against 0.011 for the line transmission reactance compensation. However, the maximum load factor λ max has increased in value in the two compensation cases with 0.45 pu for the shunt compensation against 0.48 pu for the line transmission reactance compensation and this increases the demand for active power.
The evaluation of the repercussions of each of the two types of compensation made separately in the entire CEB's power grid and presented in Figures 7 and 8 with tables 4 and 5 clearly expresses the acceptable values of the critical voltage amplitude. V critical to all PQ buses except bus 6 (0.79134 pu) for line transmission reactance compensation. Figure 9 indicates for the critical voltages V critical that the values of the RMSE and of the correlation coefficient R² of the shunt compensation equal to 0.017661 and 0.93366 are clearly better than those of the line transmission reactance compensation respectively 0.069879 and 0.61291. The sensitivity factors of the voltage stability S v of the shunt compensation are lower than those of the line transmission reactance to all the PQ buses (Tables 4 and 5). Thus in this CEB's power grid where the increase in active power demand on bus 6 is noted, shunt compensation offers better voltage stability in view of the results obtained compared to the line transmission reactance.

Conclusion
The study proposed in this paper is the comparison of two types of compensation such as shunt compensation and line transmission reactance compensation. These two compensations follow the presence of voltage instability due to increased demand for active power causing voltage drops to certain buses in the electrical network. The Newton-Raphson method coupled with Predictor-Corrector methods was used for Continuation Power Flow (CPF) on the electricity transmission network of the Benin Electric Community (CEB) to obtain voltage profiles. The results from the node voltage profile curves for the case of the CEB network showed that the two types of compensation make it possible to recover the lost voltage stability and to increase the demand for active power. The load factor λ max goes from 0.2705 pu to 0.45 pu for the shunt compensation and to 0.48 pu for the line transmission reactance.
For the magnitudes of the critical voltages V critical compared to those of the stresses, the values of the RMSE and of the correlation coefficient R² of the shunt compensation are respectively 0.017661 and 0.93366 and are clearly better than those of the line transmission reactance compensation respectively 0.069879 and 0.61291. In addition to this we note that the sensitivity factors of the voltage stability S v of the shunt compensation are lower than those of the line transmission reactance to all the PQ buses. In short, the stability of the voltages is much more guaranteed by the shunt compensation than the line transmission reactance compensation for the case of the CEB's electrical transmission network. This study will allow the grid operator to predict and anticipate the instability of the voltages in his power grid.
This study allows the manager to anticipate instability and to properly plan compensation for good voltage stability in its power grid.