Optimization of Voltage Security with Placement of FACTS Device Using Modified Newton–Raphson Approach: A Case Study of Nigerian Transmission Network

Abstract

Power flow reliability, voltage security and transmission congestion management are paramount operational issues in a power system. Flexible AC transmission system (FACTS) controllers are suitable technologies that can provide compensation and dynamic control of power system transmission parameters to enhance effective performance and reliability. The interline power flow controller (IPFC), if optimally placed, can regulate the impedance of multiple lines to improve active power transfer capacity and voltage profile. This study examines the performance of IPFCs for voltage enhancement by suppressing fluctuation. A modified Newton–Raphson load flow problem with an incorporated IPFC variable has been formulated with the objective to improve voltage stability and maintain active power flow. The effectiveness of the proposed method was tested on the Nigerian 41 bus transmission network. The obtained result of the system with an IPFC placed at the weakest bus of the network was compared with Newton–Raphson load flow analysis of the same network without an IPFC. The results of load flow analysis for Case 1 (the system without an IPFC) showed that the transmission network without an IPFC had a real power loss of 4.699488 p.u., and reactive power loss of 4.467413 p.u., whereas the integration of an IPFC to the power flow formation in Case 2 resulted in the reduction in the transmission network’s overall losses to 0.55297 p.u. and −38.3329 p.u. The modified method proves effective as the power system network with an IPFC returns a more stable voltage profile and improves active power flow. In addition, this method, similar to all other mathematical optimization approaches, returns a strong accurate result but may be a drawback in terms of longer computational time compared with metaheuristic methods which are preferred for a larger network system.

1. Introduction

The power system network generally is constantly faced with the potential challenges of voltage fluctuation, reactive power imbalances and several other factors that lead to instability. The effect of power system instability could expand from low power quality to complete shutdown. Therefore, a quality and reliable power system is the desire of most utility companies and electric power users [1]. A quality power system is the delivery of power to the end-user as required. The consistent and uninterrupted supply of power defines its reliability [2,3]. The demand for quality and reliable power is constantly increasing with the increase in population. This increasing demand places a burden on the existing power system transmission network resulting in an overstretch of its infrastructure. Furthermore, most of the existing power system infrastructures are both old and already overstretched. This trend leaves power utility industries with only a few options: either an overhaul of the existing infrastructure and replacing it with new ones, or optimizing the network system to enhance its capacity. The former option of replacing the existing infrastructure is very expensive; hence, the latter is a more sustainable option [4].

Usually, in a developing country such as Nigeria, where power system infrastructures are not only old but face constant challenges that result in perpetual outages, the integration of compensating power electronics to optimize the stability of the transmission system is paramount. Flexible alternating current transmission system (FACTS) devices have long been identified as suitable tools for optimizing power system transmission performance [5]. They are power electronic controllers designed to regulate transmission parameters for improved performance [6]. Many other studies have shown the relevance of FACTS devices in tackling common power system challenges such as optimizing line available power transfer capacity and loadability [7,8]; improving transient state and system security [9,10,11,12]; reducing sub-synchronous resonance and improving system damping [13,14]; minimizing short circuit current [15]; compensate reactive power for power quality improvement [16] and voltage stability enhancement [17,18,19]. FACTS devices are an advancement of mechanically controlled inductors, capacitors and phase-shifting transformers with dynamic control capability. The dynamic control provided by FACTS devices includes compensation of lost parameters such as reactive power compensation, bus voltage improvement, rotor angle adjustment for system stability and general power quality improvement. Therefore, the main advantage of FACTS devices over the use of inductors, capacitors and phase-shifting transformers are automated control of these multiple power system parameters. Since FACTS devices perform a similar function as inductors, capacitors and phase-shifting transformers they are also classified according to the type of connection with the transmission lines. Shunt-connected FACTS devices inject controllable currents for power flow stability whereas series-connected FACTS devices regulate the line impedances to increase the active power transfer capacity of the line. A more advanced type of FACTS device has either a combination of both shunt and series controllers to modify a single line such as a unified power flow controller (UPFC) or a combination of two series controllers to modify multiple lines such as an interline power flow controller (IPFC). The mechanically controlled inductors, capacitors and phase-shifting transformers are considered the first generation of FACTS devices whereas the second generation was developed with thyristor valves replacing the mechanical switches of the first-generation. The second-generation FACTS devices include a thyristor controlled series compensator (TCSC), a static var compensator (SVC) and a dynamic flow controller (DFC). The third generation was designed with voltage source converters to replace the thyristor controlled switches of the second generation devices [20]. They include, a static synchronous series compensator (SSSC), a static synchronous compensator (STATCOM), UPFC, IPFC and a general unified power flow controller (GUPFC). Hence the UPFC, GUPFC and IPFC are the most common advanced FACTS devices. The voltage source converters of these devices help to eliminate resonant with inductive line impedance that may initiate sub-synchronous oscillation and control output voltage and real power exchange with the AC system.

In the majority of FACTS technology applications, different optimization methods are employed for the optimal placement of the devices. The optimization approach could be sensitivity based [9,21,22], use classical methods [23,24] or meta-heuristic techniques [25,26,27]. These techniques have numerous drawbacks that make researchers continue to explore new options or modify the existing methods to improve performance. The classical methods are limited by high computation times and difficulties in handling large variables and constraints, but are still very popular for accurate results. In addition, the sensitivity analysis methods entail long computation times and difficulties in handling non-linear, large constraints and multi-objective problems but are very useful to observe system dynamic behavior at different parameter set points. The meta-heuristics techniques are considered more robust in handling multi-objectives and large constraint non-linear optimization problems with the highest computational efficiency. Despite the superior advantage of fast computational times the metaheuristic methods have over classical and sensitivity approaches, the question of accuracy is still a major drawback. Inkollu and Kota [28] compared the Newton-Raphson classical method with two metaheuristic methods, particle swarm optimization (PSO) and gravitational search algorithm (GSA) and concluded that PSO and GSA performed better than the Newton–Raphson method in determining optimal settings of UPFC and IPFC for voltage stability enhancement. In addition, the effectiveness of the GSA is buttressed by Mishra and Gundavarapu [29] for optimal placement of an IPFC for congestion management and concluded that an IPFC greatly improved power transmission in the overloaded network. Furthermore, in another study, Mishra and Gundavarapu [22], presented a sensitivity analysis of a power system network and ranked the severity of contingency on the network before using differential evolution (DE) to optimize IPFC settings or enhanced power quality. It was concluded that when compared with the genetic algorithm, DE had a better performance. Amarendra et al. [30] proposed using firefly algorithms (FA) to determine the optimal location of an IPFC at a minimal cost. The proposed FA performs better than particle PSO in minimizing fuel cost with improved convergence characteristics.

The Newton–Raphson method is one of the common classical optimization methods employed for the optimal placement of FACTS devices. This method has been modified to incorporate several FACTS devices such as UPFC, SVC, STATCOM and the thyristor controlled series compensator (TCSC) [5] but an IPFC has not been considered in a similar approach despite its importance among the FACTS devices. Therefore, this study presents the evaluation of a Nigerian transmission system using modified Newton–Raphson methods with an IPFC for voltage stability improvement. The choice of IPFC as opposed to other FACTS devices is due to its multiple line improvement capacity which generates a cheaper and more effective option as fewer numbers and sizes are required compared with other FACTS devices except for GUPFC. An IPFC provides good power flow control for more than one transmission line with each of the two SSSC giving series power addition for its own transmission line. The IPFC converters can move real power to the other and by implication real power exchange between the lines may be carried out in order to create a balance between an over-loaded and under-loaded line. In addition, the Newton–Raphson method choice is due to its accuracy and it is simple to modify. Hence, despite the drawback of a low convergence rate and long computation time it is still one of the most commonly used mathematical optimization approaches.

2. Materials and Methods

Load flow analysis is very important in the planning of the future expansion of power systems and to determine the best operation of existing systems. It is used to determine voltage magnitudes at various buses and real and reactive power in each line to meet the load demands on a network at a particular time. Therefore, load flow calculations are required for the analysis of steady state as well as the dynamic performance of power systems.

The power transmission line can be represented by a two-bus system “k” and “m” in ordinary form as shown in Figure 1.

The complex power injection at bus k is used for deriving nodal active and reactive power flow equations.

$$S_k=P_k+j{Q}_k=V_k{I}_k^{*}$$

(1)

where

S_k is the complex power injection at node k, P_k is the active power injection at node k, Q_k is the reactive power injection at node k, V_k is the complex voltage at node k and I_k is the complex current injection at node k.

The injected current I_k may be expressed as a function of a branch connected to node k,

$$I_k={\displaystyle \sum _{m=I}^n{Y}_{km}}{V}_m$$

(2)

where

Y_km = G_km + B_km with Y_km, G_km and B_km being the admittance, conductance and susceptance of branch k-m, respectively.

Substituting Equation (2) into (1) gives

$$P_k+j{Q}_k=V_k{\displaystyle \sum _{m=I}^n{Y}_{km}^{*}{V}_m^{*}}$$

(3)

The expressions for the active and reactive powers are obtained by representing the complex voltage in polar form [31].

$$V_k=\left|V_k\right|{\ell }^{j\theta k}$$

(4)

and

$$V_m=\left|V_m\right|{\ell }^{j\theta m}$$

(5)

$$P_k+j{Q}_k=\left|V_k\right|{\displaystyle \sum _{m=1}^n\left|V_m\right|}\left(G_{km}-j{B}_{km}\right){\ell }^j\left({\theta }_k-{\theta }_m\right)$$

(6)

$$P_k+j{Q}_k=\left|V_k\right|{\displaystyle \sum _{m=1}^n\left|V_m\right|\left(G_{km}-j{B}_{km}\right)}\left\{\cos \left({\theta }_k-{\theta }_m\right)+j\sin \left({\theta }_k-{\theta }_m\right)\right\}$$

(7)

$$P_k=\left|V_k\right|{\displaystyle \sum _{m=1}^n\left|V_m\right|}\left\{G_{km}\cos \left({\theta }_k-{\theta }_m\right)+B_{km}\sin \left({\theta }_k-{\theta }_m\right)\right\}$$

(8)

$$Q_k=\left|V_k\right|{\displaystyle \sum _{m=1}^n\left|V_m\right|}\left\{G_{km}\sin \left({\theta }_k-{\theta }_m\right)-B_{km}\cos \left({\theta }_k-{\theta }_m\right)\right\}$$

(9)

where $\left|V_k\right| \mathrm{and} \left|V_m\right|$ are nodal voltage magnitudes at nodes k and m and $\theta$_k and $\theta$_m are nodal voltage phase angles at nodes k and m.

These equations provide adequate tools for assessing the steady state behavior of the power network. The equations are non-linear, and their solution is obtained by iteration.

2.1. Newton–Raphson Power Flow Method

Among the power flow methods, the Newton–Raphson method has been considered as the power flow solution technique for large-scale power system analysis because of its strong convergence characteristics.

Assuming that bus one is the slack bus. The linearized relationship takes the following form for an n-bus network [32]:

$$\left[\begin{array}{l}\mathsf{\Delta }{P}_2\\ \\ \mathsf{\Delta }{P}_3\\ .\\ .\\ .\\ \mathsf{\Delta }{P}_n\\ \\ \mathsf{\Delta }{Q}_2\\ \\ \mathsf{\Delta }{Q}_3\\ .\\ .\\ .\\ \mathsf{\Delta }{Q}_n\end{array}\right]\begin{array}{l}(r)\\ \\ \\ \\ \\ \\ =\\ \\ \\ \\ \\ \\ \end{array}\left[\begin{array}{l}\frac{\partial P_2}{\partial {\theta }_2}\frac{\partial P_2}{\partial {\theta }_3}\dots \frac{\partial P_2}{\partial {\theta }_n}\hspace{1em}\frac{\partial P_2}{\partial \left|V_2\right|}\frac{\partial P_2}{\partial \left|V_3\right|}\dots \frac{\partial P_2}{\partial \left|V_n\right|}\\ \\ \frac{\partial P_3}{\partial {\theta }_2}\frac{\partial P_3}{\partial {\theta }_3}\dots \frac{\partial P_n}{\partial {\theta }_n}\hspace{1em}\frac{\partial P_3}{\partial \left|V_2\right|}\frac{\partial P_3}{\partial \left|V_3\right|}\dots \frac{\partial P_3}{\partial \left|V_n\right|}\\ .\\ \\ \frac{\partial P_n}{\partial {\theta }_2}\frac{\partial P_n}{\partial {\theta }_3}\dots \frac{\partial P_n}{\partial {\theta }_n}\hspace{1em}\frac{\partial P_n}{\partial \left|V_2\right|}\frac{\partial P_n}{\partial \left|V_3\right|}\dots \frac{\partial P_n}{\partial \left|V_n\right|}\\ \\ \frac{\partial Q_2}{\partial {\theta }_2}\frac{\partial Q_2}{\partial {\theta }_3}\dots \frac{\partial Q_2}{\partial {\theta }_n}\hspace{1em}\frac{\partial Q_2}{\partial \left|V_2\right|}\frac{\partial Q_2}{\partial \left|V_3\right|}\dots \frac{\partial Q_2}{\partial \left|V_n\right|}\\ \\ \frac{\partial Q_3}{\partial {\theta }_2}\frac{\partial Q_3}{\partial {\theta }_3}\dots \frac{\partial Q_3}{\partial {\theta }_n}\hspace{1em}\frac{\partial Q_3}{\partial \left|V_2\right|}\frac{\partial Q_3}{\partial \left|V_3\right|}\dots \frac{\partial Q_3}{\partial \left|V_n\right|}\\ \\ \frac{\partial Q_n}{\partial {\theta }_2}\frac{\partial Q_n}{\partial {\theta }_3}\dots \frac{\partial Q_n}{\partial {\theta }_n}\hspace{1em}\frac{\partial Q_n}{\partial \left|V_2\right|}\frac{\partial n_n}{\partial \left|V_3\right|}\dots \frac{\partial Q_n}{\partial \left|V_n\right|}\end{array}\right]\begin{array}{l}(r)\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \end{array}\left[\begin{array}{l}\mathsf{\Delta }{\theta }_2\\ \\ \mathsf{\Delta }{\theta }_3\\ .\\ .\\ .\\ \mathsf{\Delta }{\theta }_n\\ \\ \mathsf{\Delta }\left|V_2\right|\\ \\ \mathsf{\Delta }\left|V_3\right|\\ .\\ .\\ .\\ \mathsf{\Delta }\left|V_n\right|\end{array}\right]$$

(10)

$\mathsf{\Delta }$P_k = P_k^spec − P_k^calc is the active power mismatch at node k,

$\mathsf{\Delta }{Q}_k=Q_k^{Spec}-P_k^{calc}$ is the reactive power mismatch at node k,

$P_k^{spec}=P_k^{gen}-P_k^{load}$ is the net scheduled active power at node k,

$Q_k^{spec}=Q_k^{gen}-Q_k^{load}$ is the net scheduled reactive power at node k,

$P_k^{gen} \mathrm{and} Q_k^{gen}$ are active and reactive powers generated at node k,

$P_k^{load} \mathrm{and} Q_k^{load}$ are the active and reactive power consumed by the load at node k,

$\mathsf{\Delta }{\theta }_k \mathrm{and} \mathsf{\Delta }\left|V_k\right|$ are the incremental changes in nodal phase angle and voltage magnitude at node k,

(r) represents the r-th iterative step and k = 2, 3, 4, … n.

The elements of the Jacobian matrix can be found by differentiating Equations (8) and (9) with respect to $\theta$_k, and $\left|V_k\right|$. Equations (8) and (9) can be solved efficiently using the Newton–Raphson method. It is a set of linearized equations expressing the relationship between changes in active and reactive powers and changes in bus voltage magnitudes and phase angles.

Starting the iterative solution, initial estimates of the nodal voltage magnitudes and phase angles at all the PQ nodes and voltage phase at all the PV nodes are given to calculate the active and reactive power injections using Equations (8) and (9). It is unlikely that the initial estimated voltages will agree with the voltages at the solution point; hence, the calculated power injections will not agree with the known specified powers [32].

The mismatch power vectors may be defined as

$$\mathsf{\Delta }{P}^{(r)}=\left(P_{}^{gen}-P^{load}\right)-P^{calc,(r)}=P^{spec}-P^{calc,(r)}$$

(11)

$$\mathsf{\Delta }{Q}^{(r)}=\left(Q^{gen}-Q^{load}\right)-Q^{calc,(r)}=Q^{spec}-Q^{calc,}$$

(12)

The Jacobian elements are then calculated and linearized Equation (10) is solved to obtain the vectors of voltage updates

$${\theta }^{(\mathrm{r}+1)}={\theta }^{\left(\mathrm{r}\right)}+\mathsf{\Delta }{\theta }^{\left(\mathrm{r}\right)}$$

(13)

$${\left|V\right|}^{(\mathrm{r}+1)}={\left|V\right|}^{\left(\mathrm{r}\right)}+\mathsf{\Delta }{\left|V\right|}^{\left(\mathrm{r}\right)}$$

(14)

2.2. Power Flow Model of an IPFC

The IPFC consists of two SSSCs connected back-to-back through the DC link as shown in Figure 2. The IPFC has two series converters connected to two different transmission lines. It provides very good power flow control for more than one transmission line with each of the two SSSCs giving series power addition for its own transmission line. The two converters are joined through a DC capacitor and attached to the AC network through directly connected transformers. Thus, it not only provides a reactive power addition but furthermore, any of the converters can be manipulated to inject real power to the dc joint from its own transmission line. In addition, any of the IPFC converters can move real power to the other and by implication real power exchange between the lines may be carried out in order to create a balance between an over-loaded and an under-loaded line.

The IPFC uses the circuit represented in Figure 2b, which has an active power constraint equation that links the two voltage sources which are Equations (15) and (16) [33,34].

$$E_{cR}=V_{cR}\left(\cos {\delta }_{cR}+j\sin {\delta }_{cR}\right)$$

(15)

$$E_{vR}=V_{vR}\left(\cos {\delta }_{vR}+j\sin \delta v\right)$$

(16)

The converters are controlled between the limits

$$\left(V_{cR,\min }\le V_{cR}\le V_{cR,\max }\right) \mathrm{and} \left(O\le {\delta }_{cR}\le 2\pi \right)$$

$$\left(V_{vR,\min }\le V_{vR}\le V_{vR,\max }\right) \mathrm{and} \left(O\le {\delta }_{vR}\le 2\pi \right)$$

for voltage magnitude and phase angles, respectively.

From Figure 2 and Equations (17) and (18), the active and reactive power equations are:

At bus k

$$\begin{gathered} P_k=V_k^2{G}_{kk}+V_k{V}_m\left[G_{km}\cos \left({\theta }_k-{\theta }_m\right)+B_{km}Sin\left({\theta }_k-{\theta }_m\right)\right] \\ +V_k{V}_{cR}\left[G_{km}\cos \left({\theta }_k-{\delta }_{cR}\right)+B_{km}Sin\left({\theta }_k-{\delta }_{cR}\right)\right] \\ +V_k{V}_{vR}\left[G_{vR}\cos \left({\theta }_k-{\delta }_{vR}\right)+B_{vR}Sin\left({\theta }_k-{\delta }_{vR}\right)\right] \end{gathered}$$

(17)

$$\begin{gathered} Q_k=-V_k^2{B}_{kk}+V_k{V}_m\left[G_{km}\sin \left({\theta }_k-{\theta }_m\right)-B_{km}\cos \left({\theta }_k-{\theta }_m\right)\right] \\ +V_k{V}_{cR}\left[G_{km}\sin \left({\theta }_k-{\delta }_{cR}\right)-B_{km}\cos \left({\theta }_k-{\delta }_{cR}\right)\right] \\ +V_k{V}_{vR}\left[G_{vR}\sin \left({\theta }_k-{\delta }_{vR}\right)-B_{vR}\cos \left({\theta }_k-{\delta }_{vR}\right)\right] \end{gathered}$$

(18)

At bus m:

$$\begin{gathered} P_k=V_m^2{G}_{mm}+V_m{V}_k\left[G_{mk}\cos \left({\theta }_m-{\theta }_k\right)+B_{mk}\sin \left({\theta }_m-{\theta }_k\right)\right] \\ +V_m{V}_{cR}\left[G_{mm}\cos \left({\theta }_m-{\delta }_{cn}\right)+B_{mm}\sin \left({\theta }_m-{\delta }_{cR}\right)\right] \end{gathered}$$

(19)

$$\begin{gathered} Q_k=-V_m^2{B}_{mm}+V_m{V}_k\left[G_{mk}\sin \left({\theta }_m-{\theta }_k\right)-B_{mk}\cos \left({\theta }_m-{\theta }_k\right)\right] \\ +V_m{V}_{cR}\left[G_{mm}\sin \left({\theta }_m-{\delta }_{cR}\right)-B_{mm}\cos \left({\theta }_m-{\delta }_{cR}\right)\right] \end{gathered}$$

(20)

At bus L:

$$\begin{gathered} P_k=V_L^2{G}_{LL}+V_L{V}_k\left[G_{Lk}\cos \left({\theta }_L-{\theta }_k\right)+B_{Lk}\sin \left({\theta }_L-{\theta }_k\right)\right] \\ +V_L{V}_{vR}\left[G_{LL}\cos \left({\theta }_L-{\delta }_{vR}\right)+B_{LL}\sin \left({\theta }_L-{\delta }_{vR}\right)\right] \end{gathered}$$

(21)

$$\begin{gathered} Q_k=-V_L^2{B}_{LL}+V_L{V}_k\left[G_{Lk}\sin \left({\theta }_L-{\theta }_k\right)-B_{Lk}\cos \left({\theta }_L-{\theta }_k\right)\right] \\ +V_L{V}_{vR}\left[G_{LL}\sin \left({\theta }_L-{\theta }_k\right)-B_{LL}\cos \left({\theta }_L-{\delta }_{vR}\right)\right] \end{gathered}$$

(22)

Converter 1

$$\begin{gathered} P_{cR}=V_{cR}^2{G}_{mm}+V_{cR}{V}_k\left[G_{km}\cos \left({\delta }_{cR}-{\theta }_k\right)+B_{mm}\sin \left({\delta }_{cR}-{\theta }_k\right)\right] \\ +V_{cR}{V}_m\left[G_{mm}\cos \left({\delta }_{cR}-{\theta }_m\right)+B_{mm}\sin \left({\delta }_{cR}-{\theta }_m\right)\right] \end{gathered}$$

(23)

$$\begin{gathered} Q_{cR}=-V_{cR}^2{B}_{mm}+V_{cR}{V}_k\left[G_{km}\sin \left({\delta }_{cR}-{\theta }_k\right)-B_{km}\cos \left({\delta }_{cR}-{\theta }_k\right)\right] \\ +V_{cR}{V}_m\left[G_{mm}\sin \left({\delta }_{cR}-{\theta }_m\right)-B_{mm}\cos \left({\delta }_{cR}-{\theta }_m\right)\right] \end{gathered}$$

(24)

Converter 2

$$\begin{gathered} P_{vR}=-V_{cR}^2{G}_{LL}+V_{vR}{V}_k\left[G_{kl}\cos \left({\delta }_{vR}-{\theta }_k\right)+B_{kL}\sin \left({\delta }_{vR}-{\theta }_k\right)\right] \\ +V_{vR}{V}_L\left[G_{LL}\cos \left({\delta }_{vR}-{\theta }_L\right)+B_{LL}\sin \left({\delta }_{vR}-{\theta }_L\right)\right] \end{gathered}$$

(25)

$$\begin{gathered} Q_{vR}=V_{cR}^2{B}_{LL}+V_{vR}{V}_k\left[G_{KL}\sin \left({\delta }_{vR}-{\theta }_k\right)-B_k\cos \left({\delta }_{vR}-{\theta }_k\right)\right] \\ +V_{vR}{V}_m\left[G_{LL}\sin \left({\delta }_{vR}-{\theta }_L\right)-B_{LL}\cos \left({\delta }_{vR}-{\theta }_L\right)\right] \end{gathered}$$

(26)

The linearized system of equations are as follows:

$$\left[\begin{array}{l}\mathsf{\Delta }{P}_k\\ \\ \mathsf{\Delta }{P}_m\\ \\ \mathsf{\Delta }{P}_L\\ \\ \mathsf{\Delta }{Q}_k\\ \\ \mathsf{\Delta }{Q}_m\\ \\ \mathsf{\Delta }{Q}_L\\ \\ \mathsf{\Delta }{P}_n\end{array}\right]=\left[\begin{array}{l}\frac{\partial P_k}{\partial {\theta }_k}\hspace{1em}\frac{\partial P_k}{\partial {\theta }_m}\hspace{1em}\frac{\partial P_k}{\partial V_k}{V}_k\hspace{1em}\frac{\partial P_k}{\partial V_{cR}}{V}_{cR}\hspace{1em}\frac{\partial P_k}{\partial {\delta }_{cR}}\hspace{1em}\frac{\partial P_k}{\partial V_{vR}}{V}_{vR}\hspace{1em}\frac{\partial P_k}{\partial {\delta }_{VR}}\\ \\ \frac{\partial P_m}{\partial {\theta }_k}\hspace{1em}\frac{\partial P_m}{\partial {\theta }_m}\hspace{1em}\frac{\partial P_m}{\partial V_k}{V}_k\hspace{1em}\frac{\partial P_m}{\partial V_{cR}}{V}_{cR}\hspace{1em}\frac{\partial P_m}{\partial {\delta }_{cR}}OO\\ \\ \frac{\partial P_L}{\partial {\theta }_k}\hspace{1em}\frac{\partial P_L}{\partial {\theta }_L}\hspace{1em}\frac{\partial P_L}{\partial V_k}{V}_{k}OO\frac{\partial P_L}{\partial V_{vR}}{V}_{vR}\hspace{1em}\frac{\partial P_L}{\partial {\delta }_{VR}}\\ \\ \frac{\partial Q_k}{\partial {\theta }_k}\hspace{1em}\frac{\partial Q_k}{\partial {\theta }_m}\hspace{1em}\frac{\partial Q_k}{\partial V_k}{V}_k\hspace{1em}\frac{\partial Q_k}{\partial V_{cR}}{V}_{cR}\hspace{1em}\frac{\partial Q_k}{\partial {\delta }_{cR}}\hspace{1em}\frac{\partial Q_k}{\partial V_{vR}}{V}_{vR}\hspace{1em}\frac{\partial Q_k}{\partial {\delta }_{VR}}\\ \\ \frac{\partial Q_m}{\partial {\theta }_k}\hspace{1em}\frac{\partial Q_m}{\partial {\theta }_m}\hspace{1em}\frac{\partial Q_m}{\partial V_k}{V}_k\hspace{1em}\frac{\partial Q_m}{\partial V_{cR}}{V}_{cR}\hspace{1em}\frac{\partial Q_m}{\partial {\delta }_{cR}}\hspace{1em}OO\\ \\ \frac{\partial Q_L}{\partial {\theta }_k}\hspace{1em}\frac{\partial Q_L}{\partial {\theta }_L}\hspace{1em}\frac{\partial Q_L}{\partial V_k}{V}_{k}OO\hspace{1em}\frac{\partial Q_L}{\partial V_{vR}}{V}_{vR}\hspace{1em}\frac{\partial Q_L}{\partial {\delta }_{VR}}\\ \\ \frac{\partial P_n}{\partial {\theta }_k}\hspace{1em}\frac{\partial P_n}{\partial {\theta }_n}\hspace{1em}\frac{\partial P_n}{\partial V_k}{V}_k\hspace{1em}\frac{\partial P_n}{\partial V_{cR}}{V}_{cR}\hspace{1em}\frac{\partial P_n}{\partial {\delta }_{cR}}\hspace{1em}\frac{\partial P_n}{\partial V_{vR}}{V}_{vR}\hspace{1em}\frac{\partial P_n}{\partial {\delta }_{VR}}\end{array}\right]\left[\begin{array}{l}\mathsf{\Delta }{\theta }_k\\ \\ \mathsf{\Delta }{\theta }_m\\ \\ \mathsf{\Delta }{\theta }_L\\ \\ \frac{\mathsf{\Delta }{V}_k}{V_k}\\ \\ \frac{\mathsf{\Delta }{V}_m}{V_m}\\ \\ \frac{\mathsf{\Delta }{V}_l}{V_l}\\ \\ \mathsf{\Delta }{\theta }_n\end{array}\right]$$

(27)

2.3. Case Study

In this study, the Newton–Raphson load flow method and its modification with an IPFC is formulated and implemented using MATLAB software package, 2017 version to analyse the Nigerian 41 bus, 330 kV transmision networks as Case 1 and Case 2, respectively. Figure 3 represents the line diagram of the network, whereas

Table 1. Nigerian transmission network component.

Buses	41
Lines	77
Generators	13
Loads	17
Power rate (MVA)	100

details the other components that make up the network.

3. Results

The obtained results from Case 1 from the Newton–Raphson load flow analysis of the uncompensated network (without an IPFC) are shown in

Table 2. Power flow result of the uncompensated Nigerian transmission network.

From Bus	To Bus	Line	P Flow	Q Flow	P Loss	Q Loss
			[p.u.]	[p.u.]	[p.u.]	[p.u.]
AfamPS(46)	Aja(39)	1	0.780641	−0.7207	0.006116	−1.26177
Aja(39)	Ajaokuta(17)	2	−1.01058	−12.5873	0.418722	3.110249
Aja(39)	Ajaokuta(17)	3	−1.01058	−12.5873	0.418722	3.110249
Ajaokuta(17)	Akangba(34)	4	2.900855	−1.96867	0.046359	−1.38056
Ajaokuta(17)	Akangba(34)	5	2.900855	−1.96867	0.046359	−1.38056
Ajaokuta(17)	Aladja(40)	6	−3.38736	−0.46289	0.035998	−0.80328
Ajaokuta(17)	Aladja(40)	7	−3.38736	−0.46289	0.035998	−0.80328
Ajaokuta(17)	Aladja(40)	8	−3.38736	−0.46289	0.035998	−0.80328
Alagbon(45)	Ajaokuta(17)	9	−1.09211	−1.11017	0.000387	−0.05245
Alagbon(45)	Ajaokuta(17)	10	−1.09211	−1.11017	0.000387	−0.05245
Akangba(34)	Alaoji(42)	11	1.242819	10.45204	0.335718	2.000311
Akangba(34)	Alaoji(42)	12	1.242819	10.45204	0.335718	2.000311
Akangba(34)	Ayede(23)	13	2.294793	0.188437	0.009949	−0.63041
Akangba(34)	Ayede(23)	14	2.294793	0.188437	0.009949	−0.63041
Aladja(40)	Ganmo(22)	15	−0.41409	−2.83588	0.018841	−1.79585
Aladja(40)	Ganmo(22)	16	−0.41409	−2.83588	0.018841	−1.79585
Aladja(40)	Ganmo(22)	17	−0.41409	−2.83588	0.018841	−1.79585
Aladja(40)	Ikot-Ekpene(48)	18	−2.1721	4.906201	0.186024	0.145485
Aladja(40)	Lokoja(15)	19	−6.82618	8.505772	0.283937	1.700441
Ayede(23)	B-Kebbi(1)	20	2.936046	−0.52279	0.030969	−1.12041
Ayede(23)	Benin(30)	21	−0.43084	0.732819	0.008588	−1.31713
Ayede(23)	Benin(30)	22	−0.43084	0.732819	0.008588	−1.31713
Ayede(23)	Benin(30)	23	−0.43084	0.732819	0.008588	−1.31713
Benin(30)	CalabarPS(47)	24	−0.25334	8.836865	0.511145	3.229433
Benin(30)	NewHaven(32)	25	−2.17614	−1.59881	0.02676	−1.51277
Benin(30)	NewHaven(32)	26	−2.17614	−1.59881	0.02676	−1.51277
CalabarSS(37)	CalabarPS(47)	27	6.33 × 10−11	−9.8 × 10−12	0.001442	−0.56042
Damaturu(3)	Ikot-Ekpene(48)	28	4.12323	−1.19477	0.079501	0.129835
DeltaPS(41)	CalabarPS(47)	29	8.666057	50.27428	1.573592	13.22017
DeltaPS(41)	Eastmain(14)	30	8.510994	51.95631	12.54612	106.2714
DeltaPS(41)	Maiduguri(4)	31	0.982948	−1.09293	0.007692	−0.40457
Ganmo(22)	GereguPS(18)	32	3.072454	9.072886	0.346171	1.718641
Ganmo(22)	Gwagwalada(9)	33	0.307419	−0.3439	9.46 × 10−5	−0.41759
Ganmo(22)	Gwagwalada(9)	34	0.307419	−0.3439	9.46 × 10−5	−0.41759
Ganmo(22)	Gwagwalada(9)	35	0.307419	−0.3439	9.46 × 10−5	−0.41759
Ganmo(22)	IhobvorNIPP(Eyaen)(25)	36	2.286948	13.5976	0.470673	3.139481
Ganmo(22)	IhobvorNIPP(Eyaen)(25)	37	2.286948	13.5976	0.470673	3.139481
Ganmo(22)	IhobvorNIPP(Eyaen)(25)	38	2.286948	13.5976	0.470673	3.139481
Ganmo(22)	Ikeja-west(28)	39	1.412968	20.00039	0.672858	5.104999
GereguPS(18)	Ganmo(22)	40	−2.72628	−7.35424	0.346171	1.718641
Gombe(11)	GereguPS(18)	41	41.74504	360.266	11.4754	96.87231
Gombe(11)	GereguPS(18)	42	41.74504	360.266	11.4754	96.87231
IhobvorNIPP(Eyaen)(25)	Kano(2)	43	−0.25914	−14.0245	0.837599	6.152014
IhobvorNIPP(Eyaen)(25)	Katampe(Abuja)(7)	44	−1.40316	−12.6928	0.687461	4.91669
IhobvorNIPP(Eyaen)(25)	Lekki(44)	45	0.41406	−21.6218	1.193857	9.591004
Ikeja-west(28)	Kaduna(5)	46	−4.08488	−44.1842	2.053758	17.37657
Ikot-Ekpene(48)	EgbinPS(38)	47	−0.10738	0.726023	0.002799	0.023759
Ikot-Ekpene(48)	Ganmo(22)	48	2.912938	−7.80913	0.393381	2.122327
Ikot-Ekpene(48)	Ganmo(22)	49	2.912938	−7.80913	0.393381	2.122327
Ikot-Ekpene(48)	Jos(10)	50	−1.55177	−13.5367	0.366552	2.771526
Ikot-Ekpene(48)	Jos(10)	51	−1.55177	−13.5367	0.366552	2.771526
Jalingo(13)	Ikot-Ekpene(48)	52	6.948098	58.87768	1.291704	10.85549
Jalingo(13)	Ikot-Ekpene(48)	53	6.948098	58.87768	1.291704	10.85549
JebbaPS(19)	Ikot-Ekpene(48)	54	1.117692	4.933228	0.075974	0.644369
Jos(10)	JebbaTS(8)	55	6.31 × 10−7	−0.09188	6.31 × 10−7	−0.09188
Jos(10)	JebbaTS(8)	56	6.31 × 10−7	−0.09188	6.31 × 10−7	−0.09188
Kaduna(5)	Gwagwalada(9)	57	1.558998	−0.22551	0.002553	−0.50686
KainjiPS(20)	OkeAro(43)	58	0.684128	0.536635	0.000444	−0.32525
KainjiPS(20)	Okpai(36)	59	0.47065	0.158258	0.000598	−0.79087
KainjiPS(20)	Okpai(36)	60	0.47065	0.158258	0.000598	−0.79087
Kano(2)	KainjiPS(20)	61	0.813036	0.266961	0.000177	−0.15958
Kano(2)	KainjiPS(20)	62	0.813036	0.266961	0.000177	−0.15958
Kano(2)	OmotoshoPhaseI(29)	63	0.001161	−1.35835	0.001034	−0.90237
Kano(2)	OmotoshoPhaseI(29)	64	0.001161	−1.35835	0.001034	−0.90237
Lekki(44)	Okpai(36)	65	−0.8027	−2.90036	0.008394	−0.69975
Lekki(44)	Okpai(36)	66	−0.8027	−2.90036	0.008394	−0.69975
Lokoja(15)	Damaturu(3)	67	−7.37439	10.05698	0.253378	1.872053
Lokoja(15)	JebbaPS(19)	68	0.303704	−3.08507	0.013891	−0.21051
Markudi(16)	Ayede(23)	69	2.2 × 10−11	−2.9 × 10−12	0.00016	−0.62673
NewHaven(32)	Lekki(44)	70	−2.74385	−3.72865	0.071117	−1.47901
NewHaven(32)	Lekki(44)	71	−2.74385	−3.72865	0.071117	−1.47901
NewHaven(32)	Maiduguri(4)	72	0.44677	3.668455	0.149362	−0.63324
NewHaven(32)	Maiduguri(4)	73	0.44677	3.668455	0.149362	−0.63324
OkeAro(43)	Okpai(36)	74	0.683571	0.861909	0.00044	−0.21443
Okpai(36)	Olorunsogo(21)	75	0.00013	−0.71324	0.00013	−0.71324
Okpai(36)	Olorunsogo(21)	76	0.00013	−0.71324	0.00013	−0.71324
OmotoshoPhaseI(29)	OmotoshoPhaseII(27)	77	0.000255	−0.91196	0.000255	−0.91196
Total losses					4.699488	4.467413

which contain the power flow and loss along the transmission lines. Furthermore, Figure 4, Figure 5 and Figure 6 depict the obtained voltage profile, active and reactive power, respectively. Similarly, Case 2 results are shown in

Table 3. Power flow result of the compensated Nigerian transmission network with IPFC.

From Bus	To Bus	Line	P Flow	Q Flow	P Loss	Q Loss
			[p.u.]	[p.u.]	[p.u.]	[p.u.]
Aja(39)	AfamPS(46)	1	−0.63589	−0.56586	0.004115	−1.27909
Ajaokuta(17)	Aja(39)	2	−0.69051	0.495752	0.002434	−0.32424
Ajaokuta(17)	Aja(39)	3	−0.69051	0.495752	0.002434	−0.32424
Akangba(34)	Ajaokuta(17)	4	−1.25562	−0.63784	0.012942	−0.92363
Akangba(34)	Ajaokuta(17)	5	−1.25562	−0.63784	0.012942	−0.92363
Aladja(40)	Ajaokuta(17)	6	1.450011	0.099719	0.011146	−0.60064
Aladja(40)	Ajaokuta(17)	7	1.450011	0.099719	0.011146	−0.60064
Aladja(40)	Ajaokuta(17)	8	1.450011	0.099719	0.011146	−0.60064
Ajaokuta(17)	Alagbon(45)	9	0.680241	0.655566	0.000241	−0.03245
Ajaokuta(17)	Alagbon(45)	10	0.680241	0.655566	0.000241	−0.03245
Alaoji(42)	Akangba(34)	11	0.575	0.004934	1.11 × 10−16	0.009867
Alaoji(42)	Akangba(34)	12	0.575	0.004934	1.11 × 10−16	0.009867
Ayede(23)	Akangba(34)	13	−2.24169	1.953711	0.028936	−0.1697
Ayede(23)	Akangba(34)	14	−2.24169	1.953711	0.028936	−0.1697
Ganmo(22)	Aladja(40)	15	0.732767	−0.66596	0.004165	−1.10565
Ganmo(22)	Aladja(40)	16	0.732767	−0.66596	0.004165	−1.10565
Ganmo(22)	Aladja(40)	17	0.732767	−0.66596	0.004165	−1.10565
Ikot-Ekpene(48)	Aladja(40)	18	1.62107	−0.98307	0.022485	−0.92904
Lokoja(15)	Aladja(40)	19	1.319408	−1.23973	0.010228	−0.42045
B-Kebbi(1)	Ayede(23)	20	−1.78908	−0.36804	0.019072	−0.69
Benin(30)	Ayede(23)	21	−0.291	0.292777	0.003829	−0.89589
Benin(30)	Ayede(23)	22	−0.291	0.292777	0.003829	−0.89589
Benin(30)	Ayede(23)	23	−0.291	0.292777	0.003829	−0.89589
CalabarPS(47)	Benin(30)	24	−1.02335	−1.8586	0.026512	−0.93898
NewHaven(32)	Benin(30)	25	1.197028	−0.03086	0.011632	−1.15485
NewHaven(32)	Benin(30)	26	1.197028	−0.03086	0.011632	−1.15485
CalabarPS(47)	CalabarSS(37)	27	0.002724	−1.05868	0.002724	−1.05868
Ikot-Ekpene(48)	Damaturu(3)	28	1.847674	0.358726	0.016057	−0.41019
CalabarPS(47)	DeltaPS(41)	29	1.020625	2.917277	0.005703	−0.01346
Eastmain(14)	DeltaPS(41)	30	−1.02	0.840059	0.00929	−0.39345
Maiduguri(4)	DeltaPS(41)	31	2.003508	2.805979	0.049141	−0.09251
GereguPS(18)	Ganmo(22)	32	−0.57014	−0.75007	0.002695	−0.84153
Gwagwalada(9)	Ganmo(22)	33	0.051721	−0.33551	8.23 × 10−05	−0.2242
Gwagwalada(9)	Ganmo(22)	34	0.051721	−0.33551	8.23 × 10−05	−0.2242
Gwagwalada(9)	Ganmo(22)	35	0.051721	−0.33551	8.23 × 10−05	−0.2242
IhobvorNIPP(Eyaen)(25)	Ganmo(22)	36	−0.20122	−0.8413	0.001489	−0.59351
IhobvorNIPP(Eyaen)(25)	Ganmo(22)	37	−0.20122	−0.8413	0.001489	−0.59351
IhobvorNIPP(Eyaen)(25)	Ganmo(22)	38	−0.20122	−0.8413	0.001489	−0.59351
Ikeja-west(28)	Ganmo(22)	39	1.508066	−1.16115	0.009904	−0.3403
Ganmo(22)	GereguPS(18)	40	0.572834	−0.09146	0.002695	−0.84153
GereguPS(18)	Gombe(11)	41	0.905139	0.164571	0.000139	−0.02043
GereguPS(18)	Gombe(11)	42	0.905139	0.164571	0.000139	−0.02043
Kano(2)	IhobvorNIPP(Eyaen)(25)	43	−1.28556	2.42897	0.035178	−0.35398
Katampe(Abuja)(7)	IhobvorNIPP(Eyaen)(25)	44	−4.9 × 10−16	−3.8 × 10−15	0.000398	−0.59671
Lekki(44)	IhobvorNIPP(Eyaen)(25)	45	−3.21808	4.097515	0.06445	0.173528
Kaduna(5)	Ikeja-west(28)	46	−1.48869	0.966518	0.003242	0.027427
EgbinPS(38)	Ikot-Ekpene(48)	47	0.55	−0.74567	0.004756	0.040366
Ganmo(22)	Ikot-Ekpene(48)	48	−1.14952	0.141339	0.009936	−0.77962
Ganmo(22)	Ikot-Ekpene(48)	49	−1.14952	0.141339	0.009936	−0.77962
Jos(10)	Ikot-Ekpene(48)	50	−4.2 × 10−7	0.060935	7.68 × 10−5	−0.26837
Jos(10)	Ikot-Ekpene(48)	51	−4.2 × 10−7	0.060935	7.68 × 10−5	−0.26837
Ikot-Ekpene(48)	Jalingo(13)	52	7.48 × 10−7	−0.07354	7.48 × 10−7	−0.07354
Ikot-Ekpene(48)	Jalingo(13)	53	7.48 × 10−7	−0.07354	7.48 × 10−7	−0.07354
Ikot-Ekpene(48)	JebbaPS(19)	54	1.35744	−0.0708	0.007206	0.061119
JebbaTS(8)	Jos(10)	55	7.47 × 10−15	4.85 × 10−16	4.18 × 10−7	−0.06094
JebbaTS(8)	Jos(10)	56	7.47 × 10−15	4.85 × 10−16	4.18 × 10−7	−0.06094
Gwagwalada(9)	Kaduna(5)	57	−1.48285	0.73683	0.005843	−0.22969
OkeAro(43)	KainjiPS(20)	58	−0.54001	0.109245	0.000295	−0.16767
Okpai(36)	KainjiPS(20)	59	−0.37143	−0.11087	0.00032	−0.41008
Okpai(36)	KainjiPS(20)	60	−0.37143	−0.11087	0.00032	−0.41008
KainjiPS(20)	Kano(2)	61	−0.6419	0.437664	0.000283	−0.08039
KainjiPS(20)	Kano(2)	62	−0.6419	0.437664	0.000283	−0.08039
OmotoshoPhaseI(29)	Kano(2)	63	−6.5 × 10−5	0.233782	0.00053	−0.46265
OmotoshoPhaseI(29)	Kano(2)	64	−6.5 × 10−05	0.233782	0.00053	−0.46265
Okpai(36)	Lekki(44)	65	0.641279	0.484721	0.001932	−0.39099
Okpai(36)	Lekki(44)	66	0.641279	0.484721	0.001932	−0.39099
Damaturu(3)	Lokoja(15)	67	−0.02306	−1.55114	0.004199	−0.21565
JebbaPS(19)	Lokoja(15)	68	1.350234	−0.13192	0.003569	−0.22768
Ayede(23)	Markudi(16)	69	9.85 × 10−5	−0.38597	9.85 × 10−5	−0.38597
Lekki(44)	NewHaven(32)	70	2.248389	−1.17305	0.036285	−0.97255
Lekki(44)	NewHaven(32)	71	2.248389	−1.17305	0.036285	−0.97255
Maiduguri(4)	NewHaven(32)	72	−1.00175	−1.40299	0.013322	−1.57263
Maiduguri(4)	NewHaven(32)	73	−1.00175	−1.40299	0.013322	−1.57263
Okpai(36)	OkeAro(43)	74	−0.53983	−0.00286	0.000177	−0.1121
Olorunsogo(21)	Okpai(36)	75	−5.9 × 10−12	3.7 × 10−12	6.79 × 10−5	−0.37242
Olorunsogo(21)	Okpai(36)	76	−5.9 × 10−12	3.7 × 10−12	6.79 × 10−5	−0.37242
OmotoshoPhaseII(27)	OmotoshoPhaseI(29)	77	−7 × 10−12	4.47 × 10−12	0.000131	−0.46756
Total losses					0.55297	−38.3329

and contains the line power flow and loss of the compensated network with an IPFC, whereas Figure 7, Figure 8 and Figure 9 represent the voltage profile, active and reactive power of the same network with an IPFC.

4. Discussion

The results of the load flow analysis from Case 1 show that the transmission network without an IPFC had a real power loss of 4.699488 p.u., and a reactive power loss of 4.467413 p.u. as indicated in Table 2. The resultant effect of these line losses is high reactive power imbalances leading to an unstable active power flow as reflected in the real and reactive power outputs graphs of the network buses as shown in Figure 5 and Figure 6, respectively. The voltage profile graph from Case 1, shown in Figure 4, indicates voltage drops across numerous buses such as bus 3, 4, 6, 8, 9, 10, 14 and 31–36 with bus 9 being the weakest.

The integration of an IPFC to the power flow formation in Case 2 resulted in reducing the transmission network’s overall losses to 0.55297 p.u. and −38.3329 p.u., as indicated in Table 3; hence, a more stable active and reactive power output as shown in Figure 8 and Figure 9, respectively. Furthermore, the voltage profile of Case 2 shown in Figure 7 is more stable compared with that of Case 1 shown in Figure 4.

5. Conclusions

Consistent and reliable power system operation is very important to both power utilities and consumers. To achieve this much needed reliability in a consistent manner, voltage magnitude must be set within a required operational limit. FACTS controllers are suitable technologies that can provide compensation and dynamic control of power system transmission parameters. An IPFC, if optimally placed, can regulate the impedance of multiple lines to improve active power transfer capacity and voltage profile. This study examines the performance of an IPFC for voltage enhancement by suppressing fluctuation. A modified Newton–Raphson load flow problem with an IPFC variable incorporated has been formulated with the objective to improve voltage stability and maintain active power flow. The effectiveness of the proposed method is tested on the Nigerian 41 bus transmission network with Case 1 without an IPFC and Case 2 with an IPFC. The obtained result of the system with an IPFC placed at the weakest bus of the network was compared with a Newton–Raphson load flow analysis of the same network without an IPFC. The modified method proves effective as the power system network with an IPFC returns a more stable voltage profile and improved active power flow. It is also important to state that this study is not a comparative study; hence, the effectiveness of the proposed method is not compared with any other existing approach. The proposed method returned appreciable results that can be improved in the future especially with hybrid metaheuristic techniques which are faster and have a better convergence rate. Therefore, the proposed method is better for a small network where the main output focus is accuracy.