Keywords

16.1 Introduction

The demand of electricity is increasing day-by-day and this results in an increase in per unit production cost. Again, maintaining uninterrupted power flow and providing quality power during disturbance becomes challenging. Voltage ampere reactive (VAR) compensation is the management of reactive power to improve the performance of ac power systems. The concept of VAR compensation embraces a wide and diverse field of both system and customer problems, especially related with power quality issues since most power quality problems are attenuated or solved with an adequate control of reactive power [1].

In general, the problem of reactive power compensation is viewed from two aspects: load compensation and voltage support. In load compensation the objectives are to increase the value of the system voltage (power factor improvement), balance the real power drawn from the ac supply, compensate voltage regulation and eliminate current harmonics. In voltage support, the idea is for sustenance and to maintain stable voltage flow in the network. For power flow studies the frequency should remain nearly constant, because significant drop in frequency could result in high magnetizing currents in induction motors and transformers [2, 3]. The flows of active and reactive powers in a transmission network are fairly independent of each other and are influenced by different control actions. Active power control is closely related to frequency control, and reactive power control is closely related to voltage control [2, 3]. Since constancy of frequency and voltage are important factors in determining the quality of power supply, then the control of active and reactive power is vital to the satisfactory performance of a power system [3, 4].

Since electrical energy is normally generated at the power stations far away from the urban areas where consumers are located and are delivered to the ultimate consumers through a network of transmission and distribution, the terminal voltage vary substantially. Wider variation in voltage may cause erratic operation or even malfunctioning of consumers’ appliances. The main cause for voltage variation is the variation in load on the supply system . With the increase in load on the supply system the voltage at the consumer premises falls due to increase in voltage drop in: (I) Alternator synchronous impedance, (ii) Transmission lines, (iii) Feeders and (iv) Distributors [5,6,7].

A well designed power system is one that gives good quality and reliable supply. By good quality it means the voltage levels are within reasonable limits. Naturally all the equipment on the power system are designed to operate satisfactorily only when the voltage levels in the system correspond to the rated voltage or at the most the variation are within ±5% of rated value [7]. Thus, compensation could be beneficial in in this aspect. The benefits of compensations are enormous and include the following: reactive power compensation in a transmission system improves the stability of the ac system by increasing the maximum active power that can be transmitted. It also helps to maintain a substantially flat voltage profile at all levels of power transmission if properly harnessed. It also increases transmission efficiency. It controls steady-state and temporary over-voltages and can avoid disastrous blackouts [7,8,9,10]. Objectively, reactive power loss minimization and the study of the effect of joint compensation on an interconnected network are the main issue of this work and the result obtained showed the disparity.

In an interconnected power system of n-buses the power injected into the nodes is given by a set of 2n nonlinear simultaneous equations represented in Eqs. (16.1) and (16.2).

$$ P_{i} = \sum\limits_{n = 1}^{N} {\left| {V_{i} V_{k} Y_{ik} } \right|} \,\cos (\theta_{ik} + \partial_{k} - \partial_{i} ) $$
(16.1)
$$ Q_{i} = - \sum\limits_{n = 1}^{N} {\left| {V_{i} V_{k} Y_{ik} } \right|} \,\sin (\theta_{ik} + \partial_{k} - \partial_{i} ) $$
(16.2)

Both active and reactive powers of the loads vary as a function of voltage magnitude. Compensating devices are normally added to supply or absorb reactive power and thereby control the reactive power balance in a desired form [2].

16.1.1 Reactive Power Flow on Line Voltage Drop

Voltage variation is due to imbalance in generation and consumption of reactive power in the system . If the generated reactive power is more than the consumed power, the voltage levels go up and vice versa. However, if the two are equal, then the voltage profile becomes flat and it happens only when the load is equal to the natural load. Unfortunately the reactive power in the system keeps varying because of different types of load and if the reactive power generation is simultaneously controlled, a more or less flat voltage profile could be maintained. Too wide variation of voltage causes excessive heating of distribution transformers thereby reducing the transformer capacity [2, 9].

16.2 Compensation Plans

The device commonly used for voltage and reactive power control includes: shunt reactors, shunt capacitors, synchronous generators, synchronous condensers, FACTS DEVICES etc.

Shunt Reactors are inductive current element connected between lines and neutral to compensate for inductive load/(current) from transmission lines. They are used to compensate for the effects of line capacitance, particularly to limit voltage rise on open circuit or light load [2, 6]. Shunt Capacitors supply reactive power and boost local voltages. The major advantages of shunt capacitors are their low cost and their flexibility of installation and operation [2]. They are readily applied to various points in the system , thereby contributing to efficiency of the power transmission and distribution. The disadvantage is that their reactive power output is proportional to the square of the voltage. The reactive power output is reduced at low voltages when it is likely to be needed most [2, 11, 12]. Shunt capacitors alleviate the excess loading of system and enable further active loads to be drawn from the same system . The reduction in line currents results in reduction of system losses. It is economical to carry out reactive power compensation from capacitors ideally located in the vicinity of reactive load e.g. motors and low voltage side of transformers.

16.2.1 Real and Reactive Power Control

A synchronous machine that is connected to an infinite bus has fixed speed and terminal voltage. The control variables are the field current and the mechanical torque on the shaft. The variation of the field current \( \left( {I_{f} } \right) \), referred to as excitation system control is applied to either a generator or a motor to supply or absorb a variable amount of reactive power . Since the synchronous machine runs at a constant speed, the only means of varying the real power is through control of the torque imposed on the shaft by either the prime mover in the case of a generator or the mechanical load in the case of a motor. The complex power delivered to the system by the generator is given in per unit as in Eqs. (16.3) and (16.4)

$$ S = P + jQ = V_{i} I_{a}^{*} = |V_{i} ||I_{a} |(\cos \,\theta + j\,\sin \,\theta ) $$
(16.3)

And for real and imaginary quantities we obtain

$$ P = |V_{i} ||I_{a} |\cos \,\theta ;\quad Q = \left| {V_{i} } \right|\left| {I_{a} } \right|\sin \,\theta $$
(16.4)

16.2.2 Shunt Compensation—Static-Var Compensation

Shunt compensation is the use of shunt capacitor or and shunt reactors in the line to avoid or reduce voltage instability [1, 2, 8] Shunt compensators are connected in shunt either directly to a bus bar or to the tertiary winding or to the main transformer and sometimes at mid-point of the lines (in some countries) to minimize the voltage drop and the losses. Shunt compensators are installed near the local terminals in factory substations, in the receiving substations, at switching substations etc to provide leading volt ampere-reactive (MVar) and thus to reduce the line current and total kVA loading of substation transformer [4, 6, 7, 13].

16.2.3 Control of Voltage and Reactive Power

For a transmission line where X ≫ R and R is negligibly small, therefore

$$ \left| {\Delta V} \right| = \frac{{XQ_{r} }}{X},\quad Q_{r} = \frac{{V_{r} }}{X}\left| {\Delta V} \right| $$
(16.5)

This relationship shows that the reactive power \( Q_{r} \) is proportional to the magnitude of the voltage drop in the line. Thus voltage control and reactive power control are interrelated. The reactive power generated should be exactly equal to the reactive power consumed. Any mismatch in the reactive power balance affects the bus voltage magnitudes [1, 6, 7].

16.2.4 Reactive Power Compensation in the Nigeria 330 kV Network

VAr compensation is the management of reactive power to improve the performance of ac power systems. The concept of VAr compensation encircles a wide and diverse field of both system and customer problems, especially related with power quality issues. Most of the power quality problems can be attenuated or solved with an adequate control of reactive power .

System voltage is highly dependent on the flow of reactive power . The long transmission lines in the National Grid generate considerable reactive MVars which constitute serious problems in maintaining system voltages within statutory limits especially during light loads, system disturbances and or major switching. The Nigerian PHCN has many reactors in various locations in the country, some of which are shown in Table 16.1. Some of these reactors were incorporated in the system to carry out the compensation to control the effect of reactive Mar. The major cause of voltage variation or drop in the line is the flow of reactive power . More of over reactive currents causes \( I^{2} R \) losses in the system but produces no revenue.

Table 16.1 Status of reactors in Nigeria power network PHCN 330 kV system
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16.2.5 Reactive Power Management in Electric Power System

An important factor in the control of voltage in a power system depends on the reactive power production or absorption. Reactive power is required to excite consumer’s equipment and transmission network which consists of capacitive and inductive elements. It is important that a balance of reactive power be maintained in the operation of a system because control of voltage can be lost if this is not achieved [2, 3, 14]. The reactive power flow is minimized so as to reduce \( I^{2} R \) and (I2X) losses to a practical minimum. This ensures that the transmission system operates efficiently. The rating of capacitor can be calculated with the simplified equation as;

$$ C = \frac{{Q_{C} }}{{\omega V^{2} }} $$
(16.6)

Equation 16.5 shows that the capacitance required to improve the system efficiency is inversely proportional to \( \varvec{V}^{2} \). Note that at high voltages power capacitors or capacitor bank values are rated in kilo volt-ampere reactive (kvar or mvar). For three phase system , the equation for the capacitor in delta connection, where \( ( {V}_{P} ={V}_{L}) \)

Is given by Eq. 16.7.

$$ C_{\Delta } = \frac{{Q_{C} }}{{\omega V_{P}^{2} }} = \frac{{Q_{C} }}{{\omega V_{L}^{2} }} $$
(16.7)

Compensation added to the network is given by Eq. 16.8, [7, 15]

$$ Q_{C} = \frac{P}{{Pf_{1} }} \times \,\sin (\cos^{ - 1} (pf_{1} )) - \frac{P}{{Pf_{2} }}\,\sin (\cos^{ - 1} (pf_{2} ) $$
(16.8)

where P = real power specified at the buses, Pf1 = 0.85 power factor, Pf2 = 0.95 power factor, \( Q_{C} \) = value of shunt capacitance to be added to the network to boost the system voltage. Hence the capacitor required per three phase in star connection is equal to three times the capacitance required per phase when the capacitors are connected in delta. Also, the capacitors for the star-connected bank have a working voltage equal to \( \frac{1}{\sqrt 3 } \) times that for the delta-connected bank. For this reason, the capacitors are connected in delta in three-phase systems for improvement of the system stability. The installation of a capacitor bank can be used to avoid the need to change a transformer in the event of a load increase. System behavior is affected by the characteristics of every major element of the system . The representation of these elements by means of appropriate mathematical models is critical to the successful analysis of the system behavior [16, 17].

In an ac circuit, current ‘I’ leads voltage in a capacitive circuit and I lags voltage in an inductive circuit. A capacitor is used to generate Var and an inductor is used to reduce reactive power . In power system analysis there is a strong inter relationship between the real power and angle and also between the reactive power and the voltage magnitude which aids us in compensation analysis. These relationships below exist for real and reactive powers flow.

$$ \begin{aligned} S & = P + jQ,\,P = VI\text{Cos} \,\theta \,{\text{or}}\,P = I^{2} R = \frac{{V^{2} }}{R}, \\ & \quad {\text{and}}\,Q = VI\text{Sin} \,\theta \,{\text{or}}\,Q = I^{2} X = \frac{{V^{2} }}{X}\,X_{C} = \frac{1}{j\omega C} \\ \end{aligned} $$

and with low Var in a system , the voltage becomes low and vice versa. These relationships are used for the result analysis [8, 9, 18].

16.3 Network Description

The Nigerian power network like many practical systems in developing countries consists of a few generating stations mostly sited in remote locations near the raw fuel sources which are usually connected to the load centers by long transmission lines.

The National Electric Power Authority (NEPA) now known as Power Holding Company of Nigeria (PHCN) has the sole statutory functions of generation, transmission, distribution and marketing of electricity, before the partial unbundling of the power sector. Nigeria national electricity grid at present consists of nine generating stations comprising of three (3) hydro and six (6) thermal plants with a total installed generating capacity of 6500 MW. The thermal stations are mainly in the southern part of the country located at Afam, Okpai, Delta (Ughelli), Egbin and Sapele. The hydroelectric power stations are in the country’s middle belt and are located at Kainji, Jebba and Shiroro. The transmission network is made up of 5000 km of 330 kV lines, 6000 km of 132 kV lines, 23 of 330/132 kV sub-stations and 91 of 132/33 kV substations [7, 9, 19].

Although, the installed capacity of the existing power stations is 6500 MW the maximum load ever recorded was 4000 MW. Presently, most of the generating units have broken down due to limited available resources to carry out the needed level of maintenance. The transmission lines are radial and overloaded. The switchgears are obsolete while power transformers have not been maintained for a long time. The present installed generating capacity in Nigeria is shown in Table 16.2. The PHCN has only once been able to generate a maximum of 4700 MW, for a country of more than 160 million people [20,21,22].

Table 16.2 Existing power stations
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16.3.1 Methodology

In this work, the method used is based on the work of [1]. The existing 330 kV, 30 bus system of Nigeria transmission network with Egbin power station as the slack bus is used, and an in-depth examination of the Nigeria Integrated Power Plant Network was carried out. The parameters of all the generators and other system components were obtained. Equations for the power flow analysis are then formulated incorporating these parameters. The algorithm for the Newton-Raphson’s method was developed. The Newton-Raphson’s solution method represented with Eqs. (16.9) and (16.10) was used to carry out the analysis because of its sparsity, fast convergence and simplicity attributes as compared to other solution methods using the relevant data as obtained from Power Holding Company of Nigeria (PHCN). MATLAB m-file program and SIMULINK model were developed and used for the simulation analysis.

$$ \Delta {\rm P} = J_{11}\Delta \delta $$
(16.9)
$$ \Delta Q = J_{22}\Delta |V| $$
(16.10)

16.3.1.1 Line Flows and Line Losses Model

At the end of the iterative solution of the bus voltages, the calculation of the line flows and losses for both compensated and uncompensated conditions were carried out. The following formulae are used considering a two bus system labeled 1 and 2 (or i and j, and then to nth buses). First the Line current is given as:

$$ I_{12} = y(V_{1} - V_{2} ),\quad I_{21} = - I_{12} $$
(16.11)
$$ I_{13} = y_{13} (V_{1} - V_{3} ),\quad I_{31} = - I_{13} $$
(16.12)

Line flows are calculated using

$$ S_{12} = V_{1} I_{12}^{*} ;S_{21} = V_{2} I_{21}^{*} \quad $$
(16.13)
$$ S_{13} = V_{1} I_{13}^{*} ;\quad S_{31} = V_{3} I_{31}^{*} $$
(16.14)

Line losses

$$ S_{L12} = S_{12} + S_{21} ;\quad S_{L13} = S_{13} + S_{31} $$
(16.15)

16.3.2 Draft of Nigeria 330 kV Transmission Network Used as Case Study

The single-line diagram of the existing 330 kV Nigeria transmission network used as the case study is as shown in Fig. 16.1. It has 30 buses with nine generating station. The Egbin power station was chosen as the slack bus because of its capacity and location in the network.

Fig. 16.1
figure 1

One line diagram of the PHCN 330 kV 30 bus interconnected network

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16.3.3 Data Assembly—Line Data

The input data for the power flow analysis include the bus data, transmission line data (impedance of lines), voltages and transformer/load data obtained from Power Holding Company of Nigeria (PHCN) are as presented in Tables 16.3, 16.4 and 16.5.

Table 16.3 Transmission line data (of Bison, two conductors per Phase and 2 × 350 mm2 X-section Conductor) for 330 kV lines
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Table 16.4 Bus data values
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Table 16.5 Line data
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The load and generation data expressed in per unit values are given as \( \frac{MW + jMVAr}{base.value}. \) where the Slack Bus is Egbin Generating Station. As in Table 16.4

Base value = 100 MVA

Base voltage = 330 kV, \( {\text{Per}}\,{\text{Unit}}\,{\text{Value}} = \frac {MVA}{{Base\,Voltage}} \) as presented in Table 16.4.

16.3.4 Shunt Capacitor Compensation Algorithm

The flow chart in Fig. 16.2 is the procedural method applied to achieve the desired compensation results.

Fig. 16.2
figure 2

Flow chart for the shunt capacitor compensation analysis algorithm

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First, the base solution is obtained using Newton-Raphson’s method. Check bus voltages range. Identify the problem buses by checking the bus voltages outside \( \pm 5\% \) of the normal values (that is, 0.95–1.05) per unit. Calculate the capacitor values using this equation \( \left( {C = \frac{{Q_{C} }}{{\omega V^{2} }}} \right) \) and apply compensation using this \( Q_{C} = \frac{P}{{Pf_{1} }} \times \,\sin (\cos^{ - 1} (pf_{1} )) - \frac{P}{{Pf_{2} }}\,\sin (\cos^{ - 1} (pf_{2} )) \). Where P is real power specified at the buses, \( Pf_{1} \) and \( Pf_{2} \) are power factors, while \( Q_{C} \) is value of shunt capacitance to be added to the network to boost the system voltage. Finally output result and stop. These procedures were simulated using MATLAB/SIMULINK. The results from the Newton-Raphson iterative method give the bus voltages, line flows, and power losses under normal (uncompensated) condition as shown in Table 16.7. The voltages at buses 14, 17, 18, 19, 22, 29 and 30 are outside the limit, and in order to ensure that they are within acceptable limits shunt capacitive compensation were introduced into the buses. Based on Power Holding Company of Nigeria (PHCN) power factor of 0.85 and 0.95 for transmission lines are used. The MVAr capacities of the various capacitors required to carry out compensation of the network at the buses were determined using

$$ Q_{C} = \frac{P}{{Pf_{1} }} \times \,\sin (\cos^{ - 1} (pf_{1} )) - \frac{P}{{Pf_{2} }}\,\sin (\cos^{ - 1} (pf_{2} )) $$

The following capacitor sizes were selected for the various lines. Jos bus (30 MVAr), Gombe bus (30 MVAr), Abuja bus (60 MVAr), Kano bus (40 MVAr), Kaduna bus (40 MVAr), and Makurdi bus (30 MVAr). These were introduced into the network to examine their effect on the system . The weak buses were identified as presented in Table 16.6, and the plots of the results are as shown in Figs. 16.3 and 16.4.

Table 16.6 Bus voltages for compensated and uncompensated
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Fig. 16.3
figure 3

Plot of bus voltages under normal (un compensated) condition

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Fig. 16.4
figure 4

Bar plot of bus voltages with compensation

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16.4 Results

It is also recorded during the compensation of the entire system jointly, that some buses that were normal are affected. Some buses values decreased from tolerable values while some over increased. Some of the pictorial graphs were as presented in Fig. 16.5a–f.

Fig. 16.5
figure 5

a–f Graph of voltage versus bus nos. at different levels of percentage compensation

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16.4.1 Discussions

The analysis of Nigeria 330 kV 30 bus network using Newton-Raphson’s power flow solution algorithm with MATLAB/SIMULINK software was effectively completed. The results obtained revealed the weak buses with values outside the standard limit of 0.95 p.u. (313.5 kV) and 1.05 p.u. (346.kV). The recorded results are; Bus 14 (Jos) with value 0.8171 p.u., bus 17 (Gombe) 0.8144 p.u bus 18 (Abuja) 0.9402 p.u., bus 19 (Maiduguri) 0.8268 p.u., bus 22 (Kano) 0.7609 p.u., bus 29 (Kaduna) 0.8738 p.u., and bus 30 (Makurdi) 0.8247 p.u. under normal uncompensated condition as presented in Fig. 16.4.

The compensation was carried out on the weak buses. At 45% capacitive shunt compensation on those buses indicated improved performance, Kano and Jos were still at the weak positions due to their distances in the national grid. With sixty (60) per cent compensation a better result was recorded as buses 14 (Jos) improved to 0.9823 and 22 (Kano) 0.9338. The compensated results are as displayed in Fig. 16.5a–f. It was observed that the application of compensation on the entire system jointly has negative effect on the other buses—which is part of the main aim of this work. This was verified by a step-by-step application of compensation at 5% intervals. It was observed that compensating the whole network jointly affects some of the other buses that were within the tolerable range. For instance, at Bus (20) the value decreased from 0.9568–0.9329 p.u. about 2.39% decrease. This can cause damage to the “system’s equipment and consumers’ appliance” if no proper security for contingency analysis control was installed. Also, bus (17) increased from 0.9786 to 1.0799 p.u. and bus (19) from 0.998  to 1.1035 p.u. and so on, which show undershoot and overshoot respectively which may lead to system collapse if not monitored. System efficiency was improved from 65% (uncompensated) to 85% after compensation.

Table 16.7 Bus voltages at various compensation levels for the compensated buses
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16.5 Conclusion

In this work compensation of interconnected electric power network was studied using Shunt capacitive compensation. Shunt capacitive compensation is widely recognized as one of the prevailing methods to combat the problems of voltage drops, power losses, and voltage flicker in power system networks. Though each compensating technique has its area of proficiency and limit of application , but shunt capacitor compensation method was used because of its outstanding performance especially in long transmission lines and its control of reactive power flow. Though they associated with high cost implication but control voltage directly and also control temporary over voltage fast. It was observed also that application of compensation on the interconnected system jointly has side effect on the other buses. The results indicated that control of active and reactive power has high influence on the Nigeria network, hence adequate attention must be placed on it. Also with innovation/advent of renewable energy integration into the grid, if adequate control measure of reactive power is not put in place there will be no much success. Thus, it is advised that concentrating the compensation on the problem buses gives best result like buses 14, 17, 18, 19, 22, 29, 30, at 65% recorded 0.976966, 1.009312, 0.954925. 1.029198, 0.834761, 0.93879 and 0.98823 and others as shown in Table 16.7. This reduces cost as well. Losses in the system were minimized as can be seen through Tables 16.8, 16.9, 16.10 and 16.11.

Table 16.8 Line current, line flow, and line losses (uncompensated)
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Table 16.9 Summary of total loss (uncompensated)
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Table 16.10 Line current, line flows, and line losses (compensated)
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Table 16.11 Summary of total loss (compensated)
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