4. Var compensator with thyristor-controlled reactor

The problem of maintaining a good power factor (greater than 0.85) is a challenge faced by most of the industries. Some of these bulk consumers are penalized for operation under a low power factor. The application of cubic splines is presented to topic address this challenge and provide a commendable solution. Rolling mills and electric arc furnaces constitute very large loads in the power system. When the problem of power factor is considered, these loads (particularly electric arc furnaces) may be termed as a necessary evil because of their extremely low power factor. The variation in the arc length during operation results in the introduction of severe and rapid fluctuations in the reactive power and the voltage, and when the short circuit occurs, the power factor drops to values as low as 0.1. The large impact current and reactive power generated result in significant waste of energy and may also cause the power system to lose its stability. This may cause decadence in the quality of load and endanger the users. The solution to the abovementioned problem is cited in a process called reactive power consumption. Currently, the reactive power compensation is achieved dynamically by the placement of a dynamic reactive power compensation device at access points of such interference loads. These devices are usually represented as a fixed capacitor and thyristor-controlled reactor (FCT). The FCT enables smooth control of the reactive power and also has the desirable feature of maintaining its voltage unchanged. In addition, the FCT can effectively suppress voltage fluctuation and solve the voltage distortion and flicker problem and improve the power quality. The continuous and smooth variation in the reactive power is obtained by the variation of the thyristor conduction angle. For the FCT to effectively carry out its task, it is imperative that it calculates the control angle quickly and accurately in a real-time environment. However, the existence of a nonlinear relation between the control angle and the reactor amplification factor makes the real-time calculation a cumbersome process.

value of the fundamental current changes, which in turn results in the variation in reactive

The relationship between the delay angle, α, and the amplification factor of the equivalent

As can be seen, there exists a nonlinear relation between the delay angle and the amplification factor. The application of cubic splines makes the control of delay angle with reactive power variation easier, quicker and smoother. It must be observed that the inductance current in the TCR does not depend on the inductance but is rather governed by the thyristor conduction angle. The accurate control of the thyristor angle makes the accurate control of inductor

The problem is formulated as follows. Eq. (9) suggests that in order to control the delay angle using the amplification factor 'k', we need to find a solution to the nonlinear equation. Albeit possible, it is a cumbersome task to solve nonlinear equations, not to mention the considerable amount of time the solving takes which make real-time application and control a tedious task. Moreover, of paramount importance is the accuracy of the controlling action since it determines the reactive power compensation. The cubic spline interpolation is employed as a means to calculate the control angle 'α' having known the amplification factor 'k'. Cubic spline is used as the method of interpolation because of the advantages it provides in terms of simplicity of

As previously stated, the existence of a nonlinear relation between the control angle and the amplification factor makes the real-time computation a lengthy process. In order to achieve the aim using cubic spline interpolation, the control angle is regarded as the dependent variable and the amplification factor 'k' as the independent variable. According to [3], the author generated 158 data points using Eq. (2). The aim is now to generate 157 interpolating cubic polynomials that fit in between these points. The form of these cubic polynomials is shown in Eq. (17)

<sup>3</sup> <sup>þ</sup> a1ð Þ <sup>k</sup> � ki

The fitting coefficients a0, a1, a2 and a3 need to be computed for each interval which gives a unique cubic polynomial for each interval. When carrying out the control procedure first, the amplification is determined, and then the corresponding cubic polynomial is used to arrive at

Inference: The results shown in [3] suggest that the conduction angle determined by using cubic splines shows significant match with the values obtained by simulation. Also, the use of cubic splines yields quicker results—a trait which would be beneficial for real-time applications.

calculation, numerical stability and smoothness of the interpolated curve.

α ¼ a0ð Þ k � ki

In Eq. (10), i = 1, 2, …., 157 and ki ≤ k ≤ ki + =1.

the required conduction angle value.

<sup>k</sup> <sup>¼</sup> <sup>π</sup>

2π � 2α þ sin 2α πX<sup>L</sup>

<sup>2</sup>ð Þþ <sup>π</sup> � <sup>α</sup> sin 2<sup>α</sup> (16)

Application of Cubic Spline Interpolation Technique in Power Systems: A Review

http://dx.doi.org/10.5772/intechopen.74853

<sup>2</sup> <sup>þ</sup> a2ð Þþ <sup>k</sup> � ki a3 (17)

(15)

139

power. The equivalent fundamental susceptance of the TCR is given in Eq. (15)

BLð Þ¼ α

impedance is shown in Eq. (16)

current and hence reactive power possible.

The remainder of this section reviews the calculation and the corresponding control of the control angle via employing cubic spline interpolation technique. The calculation of the control angle, α, is based on the reactor amplification factor for each phase. Cubic spline interpolation is employed because it provides a low-order polynomial interpolation polynomial and also increases the smoothness of the interpolation function.

The basic thyristor-controlled reactor (TCR) consists of a pair of anti-parallel thyristors in series with an inductor as shown in Figure 5. The thyristor delay angle varies between 90 and 180. As a result, the fundamental current is completely reactive. An increase in the delay angle leads to a decrease in the fundamental reactive current, which is equivalent to increasing the reactance or reducing the susceptance and hence results in a decrease in the reactive power. Hence, the TCR can be seen as being equivalent to a variable susceptance which can be controlled using the delay angle 'α'. This is because the AC voltage remains constant but the

Figure 5. Thyristor controlled reactor.

value of the fundamental current changes, which in turn results in the variation in reactive power. The equivalent fundamental susceptance of the TCR is given in Eq. (15)

$$\mathbf{B}\_{\rm L}(\alpha) = \frac{2\pi - 2\alpha + \sin 2\alpha}{\pi X\_{\rm L}} \tag{15}$$

The relationship between the delay angle, α, and the amplification factor of the equivalent impedance is shown in Eq. (16)

$$\mathbf{k} = \frac{\pi}{2(\pi - \alpha) + \sin 2\alpha} \tag{16}$$

As can be seen, there exists a nonlinear relation between the delay angle and the amplification factor. The application of cubic splines makes the control of delay angle with reactive power variation easier, quicker and smoother. It must be observed that the inductance current in the TCR does not depend on the inductance but is rather governed by the thyristor conduction angle. The accurate control of the thyristor angle makes the accurate control of inductor current and hence reactive power possible.

The problem is formulated as follows. Eq. (9) suggests that in order to control the delay angle using the amplification factor 'k', we need to find a solution to the nonlinear equation. Albeit possible, it is a cumbersome task to solve nonlinear equations, not to mention the considerable amount of time the solving takes which make real-time application and control a tedious task. Moreover, of paramount importance is the accuracy of the controlling action since it determines the reactive power compensation. The cubic spline interpolation is employed as a means to calculate the control angle 'α' having known the amplification factor 'k'. Cubic spline is used as the method of interpolation because of the advantages it provides in terms of simplicity of calculation, numerical stability and smoothness of the interpolated curve.

As previously stated, the existence of a nonlinear relation between the control angle and the amplification factor makes the real-time computation a lengthy process. In order to achieve the aim using cubic spline interpolation, the control angle is regarded as the dependent variable and the amplification factor 'k' as the independent variable. According to [3], the author generated 158 data points using Eq. (2). The aim is now to generate 157 interpolating cubic polynomials that fit in between these points. The form of these cubic polynomials is shown in Eq. (17)

$$\mathbf{a} = \mathbf{a}\_0(\mathbf{k} - \mathbf{k}\_i)^3 + \mathbf{a}\_1(\mathbf{k} - \mathbf{k}\_i)^2 + \mathbf{a}\_2(\mathbf{k} - \mathbf{k}\_i) + \mathbf{a}\_3 \tag{17}$$

In Eq. (10), i = 1, 2, …., 157 and ki ≤ k ≤ ki + =1.

provide a commendable solution. Rolling mills and electric arc furnaces constitute very large loads in the power system. When the problem of power factor is considered, these loads (particularly electric arc furnaces) may be termed as a necessary evil because of their extremely low power factor. The variation in the arc length during operation results in the introduction of severe and rapid fluctuations in the reactive power and the voltage, and when the short circuit occurs, the power factor drops to values as low as 0.1. The large impact current and reactive power generated result in significant waste of energy and may also cause the power system to lose its stability. This may cause decadence in the quality of load and endanger the users. The solution to the abovementioned problem is cited in a process called reactive power consumption. Currently, the reactive power compensation is achieved dynamically by the placement of a dynamic reactive power compensation device at access points of such interference loads. These devices are usually represented as a fixed capacitor and thyristor-controlled reactor (FCT). The FCT enables smooth control of the reactive power and also has the desirable feature of maintaining its voltage unchanged. In addition, the FCT can effectively suppress voltage fluctuation and solve the voltage distortion and flicker problem and improve the power quality. The continuous and smooth variation in the reactive power is obtained by the variation of the thyristor conduction angle. For the FCT to effectively carry out its task, it is imperative that it calculates the control angle quickly and accurately in a real-time environment. However, the existence of a nonlinear relation between the control angle and the reactor amplification

The remainder of this section reviews the calculation and the corresponding control of the control angle via employing cubic spline interpolation technique. The calculation of the control angle, α, is based on the reactor amplification factor for each phase. Cubic spline interpolation is employed because it provides a low-order polynomial interpolation polynomial and also

The basic thyristor-controlled reactor (TCR) consists of a pair of anti-parallel thyristors in series with an inductor as shown in Figure 5. The thyristor delay angle varies between 90 and 180. As a result, the fundamental current is completely reactive. An increase in the delay angle leads to a decrease in the fundamental reactive current, which is equivalent to increasing the reactance or reducing the susceptance and hence results in a decrease in the reactive power. Hence, the TCR can be seen as being equivalent to a variable susceptance which can be controlled using the delay angle 'α'. This is because the AC voltage remains constant but the

factor makes the real-time calculation a cumbersome process.

increases the smoothness of the interpolation function.

Figure 5. Thyristor controlled reactor.

138 Topics in Splines and Applications

The fitting coefficients a0, a1, a2 and a3 need to be computed for each interval which gives a unique cubic polynomial for each interval. When carrying out the control procedure first, the amplification is determined, and then the corresponding cubic polynomial is used to arrive at the required conduction angle value.

Inference: The results shown in [3] suggest that the conduction angle determined by using cubic splines shows significant match with the values obtained by simulation. Also, the use of cubic splines yields quicker results—a trait which would be beneficial for real-time applications. Hence, it can be said that the cubic spline interpolation method is able to solve the problem of quick computation of the conduction angle with accuracies relevant to engineering applications.

#### 5. Estimation of voltage stability margin

In this section, the application of cubic splines is of paramount importance since it marks the limit of a power system—the maximum amount of power that can be supplied before the continuous uncontrolled drop of voltage. The impact of voltage instability on the power system has been so damaging that significant research has been conducted in this direction. The phenomenon of voltage instability is characterized by a sudden and uncontrollable drop in voltage as a response to a disturbance that has occurred on the power system. This disturbance could be anything from the variation of load to loss of a line or lightning strike, and so on. Most of the studies have been focused on the steady-state aspect of voltage stability. In order to determine the proximity of the system to voltage collapse, we need to estimate or find the voltage stability margin (VSM). Many methods have been proposed in the past to estimate the steady-state voltage stability margin. One of the popular methods is the continuous power flow method. A major disadvantage of the continuous power flow is that it requires a considerable amount of time and hence it cannot be employed in real-time applications.

An alternative approach is to use the time-synchronized voltage and current phasor measurements obtained from phasor measurement units (PMUs). In the presented method, the author proposed a combination of a coupled single-port Thevenin equivalent model and cubic spline extrapolation in order to find the point of voltage collapse or the voltage stability margin (VSM). The concept is based on the fact that the voltage collapse point of the load impedance equals the Thevenin equivalent impedance.

Thevenins' equivalent voltage and equivalent impedance as seen from load bus 'i' as given in Figure 6 can be written as shown in Eq. (18)

$$\mathbf{E}\_{\rm thi} = \begin{bmatrix} \mathbf{K} \mathbf{V}\_{\rm G} \end{bmatrix}\_{\rm i} \quad - \sum\_{\substack{\mathbf{j} = \mathbf{I}\_{\prime} \ \mathbf{j} \neq \mathbf{i}}}^{\rm n} \mathbf{Z}\_{\rm i} \mathbf{I}\_{\rm Lj} \tag{18}$$
 
$$\mathbf{Z}\_{\rm thi} = \mathbf{Z}\_{\rm ii}$$

The definitions for the abovementioned variables can be found in [4].

The load impedance ZLi of bus 'i' can be arrived by using Eq. (19)

$$\mathbf{Z\_{Li}} = \frac{\mathbf{V\_{Li}}}{\mathbf{I\_{Li}}} \tag{19}$$

Figure 7 shows the variation of Zli and Zth as a function of the load parameter λ. As can be seen, the Zli equals the Zthi at the point of maximum critical loading, and this point gives the maximum value of the load parameter—λmax. This maximum value of load parameter can be arrived at by equating an approximate function that extrapolates the Zli versus λ curve to

Application of Cubic Spline Interpolation Technique in Power Systems: A Review

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141

The choice of cubic spline extrapolation is justified by the author as the superior fitting to the impedance trajectory as evidenced by extensive simulation results. The cubic spline extrapolation proceeds by developing different cubic polynomials for the interval between measure-

the point that it meets the Zthi line.

Figure 6. Coupled single port Thevenins' equivalent.

ments based on certain constraints.

Figure 7. Variation of Thevenin and load impedance with λ [4].

The load bus voltage VLi and the load bus current ILi are obtained through the PMU measurements.

Figure 6. Coupled single port Thevenins' equivalent.

Hence, it can be said that the cubic spline interpolation method is able to solve the problem of quick computation of the conduction angle with accuracies relevant to engineering applications.

In this section, the application of cubic splines is of paramount importance since it marks the limit of a power system—the maximum amount of power that can be supplied before the continuous uncontrolled drop of voltage. The impact of voltage instability on the power system has been so damaging that significant research has been conducted in this direction. The phenomenon of voltage instability is characterized by a sudden and uncontrollable drop in voltage as a response to a disturbance that has occurred on the power system. This disturbance could be anything from the variation of load to loss of a line or lightning strike, and so on. Most of the studies have been focused on the steady-state aspect of voltage stability. In order to determine the proximity of the system to voltage collapse, we need to estimate or find the voltage stability margin (VSM). Many methods have been proposed in the past to estimate the steady-state voltage stability margin. One of the popular methods is the continuous power flow method. A major disadvantage of the continuous power flow is that it requires a consid-

erable amount of time and hence it cannot be employed in real-time applications.

An alternative approach is to use the time-synchronized voltage and current phasor measurements obtained from phasor measurement units (PMUs). In the presented method, the author proposed a combination of a coupled single-port Thevenin equivalent model and cubic spline extrapolation in order to find the point of voltage collapse or the voltage stability margin (VSM). The concept is based on the fact that the voltage collapse point of the load impedance

Thevenins' equivalent voltage and equivalent impedance as seen from load bus 'i' as given in

Zthi ¼ Zii

ZLi <sup>¼</sup> VLi ILi

The load bus voltage VLi and the load bus current ILi are obtained through the PMU measure-

<sup>j</sup>¼<sup>1</sup>, <sup>j</sup>6¼<sup>i</sup>

ZijILj (18)

(19)

Ethi <sup>¼</sup> ½ � <sup>K</sup>V<sup>G</sup> <sup>i</sup> � <sup>X</sup><sup>n</sup>

The definitions for the abovementioned variables can be found in [4].

The load impedance ZLi of bus 'i' can be arrived by using Eq. (19)

5. Estimation of voltage stability margin

140 Topics in Splines and Applications

equals the Thevenin equivalent impedance.

Figure 6 can be written as shown in Eq. (18)

ments.

Figure 7 shows the variation of Zli and Zth as a function of the load parameter λ. As can be seen, the Zli equals the Zthi at the point of maximum critical loading, and this point gives the maximum value of the load parameter—λmax. This maximum value of load parameter can be arrived at by equating an approximate function that extrapolates the Zli versus λ curve to the point that it meets the Zthi line.

The choice of cubic spline extrapolation is justified by the author as the superior fitting to the impedance trajectory as evidenced by extensive simulation results. The cubic spline extrapolation proceeds by developing different cubic polynomials for the interval between measurements based on certain constraints.

Figure 7. Variation of Thevenin and load impedance with λ [4].

For 'm' PMU-measured data points obtained, there will be 'm-1' intervals present, each of which is represented by Eq. (20)

$$\mathbf{f}\_{\mathbf{j}}\left(\mathbf{x}\right) = \mathbf{a}\_{\mathbf{j}} + \mathbf{b}\_{\mathbf{j}}\left(\mathbf{x} - \mathbf{x}\_{\mathbf{j}}\right) + \mathbf{c}\_{\mathbf{j}}\left(\mathbf{x} - \mathbf{x}\_{\mathbf{j}}\right)^{2} + \mathbf{d}\_{\mathbf{j}}\left(\mathbf{x} - \mathbf{x}\_{\mathbf{j}}\right)^{3} \tag{20}$$

The algorithm for proceeding with the described method in order to compute the VSM is as

Application of Cubic Spline Interpolation Technique in Power Systems: A Review

http://dx.doi.org/10.5772/intechopen.74853

Step 1. Obtain three sets of PMU measurements and use them to compute three sets of |ZLi|, λ

Step 4. If there is no violation of the reactive power limits of any generator, then proceed to

Step 5. Change the bus type from PV to PQ bus for the bus whose reactive power limit has

Step 8. Estimate the value of λmaxi by using the cubic spline extrapolation technique for each

Inference: The author of [4] has performed test runs of the proposed algorithm on the following test system—IEEE 30 bus system, IEEE 118 bus system and IEEE 300 bus system. Further, the obtained results have been compared with two other previously available methods. The obtained results indicate the superior performance of the proposed cubic spline technique when compared to the other methods. Upon comparison of the percentage error, it may be observed from [4] that the cubic spline method is almost 10 times more accurate than the other

In this final section, the application of cubic spline technique has been illustrated in the field of deregulated electricity market. In this market, market power issues predominantly spoil the

Market power is the ability of showing one's monopolistic nature on the price of the commodities in the market. This has become a challenging issue in the context of the present electricity market and will become more challenging and play a significant role when private generation companies start participating in buying/selling the power [5]. Due to the increase in demand and the regulatory policies, private parties have started investing in the power sector, especially in the renewable energy sources. Thus, it is inevitable for the independent system

Step 9. Find the value of λsys which is the minimum of all values computed in step 8.

6. Estimation of market power in deregulated electricity market

basic idea of maintaining equilibrium within the market players.

operator (ISO) to estimate market power for taking crucial decisions [6].

Gi

143

Step 2. Compute the parameters αi, β<sup>i</sup> and γ<sup>i</sup> for each of the generator bus of the system. Step 3. Compute the estimate of the extreme of the reactive power for each generator Qex

and QGi. Also, we possess the knowledge of the admittance matrix.

step 6. Otherwise, proceed to step 5.

Step 6. Compute the impedance matrix as shown in [4]. Step 7. Compute the Thevenin equivalent impedance.

follows:

been violated.

methods.

load bus of the system.

Step 10. Determine the VSM.

The inputs for extrapolation are taken as three sets of |ZLi| and λ and the point of |Zli| where λ is to be extrapolated and found. The λ thus found corresponds to the point where Zli equals Zth<sup>i</sup> and hence is equal to λmaxi. Each bus in the considered system will have its own corresponding λmaxs. The λmax = λsys for the entire system is found by taking the minimum of all λmaxi obtained for each of the individual load busses. This value of λsys corresponds to the value of λmaxi for the weakest load bus in the system. The proximity of the load bus to the voltage collapse point is given in Eq. (21)

$$\text{VSM} = \frac{\lambda\_{\text{sys}} - \lambda\_0}{\lambda\_0} \times 100 \tag{21}$$

After the computation of VSM, the author of [4] proceeds to find an index which helps in determining whether the load increase results in the violation of the reactive power limits at the generator busses. The author begins with the simple power equation and arrives at the equation of a surface given by Eq. (22)

$$\mathbf{Q}\_{\rm G}^{2} - \frac{|\mathbf{V}\_{\rm G}|^{2}}{X} \mathbf{Q}\_{\rm G} + \frac{|\mathbf{V}\_{\rm G}|^{2}}{X} \mathbf{Q}\_{\rm L} + \mathbf{P}\_{\rm L}^{2} = \mathbf{0} \tag{22}$$

where PL, load real power; QL, load reactive power; QG, generator reactive power.

The surface defined lies in the (PL, QL, QG) space. Upon cutting the surface with the constant power factor planes, we get the PL–QG curves which are similar to PV curves or λV curves. The existence of an approximate quadratic relation between λ and V is extended to find a quadratic relation between λ and QG. This approximate quadratic relation is modeled and given in Eq. (23)

$$
\lambda = \alpha\_{\rm i} \mathbf{Q}\_{\rm Gi}^2 + \beta\_{\rm i} \mathbf{Q}\_{\rm Ci} + \gamma\_{\rm i} \tag{23}
$$

In Eq. (16), αi, β<sup>i</sup> and γ<sup>i</sup> are parameters that are different for each generator bus and need to be determined. The parameters can be determined using three sets of PMU readings. Then, using the computed values of the parameters, an estimate for the extreme of QGi is arrived using the condition that at the extreme limit dλ/dQGi = 0 holds. Enforcing this condition gives us the following value as the estimate shown in Eq. (24)

$$\mathbf{Q}\_{\rm Ci}^{\rm ex} = -\frac{\beta\_{\rm i}}{2\alpha\_{\rm i}}\tag{24}$$

The index i stands for all the generator busses of the system. At each load step, it is ensured that the estimated value of generator reactive power—Q Giex—remains within the bounds of the reactive power limits that are specified for a generator bus.

The algorithm for proceeding with the described method in order to compute the VSM is as follows:


Step 6. Compute the impedance matrix as shown in [4].

Step 7. Compute the Thevenin equivalent impedance.

Step 8. Estimate the value of λmaxi by using the cubic spline extrapolation technique for each load bus of the system.

Step 9. Find the value of λsys which is the minimum of all values computed in step 8.

Step 10. Determine the VSM.

For 'm' PMU-measured data points obtained, there will be 'm-1' intervals present, each of

<sup>þ</sup> cj <sup>x</sup>–xj

The inputs for extrapolation are taken as three sets of |ZLi| and λ and the point of |Zli| where λ is to be extrapolated and found. The λ thus found corresponds to the point where Zli equals Zth<sup>i</sup> and hence is equal to λmaxi. Each bus in the considered system will have its own corresponding λmaxs. The λmax = λsys for the entire system is found by taking the minimum of all λmaxi obtained for each of the individual load busses. This value of λsys corresponds to the value of λmaxi for the weakest load bus in the system. The proximity of the load bus to the

VSM <sup>¼</sup> <sup>λ</sup>sys � <sup>λ</sup><sup>0</sup>

2

where PL, load real power; QL, load reactive power; QG, generator reactive power.

<sup>λ</sup> <sup>¼</sup> <sup>α</sup>iQ<sup>2</sup>

Qex Gi ¼ � <sup>β</sup><sup>i</sup> 2α<sup>i</sup>

<sup>X</sup> QG <sup>þ</sup> j j VG

The surface defined lies in the (PL, QL, QG) space. Upon cutting the surface with the constant power factor planes, we get the PL–QG curves which are similar to PV curves or λV curves. The existence of an approximate quadratic relation between λ and V is extended to find a quadratic relation between λ and QG. This approximate quadratic relation is modeled and given in

Gi þ β<sup>i</sup>

In Eq. (16), αi, β<sup>i</sup> and γ<sup>i</sup> are parameters that are different for each generator bus and need to be determined. The parameters can be determined using three sets of PMU readings. Then, using the computed values of the parameters, an estimate for the extreme of QGi is arrived using the condition that at the extreme limit dλ/dQGi = 0 holds. Enforcing this condition gives us the

The index i stands for all the generator busses of the system. At each load step, it is ensured that the estimated value of generator reactive power—Q Giex—remains within the bounds of

λ0

2 <sup>X</sup> QL <sup>þ</sup> <sup>P</sup><sup>2</sup>

After the computation of VSM, the author of [4] proceeds to find an index which helps in determining whether the load increase results in the violation of the reactive power limits at the generator busses. The author begins with the simple power equation and arrives at the

<sup>2</sup> <sup>þ</sup> dj <sup>x</sup>–xj

<sup>3</sup> (20)

� 100 (21)

<sup>L</sup> ¼ 0 (22)

QGi þ γ<sup>i</sup> (23)

(24)

fj ð Þ¼ x aj þ bj x � xj

which is represented by Eq. (20)

142 Topics in Splines and Applications

voltage collapse point is given in Eq. (21)

equation of a surface given by Eq. (22)

Eq. (23)

Q2 <sup>G</sup> � j j VG

following value as the estimate shown in Eq. (24)

the reactive power limits that are specified for a generator bus.

Inference: The author of [4] has performed test runs of the proposed algorithm on the following test system—IEEE 30 bus system, IEEE 118 bus system and IEEE 300 bus system. Further, the obtained results have been compared with two other previously available methods. The obtained results indicate the superior performance of the proposed cubic spline technique when compared to the other methods. Upon comparison of the percentage error, it may be observed from [4] that the cubic spline method is almost 10 times more accurate than the other methods.
