2. Determination of available transfer capability

The first application of cubic splines is in finding the available transfer capability, which is an important parameter in power system operation. Ever since the advent of the deregulated power system, the computation of transfer capability has been a priority. Two quantities that require special attention in these computations are the total transfer capability (TTC) and the available transfer capability (ATC). The TTC of a power system is defined as the maximum amount of power that can be transferred over the interconnected transmission network in a reliable manner while meeting all of a specific set of defined pre- and post-contingency system conditions [1]. ATC is defined as the measure of the additional amount of power that may flow over and above the base case flows without jeopardizing power system security [1]. The system operators of the deregulated power system normally can obtain previously calculated ATC values through an open-access same-time information system. However, the need for a quick calculation of ATC poses a challenge. Some of the methods employed include DC power flow, AC power flow, optimal power flow and sensitivity. Most of the methods mentioned earlier are not sufficiently agile as far as computation speed is concerned or trades accuracy for speed. For example, DC power flow yields quick results at the expense of accuracy, whereas AC power flow compromises speed for accuracy. A solution to the abovementioned problem is found by employing curve-fitting techniques especially the cubic spline interpolation technique.

In order to compute ATC, the cubic spline is employed to trace the curve of the variation of voltage magnitude and power flow with respect to the real power transfer. The ATC is then determined by the point where the limits of voltage magnitude or power flow intersect the curves. The computation of ATC takes place in two different forms—the point-to-point ATC and area-to-area ATC. The area-to-area ATC refers to the additional power that can be mobilized from the seller area to the buyer area, whereas the point-to-point ATC refers to bus-tobus transfer (usually from a generating bus to a load bus). Another factor considered of paramount importance for the computation of ATC is the effect of contingencies like line outages. Individual consideration of line outages for a large-scale power system is an impractical approach, and hence contingency analysis is carried out using contingency ranking which helps select the critical lines.

The mathematical definition of ATC is shown in Eq. (1):

All the analysis and research aimed at providing solutions or mitigating the above-mentioned problems require vast amounts of data which is discrete in nature or the analysis techniques involve responses that are discrete in nature. Regardless of the origin of this non-continuity of data, obtaining a continuous response is imperative because of the desired accuracy and the

The challenges set the stage for interpolation techniques to play an active role in mitigating the problems prevailing in the current power system studies like load forecasting, power system reactive power planning, transmission network expansion, available transfer capability (ATC) determination and market power/clearing price forecasting. Among the various interpolation techniques available, the cubic spline method has been found to be a popular method. Cubic spline also has the desirable characteristics of continuous derivatives at data points which make the design of controllers around these regions possible, and its employment has been

The measurements we take for analysis in everyday life are wrought with noise that may be caused by the surrounding environment. Especially in electrical measurements, such random noises may be caused by the magnetic field produced by the current, the presence of stray charges, heating caused by the flow of eddy currents, and so on. One of the possible solutions to the problems caused by noise is to take a large number of measurements. This ensures that the random noise gets canceled out on an average, and hence the integrity of the data is maintained. Other complicated methods of tackling the errors caused by noise are available, but the detailed and in-depth analysis of these methods falls beyond the scope of this chapter.

The first application of cubic splines is in finding the available transfer capability, which is an important parameter in power system operation. Ever since the advent of the deregulated power system, the computation of transfer capability has been a priority. Two quantities that require special attention in these computations are the total transfer capability (TTC) and the available transfer capability (ATC). The TTC of a power system is defined as the maximum amount of power that can be transferred over the interconnected transmission network in a reliable manner while meeting all of a specific set of defined pre- and post-contingency system conditions [1]. ATC is defined as the measure of the additional amount of power that may flow over and above the base case flows without jeopardizing power system security [1]. The system operators of the deregulated power system normally can obtain previously calculated ATC values through an open-access same-time information system. However, the need for a quick calculation of ATC poses a challenge. Some of the methods employed include DC power flow, AC power flow, optimal power flow and sensitivity. Most of the methods mentioned earlier are not sufficiently agile as far as computation speed is concerned or trades accuracy for speed. For example, DC power flow yields quick results at the expense of accuracy, whereas AC power flow compromises speed for accuracy. A solution to the abovementioned problem is found by employing

continuity of real-time operation.

130 Topics in Splines and Applications

seconded by the high accuracy obtained.

2. Determination of available transfer capability

curve-fitting techniques especially the cubic spline interpolation technique.

$$\text{ATE} = \text{TTC} - \text{TRM} - \text{ETC} - \text{CBM} \tag{1}$$

The definition of the abovementioned terms is as follows:

Transmission reliability margin (TRM): the amount of transmission capability necessary to ensure the security of the interconnected system under a reasonable range of uncertainties in the system.

Capacity benefit margin (CBM): the amount of transmission capability reserved by loadserving entities to ensure access to generation from interconnected systems to meet generation reliability requirements.

Existing transmission commitments (ETC): normal transmission flows.

Traditionally, ATC computation involves the recursive application of AC power flow with increasing power transfers, thereby tracing voltage magnitudes and MVA power flow variations with respect to real power formally called the P-V and the P-S curves as shown in Figure 1. These curves are then employed in conjunction with the limits imposed by acceptable voltage magnitudes and power flows to calculate the ATC. In the cubic spline-based approach, the cubic spline interpolation technique is used to trace the P-V and P-S curves which are then

Figure 1. Variation of voltage with real power transfer.

employed to calculate the ATC. This is done by first determining four known points on the curve using AC power flow and then using cubic spline interpolation to trace the curves in between those points. The four known points are denoted as Vi (Pn) and Sij (Pn), where Vi is for voltage of bus 'I', Sij is the MVA power flow between bus 'i' and 'j', Pn is the real power transfer and n is the index for the four points and has values of n = 1, 2, 3 and 4. The incremental steps for tracing the P-V or the P-S curve will be of 1 MW each.

noting the change in voltage and MVA transfer. The fourth known point P4 is selected as the minimum of all three values computed above. Then, based on this, P2 and P3 are chosen using

P2 <sup>¼</sup> P4

Step 6. Now, find the MVA and bus voltage Vi for each of the newly computed real power

Step 7. Now, use the cubic spline to trace the curve in between these points and obtain the P-V

Now, the area-to-area and point-to-point ATCs are calculated by obtaining the point where the

Inference: The author of [1] has employed the proposed cubic spline-based method to compute the ATC of the Malaysian power system. The performance of the method is measured in terms of its accuracy and speed (which are the main reasons for the employment of the method). For testing the Malaysian power system, it has been simplified into a 241-bus system and further classified into three regions namely north, east, central, south and PUB. The lower and upper voltage limits are taken as 0.9 and 1.1 p.u., respectively. The results obtained have

It is observed from the results obtained in [1] that all the ATC values obtained are due to the MVA limitation. It is further observed that the performance of the cubic spline interpolation method is comparable to the AC power flow method in terms of accuracy. The observation of paramount importance is that while the proposed method obtains the high accuracy as found in the recursive AC power flow method, the time it requires to compute the ATC is much smaller than the AC recursive power flow method—to the extent that in some cases, the time required is up to 30 times lesser. Therefore, it may be concluded that the cubic spline method is superior to

In this section, cubic spline is applied in modeling a very complex load—electric arc furnace (EAF). This example presents the true power of cubic splines in tracking extremely complex trajectories. The electric arc furnace's (EAF) ability to efficiently smelt scrap iron raw materials has made it the backbone of the steel-making industries. The EAF employs high temperatures produced by low-voltage and high-current electric arcs to smelt scrap iron raw materials. The increase in productivity requirements has led to EAFs being designed for high-power applications. The operation of EAFs introduces a significant amount of harmonics, inter-harmonics and flickers in the supply system. Therefore, it is mandatory that the operators pay much attention to the power-quality considerations. In order to study these problems, it is of paramount

AC power flow in terms of speed and superior to DC power flow in terms of accuracy.

voltage limit or the MVA limit line intersects the P-V and P-S curves, respectively.

been compared with the traditional recursive AC power flow results.

<sup>3</sup> (5)

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133

P3 ¼ P2 � 2 (6)

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

the formulae given in Eqs. (5) and (6)

3. Electric arc furnace modeling

values—P2, P3 and P4.

and the P-S curves.

The first and foremost step in the process of curve tracing consists of the determination of the four points on the P-V and P-S curves. This is of paramount importance because the power system undergoes voltage collapse once the real power transfer crosses a certain limit. In the voltage collapse process, the particular bus faces a continuous drop in bus voltage once the critical point load is exceeded. Therefore, the points must be selected carefully making sure that the critical point is not exceeded.

## 2.1. Procedure to calculate ATC


$$\text{PT}\_{\text{i,V}\_{\text{L}}} = \left| \frac{\partial \lambda}{\partial \mathbf{V}\_{\text{i}}} \times \left( \mathbf{V}\_{\text{L}} - \mathbf{V}\_{\text{i}}^{0} \right) \right| \tag{2}$$

$$\text{PT}\_{\text{i}, \text{V}\_{\text{U}}} = \left| \frac{\partial \lambda}{\partial \text{V}\_{\text{i}}} \times \left( \text{V}\_{\text{U}} - \text{V}\_{\text{i}}^{0} \right) \right| \tag{3}$$

$$\text{PT}\_{\text{i\"}\text{S}} = \left| \frac{\partial \lambda}{\partial \mathbf{S}\_{\text{i\"}}} \times \left( \mathbf{S}\_{\text{i\"}\text{j}}^{\text{limit}} - \mathbf{S}\_{\text{i\"}\text{j}}^{0} \right) \right| \tag{4}$$

In Eqs. (2)–(4), PTi,VL, PTi,VU and PTij,S are the estimates of P4 which are arrived at by using the linear estimates based on the lower voltage limit, the upper voltage limit and the thermal limit of each line, respectively. The lower limit of voltage VL is taken as 0.9 p.u. and the upper limit Vu is taken as 1.1 p.u. Slimit ij is the thermal limit of each line. Vi <sup>0</sup> and S0 ij are the base case values computed in step (ii). The derivative terms are the reciprocal of the rate of change of voltage and MVA power flow with real power. This can be obtained by solving the AC power flow starting from the base case and making a small change in the transferred real power and noting the change in voltage and MVA transfer. The fourth known point P4 is selected as the minimum of all three values computed above. Then, based on this, P2 and P3 are chosen using the formulae given in Eqs. (5) and (6)

$$\mathbf{P\_2} = \frac{\mathbf{P\_4}}{3} \tag{5}$$

$$\mathbf{P}\_3 = \mathbf{P}\_2 \times \mathbf{2} \tag{6}$$

Step 6. Now, find the MVA and bus voltage Vi for each of the newly computed real power values—P2, P3 and P4.

Step 7. Now, use the cubic spline to trace the curve in between these points and obtain the P-V and the P-S curves.

Now, the area-to-area and point-to-point ATCs are calculated by obtaining the point where the voltage limit or the MVA limit line intersects the P-V and P-S curves, respectively.

Inference: The author of [1] has employed the proposed cubic spline-based method to compute the ATC of the Malaysian power system. The performance of the method is measured in terms of its accuracy and speed (which are the main reasons for the employment of the method). For testing the Malaysian power system, it has been simplified into a 241-bus system and further classified into three regions namely north, east, central, south and PUB. The lower and upper voltage limits are taken as 0.9 and 1.1 p.u., respectively. The results obtained have been compared with the traditional recursive AC power flow results.

It is observed from the results obtained in [1] that all the ATC values obtained are due to the MVA limitation. It is further observed that the performance of the cubic spline interpolation method is comparable to the AC power flow method in terms of accuracy. The observation of paramount importance is that while the proposed method obtains the high accuracy as found in the recursive AC power flow method, the time it requires to compute the ATC is much smaller than the AC recursive power flow method—to the extent that in some cases, the time required is up to 30 times lesser. Therefore, it may be concluded that the cubic spline method is superior to AC power flow in terms of speed and superior to DC power flow in terms of accuracy.

#### 3. Electric arc furnace modeling

employed to calculate the ATC. This is done by first determining four known points on the curve using AC power flow and then using cubic spline interpolation to trace the curves in between those points. The four known points are denoted as Vi (Pn) and Sij (Pn), where Vi is for voltage of bus 'I', Sij is the MVA power flow between bus 'i' and 'j', Pn is the real power transfer and n is the index for the four points and has values of n = 1, 2, 3 and 4. The incremental steps

The first and foremost step in the process of curve tracing consists of the determination of the four points on the P-V and P-S curves. This is of paramount importance because the power system undergoes voltage collapse once the real power transfer crosses a certain limit. In the voltage collapse process, the particular bus faces a continuous drop in bus voltage once the critical point load is exceeded. Therefore, the points must be selected carefully making sure

Step 1. Perform the line contingency ranking using line-loading performance index (PIMW) and bus voltage performance (PIv). This is done to identify the critical lines. The critical lines will

Step 2. Find a base case by solving the AC load flow. This will act as the first point.

the limiting point of system constraints which is given in Eqs. (2)–(4)

PTi,VL <sup>¼</sup> <sup>∂</sup><sup>λ</sup>

PTi,VU <sup>¼</sup> <sup>∂</sup><sup>λ</sup>

PTij,<sup>S</sup> <sup>¼</sup> <sup>∂</sup><sup>λ</sup>

Step 3. Perform the simulation for line outage for one of the critical lines found in step 1.

Step 4. Specify the point or area of transfers. For the point-to-point transfer, generally a generator bus is taken as the selling bus and a load bus as the buying bus. On the other hand, for area-toarea transfer, all the generator busses in a particular area called the selling are and all the load

Step 5. Now, the next three points are determined. The first step is to determine the fourth and final point P4. In order to do so, the sensitivity method is employed. This method is based on

∂Vi

∂Vi

 

 

∂Sij

ij is the thermal limit of each line. Vi

  � VL � <sup>V</sup><sup>0</sup>

� VU � V0

ij � <sup>S</sup><sup>0</sup> ij

� Slimit

In Eqs. (2)–(4), PTi,VL, PTi,VU and PTij,S are the estimates of P4 which are arrived at by using the linear estimates based on the lower voltage limit, the upper voltage limit and the thermal limit of each line, respectively. The lower limit of voltage VL is taken as 0.9 p.u. and the upper limit

computed in step (ii). The derivative terms are the reciprocal of the rate of change of voltage and MVA power flow with real power. This can be obtained by solving the AC power flow starting from the base case and making a small change in the transferred real power and

i

i

 

> 

> >

> > > <sup>0</sup> and S0

(2)

(3)

(4)

ij are the base case values

for tracing the P-V or the P-S curve will be of 1 MW each.

that the critical point is not exceeded.

have a PI value greater than PIbase case.

busses in a specific buying area are considered.

2.1. Procedure to calculate ATC

132 Topics in Splines and Applications

Vu is taken as 1.1 p.u. Slimit

In this section, cubic spline is applied in modeling a very complex load—electric arc furnace (EAF). This example presents the true power of cubic splines in tracking extremely complex trajectories. The electric arc furnace's (EAF) ability to efficiently smelt scrap iron raw materials has made it the backbone of the steel-making industries. The EAF employs high temperatures produced by low-voltage and high-current electric arcs to smelt scrap iron raw materials. The increase in productivity requirements has led to EAFs being designed for high-power applications. The operation of EAFs introduces a significant amount of harmonics, inter-harmonics and flickers in the supply system. Therefore, it is mandatory that the operators pay much attention to the power-quality considerations. In order to study these problems, it is of paramount importance to understand the nonlinear load characteristics that the EAF present to the power system. Also, in order to mitigate the power-quality issues and to further study the impact of EAFs on the power system, it is required that the EAF be modeled after obtaining its time response.

Natural boundary condition: G"

Clamped boundary condition: G'

Step 1. Set a<sup>i</sup> = yi for i = 0, 1, 2, …, n.

where h<sup>i</sup> ¼ xiþ<sup>1</sup> � xi, for i = 1, 2, …, n-1.

x1 and xn-1.

Step 2. Solve

and

and

Step 3. Set

Step 4. Set

Step 5. Set

0(x0)=G"

0(x0)=D0 and G'

where D0 and Dn are the values of the first derivatives of the unknown functions.

n-1(xn) = 0.

It is generally found that the natural boundary conditions give less accurate results than the clamped boundary conditions. Alternately, one could possibly apply a boundary condition called not-a-knot condition, which in addition to the natural boundary condition also incorporates another condition that the third derivative of the function must be continuous at

In order to find the coefficients—ai, bi, ci and di—we follow the following steps and equations:

n-1(xn)=Dn.

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

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

ð8Þ

135

ð9Þ

ð10Þ

ð11Þ

ð12Þ

ð13Þ

ð14Þ

The entire operation of the EAF involves three stages namely striking, smelting and refining. In the striking process, the electric arc is built up by lowering the electrodes of all three phases, the melting process involves the melting of the material and the process ends with the stable refining. Due to the complexities involved in modeling the EAF operation in the striking and the melting processes, most of the research has been directed towards a steady-state modeling of the EAF in the refining stage of operation. The modeling of an EAF requires several parameters such as arc voltage, arc length and arc current, which are determined by the position of electrodes. Therefore, in order to accurately model the EAF, we need to know the field measurements of the electric response which involves the variation of voltage and current. The measured responses are then employed to develop an EAF conductance model using the cubic spline interpolation method.

The EAF is modeled as a function of nonlinear conductance using the cubic spline interpolation technique which is called the cubic spline interpolation model (CSIM). In this method, a set of cubic polynomials are obtained which helps understand the voltage–current characteristic of the EAF.

The steps involved are as follows:

First, a set of 'n' measured data points of conductance is obtained for one fundamental cycle of operation. The measurements lying in the interval [a, b] such that a=x0 < x1 < … < xn = b. For the interval between two adjacent points, a cubic function is defined as shown in Eq. (7).

$$\mathbf{G}\_{\mathbf{i}}\ (\mathbf{x}) = \mathbf{a}\_{\mathbf{i}} + \mathbf{b}\_{\mathbf{i}}\ (\mathbf{x} - \mathbf{x}\_{\mathbf{i}}) + \mathbf{c}\_{\mathbf{i}}\ (\mathbf{x} - \mathbf{x}\_{\mathbf{i}})^2 + \mathbf{d}\_{\mathbf{i}}\ (\mathbf{x} - \mathbf{x}\_{\mathbf{i}})^3\tag{7}$$

where i = 0, 1, 2,…., n-1. The coefficients ai, bi, ci and di are unknown. These coefficients need to be determined based upon the following constraints:

Step 1. Each spline must pass through the given data points yi.

$$\mathbf{G\_i(x\_i) = y\_i} \text{ and } \mathbf{G\_{n-1}(x\_n) = y\_n}$$

Step 2. Interior data points between each spline must be continuous.

$$\mathbf{G}\_{\mathbf{i}\_{-}+1}(\mathbf{x}\_{i\_{-}+1}) = \mathbf{G}\_{\mathbf{i}}(\mathbf{x}\_{i\_{-}+1})$$

Step 3. The first and the second derivatives of the splines must be continuous across the interior data points. Therefore, the spline forms a smooth function.

$$\mathbf{G}'\_{\mathbf{i}+1}(\mathbf{x}\_{\mathbf{i}+1}) = \mathbf{G}'\_{\mathbf{i}}(\mathbf{x}\_{\mathbf{i}+1}).$$

$$\mathbf{G}''\_{\mathbf{i}+1}(\mathbf{x}\_{\mathbf{i}+1}) = \mathbf{G}''\_{\mathbf{i}}(\mathbf{x}\_{\mathbf{i}+1})$$

Step 4. In addition to the conditions mentioned in steps 1–3, another boundary condition must be satisfied which concerns the derivative of the functions at the boundaries (at x0 = a and xn = b). There are two types of boundary conditions that may be required to satisfy:

Natural boundary condition: G" 0(x0)=G" n-1(xn) = 0.

Clamped boundary condition: G' 0(x0)=D0 and G' n-1(xn)=Dn.

where D0 and Dn are the values of the first derivatives of the unknown functions.

It is generally found that the natural boundary conditions give less accurate results than the clamped boundary conditions. Alternately, one could possibly apply a boundary condition called not-a-knot condition, which in addition to the natural boundary condition also incorporates another condition that the third derivative of the function must be continuous at x1 and xn-1.

In order to find the coefficients—ai, bi, ci and di—we follow the following steps and equations: Step 1. Set a<sup>i</sup> = yi for i = 0, 1, 2, …, n.

Step 2. Solve

$$\begin{aligned} \mathbf{(h\_0 + 2h\_1)c\_1 + (h\_1 - h\_0)c\_2} \\\\ = \frac{3}{h\_0 + h\_1} \left[ (a\_2 - a\_1) - \frac{h\_1}{h\_0} (a\_1 - a\_0) \right], \qquad \text{for } 1 = 1 \end{aligned} \tag{8}$$

and

importance to understand the nonlinear load characteristics that the EAF present to the power system. Also, in order to mitigate the power-quality issues and to further study the impact of EAFs on the power system, it is required that the EAF be modeled after obtaining its time

The entire operation of the EAF involves three stages namely striking, smelting and refining. In the striking process, the electric arc is built up by lowering the electrodes of all three phases, the melting process involves the melting of the material and the process ends with the stable refining. Due to the complexities involved in modeling the EAF operation in the striking and the melting processes, most of the research has been directed towards a steady-state modeling of the EAF in the refining stage of operation. The modeling of an EAF requires several parameters such as arc voltage, arc length and arc current, which are determined by the position of electrodes. Therefore, in order to accurately model the EAF, we need to know the field measurements of the electric response which involves the variation of voltage and current. The measured responses are then employed to develop an EAF conductance model using

The EAF is modeled as a function of nonlinear conductance using the cubic spline interpolation technique which is called the cubic spline interpolation model (CSIM). In this method, a set of cubic polynomials are obtained which helps understand the voltage–current characteris-

First, a set of 'n' measured data points of conductance is obtained for one fundamental cycle of operation. The measurements lying in the interval [a, b] such that a=x0 < x1 < … < xn = b. For the interval between two adjacent points, a cubic function is defined as shown in Eq. (7).

where i = 0, 1, 2,…., n-1. The coefficients ai, bi, ci and di are unknown. These coefficients need to

Step 3. The first and the second derivatives of the splines must be continuous across the interior

Step 4. In addition to the conditions mentioned in steps 1–3, another boundary condition must be satisfied which concerns the derivative of the functions at the boundaries (at x0 = a and

xn = b). There are two types of boundary conditions that may be required to satisfy:

<sup>2</sup> <sup>þ</sup> di ð Þ <sup>x</sup>–xi

<sup>3</sup> (7)

Gi ð Þ¼ x ai þ bi ð Þþ x–xi ci ð Þ x–xi

response.

134 Topics in Splines and Applications

tic of the EAF.

the cubic spline interpolation method.

The steps involved are as follows:

be determined based upon the following constraints:

i(xi+1)

Gi(xi)=yi and Gn-1(xn)=yn

Gi+1(xi+1)=Gi(xi+1)

G'i+1(xi+1)=G'<sup>i</sup> (xi+1).

i+1(xi+1)=G"

G"

Step 1. Each spline must pass through the given data points yi.

Step 2. Interior data points between each spline must be continuous.

data points. Therefore, the spline forms a smooth function.

$$\begin{aligned} \mathbf{h}\_{l+1}\mathbf{c}\_{l+1} + 2\mathbf{c}\_{l}(\mathbf{h}\_{l+1} + \mathbf{h}\_{1}) + \mathbf{h}\_{l}\mathbf{c}\_{l+1} \\\\ \mathbf{h} = \frac{3}{\mathbf{h}\_{l}}(a\_{l+1} - a\_{l}) - \frac{3}{h\_{l-1}}(a\_{l} - a\_{l-1}), \text{ for } \mathbf{i} = \mathbf{2}, \dots, \mathbf{n} - \mathbf{2} \end{aligned} \tag{9}$$

and

$$\mathbf{y}\_{\text{(n-2)}} = \mathbf{y}\_{\text{(n-2)}} \begin{bmatrix} h\_{n-2} \mathbf{y}\_{n-1} \mathbf{y}\_{n-2} & \mathbf{y}\_{n-1} \mathbf{y}\_{n-1} \mathbf{y}\_{n-1} \end{bmatrix} \tag{10}$$
 
$$= \frac{3}{h\_{n-1} + h\_{n-2}} \begin{bmatrix} h\_{n-2} \left( a\_n - a\_{n-1} \right) - (a\_{n-1} - a\_{n-2}) \end{bmatrix}. \tag{10}$$
 
$$\text{where:} \mathbf{h}\_{\ell} = \mathbf{x}\_{\ell+1} - \mathbf{x}\_{\ell}, \text{ for } \mathbf{i} = 1, 2, \dots, n-1$$

where h<sup>i</sup> ¼ xiþ<sup>1</sup> � xi, for i = 1, 2, …, n-1.

Step 3. Set

$$\mathbf{c}\_0 = c\_1 + \frac{h\_0}{h\_1}(c\_1 - c\_2),\tag{11}$$

$$\mathbf{c}\_{n} = \mathbf{c}\_{n-1} + \frac{h\_{n-1}}{h\_{n-2}} (\mathbf{c}\_{n-1} - \mathbf{c}\_{n-2}),\tag{12}$$

Step 4. Set

$$\mathbf{b}\_{l} = \frac{1}{\mathbf{h}\_{l}} (a\_{l+1} - a\_{l}) - \frac{h\_{l}}{3} (c\_{l+1} + 2c\_{l}), \text{ for } \mathbf{i} = \mathbf{0}, \ 1, \ 2, \ldots, \mathbf{n} \cdot \mathbf{l} \tag{13}$$

Step 5. Set

$$\mathbf{d}\_{l} = \frac{1}{3\mathbf{h}\_{l}} (c\_{l+1} - c\_{l}), \text{ for } \mathbf{i} = \mathbf{0}, \text{ 1, 2, ..., n-l} \tag{14}$$

The coefficients a<sup>i</sup> can be found in step 1. The coefficients ci can be calculated by solving the linear equations developed in Eqs. (8)–(10). Then, finally coefficients bi and di may be obtained from steps 4 and 5 (Eqs. (13) and (14), respectively).

Inference: The author in [2] has analyzed the performance of CSIM on an actual power system model developed in MATLAB/SIMULINK whose online diagram is also presented in [2]. The results obtained via employing the CSIM (shown in Figure 4) are then compared with the results obtained via employing two traditional methods namely (i) harmonic current injection model (HCIM), shown in Figure 2, and (ii) harmonic voltage source model (HVSM), shown in Figure 3. The results described in [2] have been obtained for three different parts—early,

middle and later stages—of EAF operation during the refining period. This is done primarily because EAF does not possess a steady-state behaviour that lasts long. Since the HCIM and HVSM are unable to model dynamic behaviour, these methods are compared with the CSIM only during the latter part of the operation because of the dynamic behaviour of the EAF in the

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137

Upon employing the CSIM for EAF modeling in the early and middle stages, it is found that the performance of the EAF during these stages is highly nonlinear. The V-I curve presents a multivalued function which makes it difficult to model the EAF. The results shown in [2] suggest that upon the comparison, the HCIM method yields the largest errors in terms of both the EAF voltage and current determined. Then, HVSM, which albeit an improvement on HCIM, still contains errors. The proposed CSIM method performs better than HCIM and HVSM and provides a better fit for the voltage and current characteristics. It is also explained in the results that the error encountered during modeling using CSIM in the early and middle stages is larger when compared to that encountered when modeling the later stages of the refining period because the number of sampling points used for the EAF model is not sufficient for modeling in the early and middle stages of refining. As an extension, it is also proposed in [2] that the cubic spline interpo-

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

lation technique can also be used for modeling other nonlinear loads.

4. Var compensator with thyristor-controlled reactor

early and middle parts (Figure 4).

Figure 4. CSIM V-I characteristic (later stage) [2].

Figure 2. HVSM V-I characteristic [2].

Figure 3. HCIM V-I characteristic [2].

Figure 4. CSIM V-I characteristic (later stage) [2].

The coefficients a<sup>i</sup> can be found in step 1. The coefficients ci can be calculated by solving the linear equations developed in Eqs. (8)–(10). Then, finally coefficients bi and di may be obtained

Inference: The author in [2] has analyzed the performance of CSIM on an actual power system model developed in MATLAB/SIMULINK whose online diagram is also presented in [2]. The results obtained via employing the CSIM (shown in Figure 4) are then compared with the results obtained via employing two traditional methods namely (i) harmonic current injection model (HCIM), shown in Figure 2, and (ii) harmonic voltage source model (HVSM), shown in Figure 3. The results described in [2] have been obtained for three different parts—early,

from steps 4 and 5 (Eqs. (13) and (14), respectively).

136 Topics in Splines and Applications

Figure 2. HVSM V-I characteristic [2].

Figure 3. HCIM V-I characteristic [2].

middle and later stages—of EAF operation during the refining period. This is done primarily because EAF does not possess a steady-state behaviour that lasts long. Since the HCIM and HVSM are unable to model dynamic behaviour, these methods are compared with the CSIM only during the latter part of the operation because of the dynamic behaviour of the EAF in the early and middle parts (Figure 4).

Upon employing the CSIM for EAF modeling in the early and middle stages, it is found that the performance of the EAF during these stages is highly nonlinear. The V-I curve presents a multivalued function which makes it difficult to model the EAF. The results shown in [2] suggest that upon the comparison, the HCIM method yields the largest errors in terms of both the EAF voltage and current determined. Then, HVSM, which albeit an improvement on HCIM, still contains errors. The proposed CSIM method performs better than HCIM and HVSM and provides a better fit for the voltage and current characteristics. It is also explained in the results that the error encountered during modeling using CSIM in the early and middle stages is larger when compared to that encountered when modeling the later stages of the refining period because the number of sampling points used for the EAF model is not sufficient for modeling in the early and middle stages of refining. As an extension, it is also proposed in [2] that the cubic spline interpolation technique can also be used for modeling other nonlinear loads.
