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A new approach for voltage harmonic distortion minimization
Electric Power Systems Research 70 (2004) 253–260
A new approach for voltage harmonic distortion minimization A.F. Zobaa Electrical Power & Machines Department, Faculty of Engineering, Cairo University, Giza, Egypt Received 29 July 2003; received in revised form 4 November 2003; accepted 9 December 2003
Abstract The problem of harmonic reduction is alternatively approached as an energy user’s problem. The user’s objective is to eliminate his voltage harmonic distortion problem locally without considering the effects of voltage distortion at neighboring buses. The remedy is insertion of a reactor in series with the local compensating capacitor. A method is presented for finding the optimum fixed LC combination to minimize voltage harmonic distortion at a load bus while holding the power factor at a desired value and constraining the compensator values which would create resonant conditions and the manufacturer’s standard values for power shunt capacitors. © 2004 Elsevier B.V. All rights reserved. Keywords: Harmonics; Power factor; Reactive power optimization
1. Introduction The application of capacitors in electric power systems is intended for the control of power flow, improvement of stability, voltage profile management, power factor (PF) correction, and loss minimization. Many publication on the subject of PF correction in sinusoidal systems [1] advocate the need for unity PF. A comprehensive search over the last 30 years of publications [2,3] proves that a lot of successful research and development of compensators has taken place. The assumption made and brief descriptions of the solution methods are presented. The different types of reactive compensation strategies on the power system losses and voltage distortion are: A. Compensation of the 60 Hz reactive power. In this case the compensation current is sinusoidal. The line current however will remain nonsinusoidal. This method has the advantage that the compensator does not generate additional harmonics like the Thyristor-Controlled Reactors (TCRs) do, neither may cause resonances as the shunt capacitances sometimes will do. B. Provide a sinusoidal line current in phase with the fundamental voltage. Hence, if a sufficient number of dominant nonlinear loads in a power network will be equipped with such compensators the harmonic power flow can be reduced to inconsequential levels. 0378-7796/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2003.12.014
C. Produce unity PF at the load bus. This condition requires the line current and the bus voltage to have identical waveforms. D. Minimization of power loss in the power network. The compensator acts like a filter; it stops the harmonic current to flow in the network and “sink” a part of the harmonic currents to reduce the power loss. E. Minimization of voltage distortion at the user’s end or at a large customer bus. The goal of Ref. [4] is to evaluate the effect of PF correction, implemented by different types of reactive power compensators and compensation strategies, on the power system losses and voltage distortion. Neither the case when the line current is sinusoidal, nor the case when PF is unity, provides minimum power system loss or minimum voltage distortion. Tuned LC compensators proved to be an effective means of correcting PF, yielding reduction in the power system losses and lowering the voltage distortion to reasonable levels [5,6]. In such an arrangement, the LC compensator may actually have a lower volt–ampere rating (or cost) than that of a pure capacitive compensator [7]. In other attempts at optimizing the LC compensators [8,9], the main objective has been to maximize the load PF with minimum transmission loss (TL). This may also reduce the total harmonic distortion of voltage and current, but it may not minimize them. In other words, maximizing PF does not solve the problem of minimizing voltage harmonic distortion. In Ref. [8], both the equivalent source and load are
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considered to generate harmonics. The different criteria for the design of the LC compensator, (a) maximizing PF (b) minimizing TL (c) maximizing transmission efficiency (η), are discussed taking into consideration the non-linearity of the load by using Direct Polytope Search method. In Ref. [9], the manufacturer’s standard values for power shunt capacitors are taken into consideration. These values are considered as constraints in the sense that the solution for the capacitor should be one of the standard values. The different criteria for the design of the LC compensator are discussed by using Golden Section Search method. The reason for doing this is to compare the values obtained in Ref. [4] with real practical values in the market [10]. Ref. [10] shows the voltage and reactive power ratings of shunt capacitors. The inductive reactive values are almost continuous and there is little limitation on the manufacturer’s values. In this study, both the equivalent source and load are considered to generate harmonics. It is assumed that the load harmonics are not sufficiently serious to suggest tuned filters, but when combined with source harmonics, the use of a pure capacitive compensator would degrade PF and overload the equipment. In this paper, a method is presented for minimizing the voltage total harmonic distortion (VTHD) at the load bus where it is desired to maintain the PF at a desired level. An optimum fixed LC compensator will be selected that will minimize the expected value of VTHD for a specified range of source harmonic and impedance values taking into consideration. The major attribute of the method is that it, unlike conventional approaches, guarantees convergence to the optimal solution. This accomplishment is a direct result from the improvement in: A. The algorithm in which provisions are made to identify and to avoid compensator values which would create resonance conditions. B. The problem formulation in which the effect of the transmission line impedance on the load voltage is included while calculating the optimal compensator value. C. Frequency dependent nature of the supply source. D. The manufacturer’s standard values for power shunt capacitors are taken into consideration. Finally, the contribution of the newly developed method is demonstrated in examples taken from previous publications.
The rms value of the total waveform is not the sum of the individual components, but is the square root of the sum of the squares. THD is a very useful quantity for many applications, but its limitations must be realized. It can provide a good idea of how much extra heat will be realized when a distorted voltage is applied across a resistive load. Likewise, it can give an indication of the addition losses caused by the current flowing through a conductor. However, it is not a good indicator of the voltage stress with a capacitor because that is related to the peak value of the voltage waveform, not its heating value.
3. Basic approach to harmonic reduction Fig. 1 is a single-phase equivalent circuit of a bus with LC compensator, experiencing voltage harmonic distortion at harmonic order K because of a voltage source, V SK , and harmonic current sources within the load itself, ILK . Thevenin voltage source representing the utility supply and the harmonic current source representing the nonlinear load are VS (t) = VSK (t) (1) K
and ILK (t) =
ILK (t)
(2)
K
where K is the order of harmonic present. The Kth harmonic Thevenin source and load impedances are ZTK = RTK + jXTK
(3)
and ZLK = RLK + jXLK
(4)
In Ref. [12], a new model for the distribution system including nonlinear loads is introduced. The model is based on measurements, where current and voltage measurements at two different operating conditions are used to calculate ZLK from the following (1)
ZLK =
(2)
VLK − VLK (1)
(2)
ILK − ILK
(5)
2. Total harmonic distortion There are several measures commonly used for indicating the harmonic content of a waveform with a single number [11]. One of the most common is total harmonic distortion (THD), which can be calculated for either voltage or current (ITHD). THD is a measure of the effective value of the harmonic components of a distorted waveform, that is, the potential heating of the harmonics relative to the fundamental.
Fig. 1. Single-phase equivalent circuit for Kth harmonic with shunt LC compensator.
A.F. Zobaa / Electric Power Systems Research 70 (2004) 253–260
The reader should refer to Ref. [12] for detail discussion on the derivation of (5). To simplify the analysis, only the load model using the respective active and reactive powers at the fundamental frequency is considered while sizing the compensators. This model, Fig. 1, is adequate where VTHD is less than 10 % [11]. The approach will be to minimize voltage harmonic distortion on the load by adjusting XC and XL . The voltage harmonic distortion at the compensated load terminals is defined as 2 K>1 VLK (6) VTHD = VL1 and VLK
VSK (CR) − ILK (DR × ER) = AIK + jAJK
(7)
where AR = R + RLK , BR = (XLK + KXL − XC /K), CR = RCLK + jXCLK , DR = R + j(KXL –XC /K), ER = RTLK + jXTLK , AIK = RTLK + R(RLK + RTK ) − (XLK + XTK )(KXL − XC /K), AJK = XTLK + R(RLK + XTK ) + (RLK + RTK )(KXL − XC /K). RCLK + jXCLK ZCLK = (8) ZLK + ZCK where RCLK = RRLK − XLK (KXL − XC /K), XCLK = RXLK + RLK (KXL − XC /K). RTLK + jXTLK (9) ZTLK = ZTK + ZLK where RTLK = RTK RLK − XTK XLK , XTLK = RTK XLK + XLK RLK . The compensated PF at the load is given as 2 GLK VLK PL = (10) PF = 2 V L IS 2 ISK VLK
255
impedance to the flow of harmonic current, while series resonance is low impedance to the flow of harmonic current [13]. In actual electrical systems utilizing PF correction, both types of resonance or a combination of both may occur if the resonant point happens to be close to one of the frequencies generated by harmonic sources in the system.The expected impedance seen from the Thevenin source is given by Z = RTK + jXTK +
RCLK + jXCLK ZLK + ZCK
(15)
The resonance peaks can be obtained by setting the imaginary part of (15) to zero, resulting in a quadratic equation in XC and XL for any given harmonic order K, XC 2 XC + A2 KXL − A1 KXL − + A3 = 0 K K
(16)
2 + 2X X , where A1 = XTK + XLK , A2 = R2LK + XLK LK TK 2 2 2 A3 = R XLK + XTK [(R + RLK ) + XLK ] and by taking the solution of (16) where the square root of the discriminant is positive. (The other solution corresponds to resonance between the load and the combination of source impedance and compensator.) Note that for sufficiently large load resistance and/or load reactance, (16) reduces to:
KXL −
XC + XTK = 0 K
(17)
which then represents only the series resonance, which represents all possible combinations of XC and XL values which result in resonance between the transmission impedance and compensated load. Under these conditions, the PF will reach a minimum. It is evident that the number of series resonance lines will depend on the number of harmonics presents in the transmission source.
where ISK =
VSK (AR + jBR) + ILK CR AIK + jAJK
5. Frequency dependent nature of the supply source (11)
YLK = GLK − jBLK
(12)
The TL is given as 2 ISK RTK TL =
(13)
K
The network η is given as 2 GLK VLK PL = 2 η= 2 PS ISK RTK + GLK VLK
(14)
4. Harmonic resonance constraint System resonant conditions are the most important factors affecting system harmonic levels. Parallel resonance is high
Skin effect is an alternating current phenomenon where the current in a conductor tends to flow more densely near the outer surface of a conductor that in the center area. Skin effect will be applied in the analysis to account for the impact on the system impedance of the frequency dependence of the resistive components. In most power systems, one can generally assume that the resistance does not change significantly when studying the effects of harmonics less than the ninth [14]. The response of various types of equipment to distorted waveforms is by no means uniform and some components are of more concern than others [15]. Ref. [16] shows a detailed analysis of distribution systems, loads and other system elements. It consists basically of representing the dominant characteristics of the network using alternative configurations and models. Also simpler equivalents for extended networks are suggested.
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6. Proposed solution of the problem
X1 = Xlo + λ
VTHD and PF can be expressed as functions of XC and XL , using (6) and (10). Each value of the reactive power ratings QCi of the particular voltage [10] is used to calculate the corresponding value of XCi . This value is then substituted into the objective function to become one variable equation in XL , which can be solved by using the Penalty Function method. After formulating the objective function and the constraints, the problem addressed in this study becomes:
and
Minimize VTHD(XCi , XL ) subject to : 90% ≤ PF(XCi , XL ) ≤ 100% (18) XCi , XL is not part of solution of Eq. (16) Naturally, the solution will satisfy the upper limit of the PF and efficiency constraints. PF in distribution systems may be allowed within certain limits according to the authority, and similarly for the efficiency so the presented method will generalize the limits of the constraints. This problem can be rewritten in the form Minimize
VTHD(XCi , XL )
The suggested search algorithm is discussed below. Step 1: Choose the first value of the standard manufactured reactive power rating of capacitors in kvar [10] (20)
where n is the number of discrete values available for the particular voltage rating used and i has a starting value of 1. Step 2: Using only the selected value of QCi, calculate XCi from the following 2 VS1 QCi
(21)
The pre-calculated inductor values for series resonance are used to subdivide the entire search region into small regions. Substitute the value of XCi into the objective function to become one variable problem in XL . Step 3: Let penalty parameter µ > 1, scalar β > 0, and (J) J = 1. Starting with XL to solve the following problem Min.f(XL ) = VTHD(XCi , XL ) µm (max[0, gm (XCi , XL )]) +
(22)
m (J)
For a certain value of µm , the Golden search method [17] (J+1) can be applied for obtaining the optimal XL as follow Step 3.1: Calculate ∆ = Xup − Xlo
Step 4: According to the Penalty Function method the (J) value of µm is updated using (23)
and the previous step is repeated till convergence is achieved using (19)
XCi =
Evaluate f(X1 ) and f(X2 ). Step 3.2: If f(X1 ) ≤ f(X2 ), go to step (3.7). Step 3.3: Set Xlo = X1 and f(Xlo ) = f(X1 ). Step 3.4: Set X1 = X2 and f(X1 ) = f(X2 ). Step 3.5: Set X2 = Xup −λ (Xup −Xlo ) and evaluate f(X2 ). Step 3.6: Go to step (3.10). Step 3.7: Set Xup = X2 and f(Xup ) = f(X2 ). Step 3.8: Set X2 = X1 and f(X2 ) = f(X1 ). Step 3.9: Set X1 = Xlo +λ (Xup −Xlo ) and evaluate f(X1 ). Step 3.10: If (Xup − Xlo ) ≥∈; go to step (3.2) otherwise stop, where √ 3− 5 λ= , ∈= 10−6 2
= βµ(J) µ(J+1) m m
subject to : g1 (XCi , XL ) = 0.9 − PF(XCi , XL ) ≤ 0.0 g2 (XCi , XL ) = PF(XCi , XL ) − 1.0 ≤ 0.0 XCi , XL is not part of solution of Eq. (16)
QCi = {QC1 , QC2 , . . . , QCn }
X2 = Xup − λ
(J+1)
µ(J) m (max[0, gm (XCi , XL
)]) < ε
(24)
Step 5: If i = n stop otherwise replace i by (i + 1) and go to step 2. Step 6: After stopping, scan through to get the global minimum. Ref. [18] shows that the starting penalty parameter value µ(0) = 10. Values of β in the range 0.1–0.5 work well for most problems. The algorithm will stop when a feasible point is reached or when the relative change in the objective function is small (less than ε = 10−6 ). In the optimization process, the resistance of the compensator reactor has been neglected due to its small value with respect to its fundamental reactance (less than 5%) [7].
7. Examples and simulated results Four cases of an industrial plant were simulated using the optimization method. The numerical data were primarily taken from an example in Ref. [13] where the inductive three-phase load is 5100 kW with a displacement factor (dPF) of 0.7165. The 60-cycle supply bus voltage is 4.16 kV (line-to-line). The resistance and fundamental reactance values are given in Table 1. Comparison of the results, Table 2, shows that a lower short-circuit capacity corresponds to a higher PF at the same conditions. This has to be expected since with higher transmission impedance, less harmonic current will flow into the compensated load.
A.F. Zobaa / Electric Power Systems Research 70 (2004) 253–260 Table 1 System parameters and source harmonics Parameters & harmonics Case 1 Short circuit (MVA) RT1 (") XT1 (") RL1 (") XL1 (") VS1 (kV) VS5 (%VS1 ) VS7 (%VS1 ) VS11 (%VS1 ) VS13 (%VS1 ) IL5 (%IS1 ) IL7 (%IS1 ) IL11 (%IS1 ) IL13 (%IS1 )
Case 2
Case 3
Case 4
150 150 80 80 0.01154 0.01154 0.02163 0.02163 0.1154 0.1154 0.2163 0.2163 1.742 1.742 1.742 1.742 1.696 1.696 1.696 1.696 2.4 2.4 2.4 2.4 5 1 5 1 3 7 3 7 2 2 2 2 1 1 1 1 5 5 5 5 3 3 3 3 2 2 2 2 1 1 1 1
Table 2 Simulated results for the presented optimization method
257
Table 5 Harmonic distortions and displacement factor before and after compensation Case
ITHD (%)
VTHD (%)
dPF (%)
1 2 3 4 5 6
21.69 12.99 14.18 9.71 28.65 19.61
2.83 4.46 2.37 3.46 3.62 2.68
91.23 91.23 91.19 91.19 92.30 91.19
Finally, in the IEEE Standard 519-1992 [13] the objectives of the nonlinear load harmonic current limits are to limit the maximum individual harmonic voltage to 3% of the fundamental voltage and the total harmonic distortion of the voltage to 5%. Table 5 shows the distortion levels and the displacement factor after compensation. The resultant values all come out well within standard limits.
Case
XC (")
XL (")
PF (%)
IS (A)
η (%)
TL (kW)
8. Compensator consideration
1 2 3 4
6.78 6.78 6.78 6.78
0.3085 0.3085 0.2948 0.2948
89.12 90.38 90.26 90.71
777.58 766.275 755.45 751.415
99.57 99.59 99.22 99.23
6.98 6.78 12.34 12.21
ANSI/IEEE Standard 18-1992 [10] specifies the following continuous capacitor ratings:
Now we study another two cases, 5 and 6, by changing the %VS5 and %VS7 of cases 2 and 4 to 7 and 4%, respectively. Comparison of the results in Table 3 shows that an additional harmonic content results in lower PF. This is caused by the increase in compensated line current due to the additional harmonics.Table 4 shows the required capacitive reactance, the power factor, the supply current, the efficiency and the transmission loss for the same system data but with the load being linear. Comparing these results with Table 2, we realized a degraded PF, a high line current, and hence an increase in TL and a decrease in the η. Table 3 Simulated results for the presented optimization method Case
XC (")
XL (")
PF (%)
IS (A)
η (%)
TL (kW)
1 5 3 6
6.78 6.40 6.78 6.780
0.3085 0.3038 0.2948 0.2948
89.12 88.67 90.26 89.45
777.58 782.02 755.45 762.19
99.57 99.57 99.22 99.21
6.98 7.06 12.34 12.57
Table 4 Simulated results for the optimization method in Ref. [19] Case
XC (")
PF (%)
IS (A)
η (%)
TL (kW)
1 2 3 4
7.81 3.12 3.02 2.85
85.24 93.84 91.66 87.22
807.64 751.45 770.77 812.64
99.54 99.28 99.25 99.18
7.53 12.21 12.85 14.28
A. One hundred and thirty-five percent of nameplate kvar (QC ). B. One hundred and ten percent of rated rms voltage (VC ) including harmonics but excluding transients. C. One hundred and eighty percent of rated rms current (IC ) including fundamental and harmonic current. D. One hundred and twenty percent of peak voltage including harmonics. The following is the illustration of the calculation of important design parameters of LC compensator required for case 1: A. System information: Capacitor bank rating = 2550 kvar, rated bank current = 353.9 A, rated capacitor voltage = 4160 V, power system frequency = 60 Hz, tuning frequency = 282 Hz (4.7th). B. Capacitor duty calculation: Capacitor current distortion = 44.93%, rms capacitor current = 397.28 A, harmonic capacitor current = 158.96 A, fundamental capacitor current = 364.1 A, maximum peak current = 564 A, capacitor voltage distortion = 8.87%, rms capacitor voltage = 4289.1 V, harmonic capacitor voltage = 369.02 V, fundamental capacitor voltage = 4273.2 V, maximum peak voltage = 4704.1 V. C. Capacitor limits (IEEE Std 18-1992): Item
Calculated (%)
Limit (%)
Exceeds limit
Peak voltage rms voltage rms current kvar
113.08 103.10 112.25 115.74
120.00 110.00 180.00 135.00
No No No No
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D. Reactor design specification: Reactor impedance = 0.3085 ", reactor rating = 0.818 mH, harmonic current = 158.96 A, fundamental current = 364.1 A. It is usually a good idea to use capacitors with a higher voltage rating in some cases because of the voltage rise across the reactor at the fundamental frequency and due to the harmonic loading. Now, we test the method if we take into consideration VC , IC , QC as constranits. 2 IC = ICK (25) K
VC =
2 VCK
(26)
K
QC = VC × IC
(27)
where the compensator current ICK is given by ICK =
VSK (RLK + jXLK ) − ILK (RTLK + jXTLK ) AIK + jAJK
(28)
and the capacitor voltage VCK is given by VCK =
ICK × XC K
(29)
Then, the problem becomes Minimize subject to :
VTHD(XCi , XL )
(30)
90% ≤ PF(XCi , XL ) ≤ 100% IC (XCi , XL ) ≤ 180% VC (XCi , XL ) ≤ 110% QC (XCi , XL ) ≤ 135% XCi , XL is not part of solution of (16)
Same results are obtained. This has to be expected since without involving these constraints, the results are within the standard limits [10] as shown in Section 8.
9. Problems with three-phase four-conductor systems 9.1. Practical considerations In installations where the neutral is distributed, non-linear loads may cause significant overloads in this conductor due to the presence of the third harmonic. The analysis in this section is based on often used modern electronic devices fed by a single-phase rectifier with shunt condensator. These devices are normally connected to one phase and the neutral conductor. The phase currents are nonsinusoidal and therefore contain harmonics. Let us assume that the current waves of the three phases do not overlap. For a period T of the fundamental, a phase current consists of a positive wave and a negative wave separated by an interval where the current
is zero. The rms value of the line current can be calculated using the formula: 1 T 2 i dt (31) IS = T 0 S The rectifier takes all the current needed to load the condensator during the peak region of the voltage. At this moment the rectifier connected to the two other phases do not take any current, therefore the contributions of the three-phase current to the neutral current do not cancel themselves even if the single-phase loads are equally distributed resulting in a symmetrical three-phase load. Now, the rms of the neutral current can be calculated over an interval equal to T/3. During this interval, the neutral current also consists of a positive wave and a negative wave, identical to those of the phase current. The rms value of the neutral current can therefore be calculated as follows: T/3 1 IN = i2N dt (32) T/3 0 √ 1 T/3 2 IN = 3 iN dt (33) T 0 √ √ 1 T 2 iS dt = 3IS (34) IN = 3 T 0 Here, therefore, the current in the neutral conductor has an √ rms value 3 times greater than that of the current in a phase [20]. When the current wave of all three phases overlaps, the √ rms value of the current in the neutral is less than 3 times the rms value of the current in a phase. So, a lower supply current from the presented method corresponds to a lower neutral current compared with the uncompensated case. If we consider that the third harmonic is dominant, the ITHD is very close to the third harmonic ratio. So: IS3 ITHD ∼ (35) = IS1 ∼ 3 × IS3 (36) IN = This can be expressed as: IN ∼ = 3 × ITHD × IS1 Using the general formula: IS IS1 = 1 + ITHD2 We can obtain: 3 × ITHD IN = IS 1 + ITHD2
(37)
(38)
(39)
This approximate formula is only valid when the result is √ less than 3. The example above has proved that the rms value of the current flowing in the neutral conductor may be even larger than the rms value of each phase current.
A.F. Zobaa / Electric Power Systems Research 70 (2004) 253–260 Table 6 Load harmonic contents
and
K
ILK (A)
3 5 7 9 11 13
304 33 25 26 8 9
SL (kVA) =
ICK KXL
K
Table 7 Comparision of results before and after compensation for the case under study Case
ITHD (%)
VTHD (%)
IN (A)
Before After
29.92 10.33
8.19 5.13
820.12 161.28
259
1/2 2 ICK
(43)
K
In Eqs. (42) and (43), the harmonic voltages are added linearly in the first summation to empahsize the effect of peak voltage on insulation cost. Refs. [22,23] describe two different methods for the evaluation of the parameters of the LC compensator for nonlinear loads taking into account cost constraints. The results are compared with the maximum possible power factor which can be achieved by pure capacitive compensation [19]. It is observed that the same power factor can be obtained by LC compensation at a lower cost, or a high power factor can be achieved by the LC compensator at the same cost. These is an agreement with results of Ref. [7], but with the load being linear.
9.2. Theoretical considerations 11. Conclusions The theoretical limit of IN is three times the rms value of each phase current caused by a pure zero-sequence current, IO . IN = 3 × IO
(40)
Now, we test case 3 with load harmonic contents given in Table 6. Table 7 shows the values of ITHD, VTHD and IN before and after compensation for case 3. The general performance, practical and theoretical, of the proposed method is satisfactory, providing neutral current reduction. Comparing the overloading of the neutral current, we realized a decrease in its value from 85.17 to 21.95%. This has to be expected because the notch frequency is near the third. So, the presented method gives safe operation for most power systems. Finally, in installation where a large number of non-linear loads, such as switch mode power supplies for computer equipment, the current in the neutral may therefore exceed the current in each phase. This situation, although rare, requires the use of the reinforced neutral conductor.
10. Economical aspects
Acknowledgements
The compensator cost is defined as C = UC × SC + UL × SL
(41)
where UC , UL are the unit cost of capacitor and inductor, and considered to be constant parameters. For capacitors and reactors, volt–ampere ratings are defined as [21]:
1/2
ICK XC 2 SC (kVA) = ICK (42) K K
The power factor can be maintained while reducing voltage harmonics if a reactance is placed in series with the compensating capacitor to form an LC compensator. Such compensators have dual purposes. The first is that it acts as a compensator to improve the power factor of the nonlinear loads. Secondly, it acts as a filter of the harmonic load currents thus preventing the proliferation of the network with these currents. A mathematical model is developed and a solution method is presented for minimizing the voltage total harmonic distortion at the load bus where it is desired to maintain the power factor at a desired level. It is shown that the LC compensator sizes can be quite different when nonlinear loads are present in a system when compared to those found by neglecting harmonic components of the load. Also, it is shown that significant improvement in distortion levels can be achieved. Four cases are tested, and the general performance of the proposed method is satisfactory, providing improvement of distortion levels, neutral current reduction, and power factor correction, compared with other published results.
K
The author wishes to thank Prof. Dr. Mohamed Mamdouh Abdel Aziz, Professor of Electrical Power Engineering, Electrical Power & Machines Department, Faculty of Engineering, Cairo University, Giza, Egypt, for his willing assistance, his sincere efforts and kind advices.
Appendix A. List of symbols GLK , BLK
load conductance and susceptance at harmonic number K (")
260
ICK IlK ILK IS ISK IN PL PS R RLK , XLK RTK , XTK VLK VSK VL XL , X C
A.F. Zobaa / Electric Power Systems Research 70 (2004) 253–260
capacitor current at harmonic number K (A) load current at harmonic number K (A) load harmonic current (A) rms value of supply current (A) supply current at harmonic number K (A) rms value of neutral current (A) load power (W) supply power (W) resistance of the compensator reactor (") load resistance and reactance at harmonic number K (") transmission system resistance and reactance at harmonic number K (") load voltage at harmonic number K (V) supply voltage at harmonic number K (V) rms value of load voltage (V) fundamental inductive and capacitive reactance of the compensator (")
References [1] W. Shepherd, P. Zand, Energy Flow and Power Factor in Nonsinusoidal Circuits, Cambridge University Press, Cambridge, 1979. [2] W.M. Grady, M.J. Samotyj, A.H. Noyola, Survey of Active Power Line Conditioning Methodologies, IEEE, PES Winter Meething, Atlanta, GA, February 1990, Paper WM 048-9PWRD. [3] S.F. Mekhamer, M.E. El-Hawary, M.M. Mansour, M.A. Moustafa, S.A. Soliman, State-of-the-art in optimal capacitor allocation for reactive power compensation in distribution feeders, in: Proceedings of the 2002 Large Engineering Systems Conference on Power Engineering, Halifax, Nova Scotia, Canada, 26–28 June 2002, pp. 61–75. [4] A.E. Emanuel, M. Yang, On the harmonic compensation in nonsinusoidal systems, IEEE Trans. Power Delivery 8 (1) (1993) 393–399. [5] G.G. Richards, P. Klinkhachorn, O.T. Tan, R.K. Hartana, Optimal LC compensators for nonlinear loads with uncertain nonsinusoidal source and load characteristics, IEEE Trans. Power Syst. 4 (1) (1989) 30–36. [6] R.K. Hartana, G.G. Richards, Comparing capacitive and LC compensators for power factor correction and voltage harmonic reduction, Electric Power Syst. Res. 17 (1989) 57–64. [7] G.G. Richards, O.T. Tan, P. Klinkhachorn, N.I. Santoso, Costconstrained power factor optimization with source harmonics using LC compensators, IEEE Trans. Ind. Electron. 34 (2) (1987) 266–270. [8] M.M. Abdel-Aziz, E.E. Abou El-Zahab, A.M. Ibrahim, A.F. Zobaa, Comparing capacitive and LC compensators for power factor correction, in: Proceedings of the 10th International Conference on Harmonics and Quality of Power ICHQP’02, vol. 1, Rio de Janeiro, Brazil, 2002, pp. 42–45. [9] M.M. Abdel-Aziz, E.E. Abou El-Zahab, A.M. Ibrahim, A.F. Zobaa, Practical considerations on power factor correction for nonlinear loads, in: Proceedings of the 10th International Conference on Harmonics and Quality of Power ICHQP’02, vol. 1, Rio de Janeiro, Brazil, 2002, pp. 46–49. [10] IEEE Standards for Shunt Power Capacitors, IEEE Std. 18-1992, 1992. [11] M.F. McGranaghan, R.C. Dugan, J.A. King, W.T. Jewell, Distribution feeder harmonic study methodology, IEEE Trans. Power Apparatus Syst. 103 (12) (1984) 3663–3671.
[12] S. Abdelkader, M.H. Abdel-Rahman, M.G. Osman, A Norton equivalent model for nonlinear loads, in: Proceedings of the 2001 Large Engineering Systems Conference on Power Engineering, LESCOPE’01, Canada, 2001, pp. 63–67. [13] IEEE Recommended Practices and Requirements for Harmonic Control in Electrical Power Systems, IEEE Std. 519-1992, 1992. [14] R.C. Dugan, M.F. McGranaghan, H.W. Beaty, Electrical Power Systems Quality, McGraw-Hill, USA, 1996. [15] D.E. Rice, Adjustable speed drive and power rectifier harmonics: their effect on power system components, IEEE-PCIC’84, Paper no. PCIC-84-52, 1984. [16] P.F. Ribeiro, Tutorial on Harmonics Modeling and Simulation, IEEE PES Winter Meeting, Tampa, 4 February 1998. [17] S.S. Rao, Engineering Optimization Theory and Practice, Wiley, and New Age International (P) Ltd., 2000. [18] M. Ben-Daya, K.S. Al-Sultan, A new penalty function algorithm for convex quadratic programming, Euro. J. Oper. Res. 101 (1997) 155–163. [19] I.M. El-Amin, S.O. Duffuaa, A.U. Bawah, Optimal shunt compensators at nonsinusoidal bus bars, IEEE Trans. Power Syst. 10 (2) (1995) 716–723. [20] IEEE Working Group on Nonsinusoidal Situations: effects on meter performance and definitions of power, practical definitions for powers in systems with nonsinusoidal waveforms and unbalanced loads: a discussion, IEEE Trans. Power Delivery 11 (1) (1996) 79–101. [21] IEEE Guide for Harmonic Control and Reactive Compensation of Static Power Converters, IEEE Std. 519-1981, 1981. [22] M.M. Abdel Aziz, E.E. Abou El-Zahab, A.F. Zobaa, LC compensators based on cost minimization for nonlinear loads, in: Proceedings of the 2003 Large Engineering Systems Conference on Power Engineering, LESCOPE’03, Montreal, Que., Canada, 7–9 May 2003, pp. 143–147. [23] A.F. Zobaa, Cost effective applications of power factor correction for nonlinear loads, approved for publication in IEEE Transactions on Power Delivery, Paper TPWRD-00152-2003.
Biography Ahmed Faheem Zobaa graduated from and received the M.Sc. and Ph.D. degrees in Electrical Power & Machines, from Faculty of Engineering, Cairo University, Giza, Egypt, in 1992, 1997 and 2002. He was an Instructor in the Department of Electrical Power & Machines, at Faculty of Engineering, Cairo University from 1992 to 1997. He was a Teaching Assistant in the Department of Electrical Power & Machines, at Faculty of Engineering, Cairo University from 1997 to 2002. He is currently an Assistant Professor in the Department of Electrical Power & Machines, at Faculty of Engineering, Cairo University. Dr. Zobaa has been a member of the IEEE Power Engineering/Industry Applications/Industrial Electronics/Power Electronics Societies, and the International Solar Energy Society. On the technical side, he is an Editorial Board member for International Journal of Power and Energy Systems, and Electric Power Components & Systems Journal. He regularly reviews papers for most IEEE Transactions in his area.. He is author or co-author of many refereed Journal and Conference papers. Areas of research include harmonics, compensation of reactive power, power quality, photovoltaics, wind energy, education and distance learning.
A new approach for voltage harmonic distortion minimization A.F. Zobaa Electrical Power & Machines Department, Faculty of Engineering, Cairo University, Giza, Egypt Received 29 July 2003; received in revised form 4 November 2003; accepted 9 December 2003
Abstract The problem of harmonic reduction is alternatively approached as an energy user’s problem. The user’s objective is to eliminate his voltage harmonic distortion problem locally without considering the effects of voltage distortion at neighboring buses. The remedy is insertion of a reactor in series with the local compensating capacitor. A method is presented for finding the optimum fixed LC combination to minimize voltage harmonic distortion at a load bus while holding the power factor at a desired value and constraining the compensator values which would create resonant conditions and the manufacturer’s standard values for power shunt capacitors. © 2004 Elsevier B.V. All rights reserved. Keywords: Harmonics; Power factor; Reactive power optimization
1. Introduction The application of capacitors in electric power systems is intended for the control of power flow, improvement of stability, voltage profile management, power factor (PF) correction, and loss minimization. Many publication on the subject of PF correction in sinusoidal systems [1] advocate the need for unity PF. A comprehensive search over the last 30 years of publications [2,3] proves that a lot of successful research and development of compensators has taken place. The assumption made and brief descriptions of the solution methods are presented. The different types of reactive compensation strategies on the power system losses and voltage distortion are: A. Compensation of the 60 Hz reactive power. In this case the compensation current is sinusoidal. The line current however will remain nonsinusoidal. This method has the advantage that the compensator does not generate additional harmonics like the Thyristor-Controlled Reactors (TCRs) do, neither may cause resonances as the shunt capacitances sometimes will do. B. Provide a sinusoidal line current in phase with the fundamental voltage. Hence, if a sufficient number of dominant nonlinear loads in a power network will be equipped with such compensators the harmonic power flow can be reduced to inconsequential levels. 0378-7796/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2003.12.014
C. Produce unity PF at the load bus. This condition requires the line current and the bus voltage to have identical waveforms. D. Minimization of power loss in the power network. The compensator acts like a filter; it stops the harmonic current to flow in the network and “sink” a part of the harmonic currents to reduce the power loss. E. Minimization of voltage distortion at the user’s end or at a large customer bus. The goal of Ref. [4] is to evaluate the effect of PF correction, implemented by different types of reactive power compensators and compensation strategies, on the power system losses and voltage distortion. Neither the case when the line current is sinusoidal, nor the case when PF is unity, provides minimum power system loss or minimum voltage distortion. Tuned LC compensators proved to be an effective means of correcting PF, yielding reduction in the power system losses and lowering the voltage distortion to reasonable levels [5,6]. In such an arrangement, the LC compensator may actually have a lower volt–ampere rating (or cost) than that of a pure capacitive compensator [7]. In other attempts at optimizing the LC compensators [8,9], the main objective has been to maximize the load PF with minimum transmission loss (TL). This may also reduce the total harmonic distortion of voltage and current, but it may not minimize them. In other words, maximizing PF does not solve the problem of minimizing voltage harmonic distortion. In Ref. [8], both the equivalent source and load are
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considered to generate harmonics. The different criteria for the design of the LC compensator, (a) maximizing PF (b) minimizing TL (c) maximizing transmission efficiency (η), are discussed taking into consideration the non-linearity of the load by using Direct Polytope Search method. In Ref. [9], the manufacturer’s standard values for power shunt capacitors are taken into consideration. These values are considered as constraints in the sense that the solution for the capacitor should be one of the standard values. The different criteria for the design of the LC compensator are discussed by using Golden Section Search method. The reason for doing this is to compare the values obtained in Ref. [4] with real practical values in the market [10]. Ref. [10] shows the voltage and reactive power ratings of shunt capacitors. The inductive reactive values are almost continuous and there is little limitation on the manufacturer’s values. In this study, both the equivalent source and load are considered to generate harmonics. It is assumed that the load harmonics are not sufficiently serious to suggest tuned filters, but when combined with source harmonics, the use of a pure capacitive compensator would degrade PF and overload the equipment. In this paper, a method is presented for minimizing the voltage total harmonic distortion (VTHD) at the load bus where it is desired to maintain the PF at a desired level. An optimum fixed LC compensator will be selected that will minimize the expected value of VTHD for a specified range of source harmonic and impedance values taking into consideration. The major attribute of the method is that it, unlike conventional approaches, guarantees convergence to the optimal solution. This accomplishment is a direct result from the improvement in: A. The algorithm in which provisions are made to identify and to avoid compensator values which would create resonance conditions. B. The problem formulation in which the effect of the transmission line impedance on the load voltage is included while calculating the optimal compensator value. C. Frequency dependent nature of the supply source. D. The manufacturer’s standard values for power shunt capacitors are taken into consideration. Finally, the contribution of the newly developed method is demonstrated in examples taken from previous publications.
The rms value of the total waveform is not the sum of the individual components, but is the square root of the sum of the squares. THD is a very useful quantity for many applications, but its limitations must be realized. It can provide a good idea of how much extra heat will be realized when a distorted voltage is applied across a resistive load. Likewise, it can give an indication of the addition losses caused by the current flowing through a conductor. However, it is not a good indicator of the voltage stress with a capacitor because that is related to the peak value of the voltage waveform, not its heating value.
3. Basic approach to harmonic reduction Fig. 1 is a single-phase equivalent circuit of a bus with LC compensator, experiencing voltage harmonic distortion at harmonic order K because of a voltage source, V SK , and harmonic current sources within the load itself, ILK . Thevenin voltage source representing the utility supply and the harmonic current source representing the nonlinear load are VS (t) = VSK (t) (1) K
and ILK (t) =
ILK (t)
(2)
K
where K is the order of harmonic present. The Kth harmonic Thevenin source and load impedances are ZTK = RTK + jXTK
(3)
and ZLK = RLK + jXLK
(4)
In Ref. [12], a new model for the distribution system including nonlinear loads is introduced. The model is based on measurements, where current and voltage measurements at two different operating conditions are used to calculate ZLK from the following (1)
ZLK =
(2)
VLK − VLK (1)
(2)
ILK − ILK
(5)
2. Total harmonic distortion There are several measures commonly used for indicating the harmonic content of a waveform with a single number [11]. One of the most common is total harmonic distortion (THD), which can be calculated for either voltage or current (ITHD). THD is a measure of the effective value of the harmonic components of a distorted waveform, that is, the potential heating of the harmonics relative to the fundamental.
Fig. 1. Single-phase equivalent circuit for Kth harmonic with shunt LC compensator.
A.F. Zobaa / Electric Power Systems Research 70 (2004) 253–260
The reader should refer to Ref. [12] for detail discussion on the derivation of (5). To simplify the analysis, only the load model using the respective active and reactive powers at the fundamental frequency is considered while sizing the compensators. This model, Fig. 1, is adequate where VTHD is less than 10 % [11]. The approach will be to minimize voltage harmonic distortion on the load by adjusting XC and XL . The voltage harmonic distortion at the compensated load terminals is defined as 2 K>1 VLK (6) VTHD = VL1 and VLK
VSK (CR) − ILK (DR × ER) = AIK + jAJK
(7)
where AR = R + RLK , BR = (XLK + KXL − XC /K), CR = RCLK + jXCLK , DR = R + j(KXL –XC /K), ER = RTLK + jXTLK , AIK = RTLK + R(RLK + RTK ) − (XLK + XTK )(KXL − XC /K), AJK = XTLK + R(RLK + XTK ) + (RLK + RTK )(KXL − XC /K). RCLK + jXCLK ZCLK = (8) ZLK + ZCK where RCLK = RRLK − XLK (KXL − XC /K), XCLK = RXLK + RLK (KXL − XC /K). RTLK + jXTLK (9) ZTLK = ZTK + ZLK where RTLK = RTK RLK − XTK XLK , XTLK = RTK XLK + XLK RLK . The compensated PF at the load is given as 2 GLK VLK PL = (10) PF = 2 V L IS 2 ISK VLK
255
impedance to the flow of harmonic current, while series resonance is low impedance to the flow of harmonic current [13]. In actual electrical systems utilizing PF correction, both types of resonance or a combination of both may occur if the resonant point happens to be close to one of the frequencies generated by harmonic sources in the system.The expected impedance seen from the Thevenin source is given by Z = RTK + jXTK +
RCLK + jXCLK ZLK + ZCK
(15)
The resonance peaks can be obtained by setting the imaginary part of (15) to zero, resulting in a quadratic equation in XC and XL for any given harmonic order K, XC 2 XC + A2 KXL − A1 KXL − + A3 = 0 K K
(16)
2 + 2X X , where A1 = XTK + XLK , A2 = R2LK + XLK LK TK 2 2 2 A3 = R XLK + XTK [(R + RLK ) + XLK ] and by taking the solution of (16) where the square root of the discriminant is positive. (The other solution corresponds to resonance between the load and the combination of source impedance and compensator.) Note that for sufficiently large load resistance and/or load reactance, (16) reduces to:
KXL −
XC + XTK = 0 K
(17)
which then represents only the series resonance, which represents all possible combinations of XC and XL values which result in resonance between the transmission impedance and compensated load. Under these conditions, the PF will reach a minimum. It is evident that the number of series resonance lines will depend on the number of harmonics presents in the transmission source.
where ISK =
VSK (AR + jBR) + ILK CR AIK + jAJK
5. Frequency dependent nature of the supply source (11)
YLK = GLK − jBLK
(12)
The TL is given as 2 ISK RTK TL =
(13)
K
The network η is given as 2 GLK VLK PL = 2 η= 2 PS ISK RTK + GLK VLK
(14)
4. Harmonic resonance constraint System resonant conditions are the most important factors affecting system harmonic levels. Parallel resonance is high
Skin effect is an alternating current phenomenon where the current in a conductor tends to flow more densely near the outer surface of a conductor that in the center area. Skin effect will be applied in the analysis to account for the impact on the system impedance of the frequency dependence of the resistive components. In most power systems, one can generally assume that the resistance does not change significantly when studying the effects of harmonics less than the ninth [14]. The response of various types of equipment to distorted waveforms is by no means uniform and some components are of more concern than others [15]. Ref. [16] shows a detailed analysis of distribution systems, loads and other system elements. It consists basically of representing the dominant characteristics of the network using alternative configurations and models. Also simpler equivalents for extended networks are suggested.
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6. Proposed solution of the problem
X1 = Xlo + λ
VTHD and PF can be expressed as functions of XC and XL , using (6) and (10). Each value of the reactive power ratings QCi of the particular voltage [10] is used to calculate the corresponding value of XCi . This value is then substituted into the objective function to become one variable equation in XL , which can be solved by using the Penalty Function method. After formulating the objective function and the constraints, the problem addressed in this study becomes:
and
Minimize VTHD(XCi , XL ) subject to : 90% ≤ PF(XCi , XL ) ≤ 100% (18) XCi , XL is not part of solution of Eq. (16) Naturally, the solution will satisfy the upper limit of the PF and efficiency constraints. PF in distribution systems may be allowed within certain limits according to the authority, and similarly for the efficiency so the presented method will generalize the limits of the constraints. This problem can be rewritten in the form Minimize
VTHD(XCi , XL )
The suggested search algorithm is discussed below. Step 1: Choose the first value of the standard manufactured reactive power rating of capacitors in kvar [10] (20)
where n is the number of discrete values available for the particular voltage rating used and i has a starting value of 1. Step 2: Using only the selected value of QCi, calculate XCi from the following 2 VS1 QCi
(21)
The pre-calculated inductor values for series resonance are used to subdivide the entire search region into small regions. Substitute the value of XCi into the objective function to become one variable problem in XL . Step 3: Let penalty parameter µ > 1, scalar β > 0, and (J) J = 1. Starting with XL to solve the following problem Min.f(XL ) = VTHD(XCi , XL ) µm (max[0, gm (XCi , XL )]) +
(22)
m (J)
For a certain value of µm , the Golden search method [17] (J+1) can be applied for obtaining the optimal XL as follow Step 3.1: Calculate ∆ = Xup − Xlo
Step 4: According to the Penalty Function method the (J) value of µm is updated using (23)
and the previous step is repeated till convergence is achieved using (19)
XCi =
Evaluate f(X1 ) and f(X2 ). Step 3.2: If f(X1 ) ≤ f(X2 ), go to step (3.7). Step 3.3: Set Xlo = X1 and f(Xlo ) = f(X1 ). Step 3.4: Set X1 = X2 and f(X1 ) = f(X2 ). Step 3.5: Set X2 = Xup −λ (Xup −Xlo ) and evaluate f(X2 ). Step 3.6: Go to step (3.10). Step 3.7: Set Xup = X2 and f(Xup ) = f(X2 ). Step 3.8: Set X2 = X1 and f(X2 ) = f(X1 ). Step 3.9: Set X1 = Xlo +λ (Xup −Xlo ) and evaluate f(X1 ). Step 3.10: If (Xup − Xlo ) ≥∈; go to step (3.2) otherwise stop, where √ 3− 5 λ= , ∈= 10−6 2
= βµ(J) µ(J+1) m m
subject to : g1 (XCi , XL ) = 0.9 − PF(XCi , XL ) ≤ 0.0 g2 (XCi , XL ) = PF(XCi , XL ) − 1.0 ≤ 0.0 XCi , XL is not part of solution of Eq. (16)
QCi = {QC1 , QC2 , . . . , QCn }
X2 = Xup − λ
(J+1)
µ(J) m (max[0, gm (XCi , XL
)]) < ε
(24)
Step 5: If i = n stop otherwise replace i by (i + 1) and go to step 2. Step 6: After stopping, scan through to get the global minimum. Ref. [18] shows that the starting penalty parameter value µ(0) = 10. Values of β in the range 0.1–0.5 work well for most problems. The algorithm will stop when a feasible point is reached or when the relative change in the objective function is small (less than ε = 10−6 ). In the optimization process, the resistance of the compensator reactor has been neglected due to its small value with respect to its fundamental reactance (less than 5%) [7].
7. Examples and simulated results Four cases of an industrial plant were simulated using the optimization method. The numerical data were primarily taken from an example in Ref. [13] where the inductive three-phase load is 5100 kW with a displacement factor (dPF) of 0.7165. The 60-cycle supply bus voltage is 4.16 kV (line-to-line). The resistance and fundamental reactance values are given in Table 1. Comparison of the results, Table 2, shows that a lower short-circuit capacity corresponds to a higher PF at the same conditions. This has to be expected since with higher transmission impedance, less harmonic current will flow into the compensated load.
A.F. Zobaa / Electric Power Systems Research 70 (2004) 253–260 Table 1 System parameters and source harmonics Parameters & harmonics Case 1 Short circuit (MVA) RT1 (") XT1 (") RL1 (") XL1 (") VS1 (kV) VS5 (%VS1 ) VS7 (%VS1 ) VS11 (%VS1 ) VS13 (%VS1 ) IL5 (%IS1 ) IL7 (%IS1 ) IL11 (%IS1 ) IL13 (%IS1 )
Case 2
Case 3
Case 4
150 150 80 80 0.01154 0.01154 0.02163 0.02163 0.1154 0.1154 0.2163 0.2163 1.742 1.742 1.742 1.742 1.696 1.696 1.696 1.696 2.4 2.4 2.4 2.4 5 1 5 1 3 7 3 7 2 2 2 2 1 1 1 1 5 5 5 5 3 3 3 3 2 2 2 2 1 1 1 1
Table 2 Simulated results for the presented optimization method
257
Table 5 Harmonic distortions and displacement factor before and after compensation Case
ITHD (%)
VTHD (%)
dPF (%)
1 2 3 4 5 6
21.69 12.99 14.18 9.71 28.65 19.61
2.83 4.46 2.37 3.46 3.62 2.68
91.23 91.23 91.19 91.19 92.30 91.19
Finally, in the IEEE Standard 519-1992 [13] the objectives of the nonlinear load harmonic current limits are to limit the maximum individual harmonic voltage to 3% of the fundamental voltage and the total harmonic distortion of the voltage to 5%. Table 5 shows the distortion levels and the displacement factor after compensation. The resultant values all come out well within standard limits.
Case
XC (")
XL (")
PF (%)
IS (A)
η (%)
TL (kW)
8. Compensator consideration
1 2 3 4
6.78 6.78 6.78 6.78
0.3085 0.3085 0.2948 0.2948
89.12 90.38 90.26 90.71
777.58 766.275 755.45 751.415
99.57 99.59 99.22 99.23
6.98 6.78 12.34 12.21
ANSI/IEEE Standard 18-1992 [10] specifies the following continuous capacitor ratings:
Now we study another two cases, 5 and 6, by changing the %VS5 and %VS7 of cases 2 and 4 to 7 and 4%, respectively. Comparison of the results in Table 3 shows that an additional harmonic content results in lower PF. This is caused by the increase in compensated line current due to the additional harmonics.Table 4 shows the required capacitive reactance, the power factor, the supply current, the efficiency and the transmission loss for the same system data but with the load being linear. Comparing these results with Table 2, we realized a degraded PF, a high line current, and hence an increase in TL and a decrease in the η. Table 3 Simulated results for the presented optimization method Case
XC (")
XL (")
PF (%)
IS (A)
η (%)
TL (kW)
1 5 3 6
6.78 6.40 6.78 6.780
0.3085 0.3038 0.2948 0.2948
89.12 88.67 90.26 89.45
777.58 782.02 755.45 762.19
99.57 99.57 99.22 99.21
6.98 7.06 12.34 12.57
Table 4 Simulated results for the optimization method in Ref. [19] Case
XC (")
PF (%)
IS (A)
η (%)
TL (kW)
1 2 3 4
7.81 3.12 3.02 2.85
85.24 93.84 91.66 87.22
807.64 751.45 770.77 812.64
99.54 99.28 99.25 99.18
7.53 12.21 12.85 14.28
A. One hundred and thirty-five percent of nameplate kvar (QC ). B. One hundred and ten percent of rated rms voltage (VC ) including harmonics but excluding transients. C. One hundred and eighty percent of rated rms current (IC ) including fundamental and harmonic current. D. One hundred and twenty percent of peak voltage including harmonics. The following is the illustration of the calculation of important design parameters of LC compensator required for case 1: A. System information: Capacitor bank rating = 2550 kvar, rated bank current = 353.9 A, rated capacitor voltage = 4160 V, power system frequency = 60 Hz, tuning frequency = 282 Hz (4.7th). B. Capacitor duty calculation: Capacitor current distortion = 44.93%, rms capacitor current = 397.28 A, harmonic capacitor current = 158.96 A, fundamental capacitor current = 364.1 A, maximum peak current = 564 A, capacitor voltage distortion = 8.87%, rms capacitor voltage = 4289.1 V, harmonic capacitor voltage = 369.02 V, fundamental capacitor voltage = 4273.2 V, maximum peak voltage = 4704.1 V. C. Capacitor limits (IEEE Std 18-1992): Item
Calculated (%)
Limit (%)
Exceeds limit
Peak voltage rms voltage rms current kvar
113.08 103.10 112.25 115.74
120.00 110.00 180.00 135.00
No No No No
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D. Reactor design specification: Reactor impedance = 0.3085 ", reactor rating = 0.818 mH, harmonic current = 158.96 A, fundamental current = 364.1 A. It is usually a good idea to use capacitors with a higher voltage rating in some cases because of the voltage rise across the reactor at the fundamental frequency and due to the harmonic loading. Now, we test the method if we take into consideration VC , IC , QC as constranits. 2 IC = ICK (25) K
VC =
2 VCK
(26)
K
QC = VC × IC
(27)
where the compensator current ICK is given by ICK =
VSK (RLK + jXLK ) − ILK (RTLK + jXTLK ) AIK + jAJK
(28)
and the capacitor voltage VCK is given by VCK =
ICK × XC K
(29)
Then, the problem becomes Minimize subject to :
VTHD(XCi , XL )
(30)
90% ≤ PF(XCi , XL ) ≤ 100% IC (XCi , XL ) ≤ 180% VC (XCi , XL ) ≤ 110% QC (XCi , XL ) ≤ 135% XCi , XL is not part of solution of (16)
Same results are obtained. This has to be expected since without involving these constraints, the results are within the standard limits [10] as shown in Section 8.
9. Problems with three-phase four-conductor systems 9.1. Practical considerations In installations where the neutral is distributed, non-linear loads may cause significant overloads in this conductor due to the presence of the third harmonic. The analysis in this section is based on often used modern electronic devices fed by a single-phase rectifier with shunt condensator. These devices are normally connected to one phase and the neutral conductor. The phase currents are nonsinusoidal and therefore contain harmonics. Let us assume that the current waves of the three phases do not overlap. For a period T of the fundamental, a phase current consists of a positive wave and a negative wave separated by an interval where the current
is zero. The rms value of the line current can be calculated using the formula: 1 T 2 i dt (31) IS = T 0 S The rectifier takes all the current needed to load the condensator during the peak region of the voltage. At this moment the rectifier connected to the two other phases do not take any current, therefore the contributions of the three-phase current to the neutral current do not cancel themselves even if the single-phase loads are equally distributed resulting in a symmetrical three-phase load. Now, the rms of the neutral current can be calculated over an interval equal to T/3. During this interval, the neutral current also consists of a positive wave and a negative wave, identical to those of the phase current. The rms value of the neutral current can therefore be calculated as follows: T/3 1 IN = i2N dt (32) T/3 0 √ 1 T/3 2 IN = 3 iN dt (33) T 0 √ √ 1 T 2 iS dt = 3IS (34) IN = 3 T 0 Here, therefore, the current in the neutral conductor has an √ rms value 3 times greater than that of the current in a phase [20]. When the current wave of all three phases overlaps, the √ rms value of the current in the neutral is less than 3 times the rms value of the current in a phase. So, a lower supply current from the presented method corresponds to a lower neutral current compared with the uncompensated case. If we consider that the third harmonic is dominant, the ITHD is very close to the third harmonic ratio. So: IS3 ITHD ∼ (35) = IS1 ∼ 3 × IS3 (36) IN = This can be expressed as: IN ∼ = 3 × ITHD × IS1 Using the general formula: IS IS1 = 1 + ITHD2 We can obtain: 3 × ITHD IN = IS 1 + ITHD2
(37)
(38)
(39)
This approximate formula is only valid when the result is √ less than 3. The example above has proved that the rms value of the current flowing in the neutral conductor may be even larger than the rms value of each phase current.
A.F. Zobaa / Electric Power Systems Research 70 (2004) 253–260 Table 6 Load harmonic contents
and
K
ILK (A)
3 5 7 9 11 13
304 33 25 26 8 9
SL (kVA) =
ICK KXL
K
Table 7 Comparision of results before and after compensation for the case under study Case
ITHD (%)
VTHD (%)
IN (A)
Before After
29.92 10.33
8.19 5.13
820.12 161.28
259
1/2 2 ICK
(43)
K
In Eqs. (42) and (43), the harmonic voltages are added linearly in the first summation to empahsize the effect of peak voltage on insulation cost. Refs. [22,23] describe two different methods for the evaluation of the parameters of the LC compensator for nonlinear loads taking into account cost constraints. The results are compared with the maximum possible power factor which can be achieved by pure capacitive compensation [19]. It is observed that the same power factor can be obtained by LC compensation at a lower cost, or a high power factor can be achieved by the LC compensator at the same cost. These is an agreement with results of Ref. [7], but with the load being linear.
9.2. Theoretical considerations 11. Conclusions The theoretical limit of IN is three times the rms value of each phase current caused by a pure zero-sequence current, IO . IN = 3 × IO
(40)
Now, we test case 3 with load harmonic contents given in Table 6. Table 7 shows the values of ITHD, VTHD and IN before and after compensation for case 3. The general performance, practical and theoretical, of the proposed method is satisfactory, providing neutral current reduction. Comparing the overloading of the neutral current, we realized a decrease in its value from 85.17 to 21.95%. This has to be expected because the notch frequency is near the third. So, the presented method gives safe operation for most power systems. Finally, in installation where a large number of non-linear loads, such as switch mode power supplies for computer equipment, the current in the neutral may therefore exceed the current in each phase. This situation, although rare, requires the use of the reinforced neutral conductor.
10. Economical aspects
Acknowledgements
The compensator cost is defined as C = UC × SC + UL × SL
(41)
where UC , UL are the unit cost of capacitor and inductor, and considered to be constant parameters. For capacitors and reactors, volt–ampere ratings are defined as [21]:
1/2
ICK XC 2 SC (kVA) = ICK (42) K K
The power factor can be maintained while reducing voltage harmonics if a reactance is placed in series with the compensating capacitor to form an LC compensator. Such compensators have dual purposes. The first is that it acts as a compensator to improve the power factor of the nonlinear loads. Secondly, it acts as a filter of the harmonic load currents thus preventing the proliferation of the network with these currents. A mathematical model is developed and a solution method is presented for minimizing the voltage total harmonic distortion at the load bus where it is desired to maintain the power factor at a desired level. It is shown that the LC compensator sizes can be quite different when nonlinear loads are present in a system when compared to those found by neglecting harmonic components of the load. Also, it is shown that significant improvement in distortion levels can be achieved. Four cases are tested, and the general performance of the proposed method is satisfactory, providing improvement of distortion levels, neutral current reduction, and power factor correction, compared with other published results.
K
The author wishes to thank Prof. Dr. Mohamed Mamdouh Abdel Aziz, Professor of Electrical Power Engineering, Electrical Power & Machines Department, Faculty of Engineering, Cairo University, Giza, Egypt, for his willing assistance, his sincere efforts and kind advices.
Appendix A. List of symbols GLK , BLK
load conductance and susceptance at harmonic number K (")
260
ICK IlK ILK IS ISK IN PL PS R RLK , XLK RTK , XTK VLK VSK VL XL , X C
A.F. Zobaa / Electric Power Systems Research 70 (2004) 253–260
capacitor current at harmonic number K (A) load current at harmonic number K (A) load harmonic current (A) rms value of supply current (A) supply current at harmonic number K (A) rms value of neutral current (A) load power (W) supply power (W) resistance of the compensator reactor (") load resistance and reactance at harmonic number K (") transmission system resistance and reactance at harmonic number K (") load voltage at harmonic number K (V) supply voltage at harmonic number K (V) rms value of load voltage (V) fundamental inductive and capacitive reactance of the compensator (")
References [1] W. Shepherd, P. Zand, Energy Flow and Power Factor in Nonsinusoidal Circuits, Cambridge University Press, Cambridge, 1979. [2] W.M. Grady, M.J. Samotyj, A.H. Noyola, Survey of Active Power Line Conditioning Methodologies, IEEE, PES Winter Meething, Atlanta, GA, February 1990, Paper WM 048-9PWRD. [3] S.F. Mekhamer, M.E. El-Hawary, M.M. Mansour, M.A. Moustafa, S.A. Soliman, State-of-the-art in optimal capacitor allocation for reactive power compensation in distribution feeders, in: Proceedings of the 2002 Large Engineering Systems Conference on Power Engineering, Halifax, Nova Scotia, Canada, 26–28 June 2002, pp. 61–75. [4] A.E. Emanuel, M. Yang, On the harmonic compensation in nonsinusoidal systems, IEEE Trans. Power Delivery 8 (1) (1993) 393–399. [5] G.G. Richards, P. Klinkhachorn, O.T. Tan, R.K. Hartana, Optimal LC compensators for nonlinear loads with uncertain nonsinusoidal source and load characteristics, IEEE Trans. Power Syst. 4 (1) (1989) 30–36. [6] R.K. Hartana, G.G. Richards, Comparing capacitive and LC compensators for power factor correction and voltage harmonic reduction, Electric Power Syst. Res. 17 (1989) 57–64. [7] G.G. Richards, O.T. Tan, P. Klinkhachorn, N.I. Santoso, Costconstrained power factor optimization with source harmonics using LC compensators, IEEE Trans. Ind. Electron. 34 (2) (1987) 266–270. [8] M.M. Abdel-Aziz, E.E. Abou El-Zahab, A.M. Ibrahim, A.F. Zobaa, Comparing capacitive and LC compensators for power factor correction, in: Proceedings of the 10th International Conference on Harmonics and Quality of Power ICHQP’02, vol. 1, Rio de Janeiro, Brazil, 2002, pp. 42–45. [9] M.M. Abdel-Aziz, E.E. Abou El-Zahab, A.M. Ibrahim, A.F. Zobaa, Practical considerations on power factor correction for nonlinear loads, in: Proceedings of the 10th International Conference on Harmonics and Quality of Power ICHQP’02, vol. 1, Rio de Janeiro, Brazil, 2002, pp. 46–49. [10] IEEE Standards for Shunt Power Capacitors, IEEE Std. 18-1992, 1992. [11] M.F. McGranaghan, R.C. Dugan, J.A. King, W.T. Jewell, Distribution feeder harmonic study methodology, IEEE Trans. Power Apparatus Syst. 103 (12) (1984) 3663–3671.
[12] S. Abdelkader, M.H. Abdel-Rahman, M.G. Osman, A Norton equivalent model for nonlinear loads, in: Proceedings of the 2001 Large Engineering Systems Conference on Power Engineering, LESCOPE’01, Canada, 2001, pp. 63–67. [13] IEEE Recommended Practices and Requirements for Harmonic Control in Electrical Power Systems, IEEE Std. 519-1992, 1992. [14] R.C. Dugan, M.F. McGranaghan, H.W. Beaty, Electrical Power Systems Quality, McGraw-Hill, USA, 1996. [15] D.E. Rice, Adjustable speed drive and power rectifier harmonics: their effect on power system components, IEEE-PCIC’84, Paper no. PCIC-84-52, 1984. [16] P.F. Ribeiro, Tutorial on Harmonics Modeling and Simulation, IEEE PES Winter Meeting, Tampa, 4 February 1998. [17] S.S. Rao, Engineering Optimization Theory and Practice, Wiley, and New Age International (P) Ltd., 2000. [18] M. Ben-Daya, K.S. Al-Sultan, A new penalty function algorithm for convex quadratic programming, Euro. J. Oper. Res. 101 (1997) 155–163. [19] I.M. El-Amin, S.O. Duffuaa, A.U. Bawah, Optimal shunt compensators at nonsinusoidal bus bars, IEEE Trans. Power Syst. 10 (2) (1995) 716–723. [20] IEEE Working Group on Nonsinusoidal Situations: effects on meter performance and definitions of power, practical definitions for powers in systems with nonsinusoidal waveforms and unbalanced loads: a discussion, IEEE Trans. Power Delivery 11 (1) (1996) 79–101. [21] IEEE Guide for Harmonic Control and Reactive Compensation of Static Power Converters, IEEE Std. 519-1981, 1981. [22] M.M. Abdel Aziz, E.E. Abou El-Zahab, A.F. Zobaa, LC compensators based on cost minimization for nonlinear loads, in: Proceedings of the 2003 Large Engineering Systems Conference on Power Engineering, LESCOPE’03, Montreal, Que., Canada, 7–9 May 2003, pp. 143–147. [23] A.F. Zobaa, Cost effective applications of power factor correction for nonlinear loads, approved for publication in IEEE Transactions on Power Delivery, Paper TPWRD-00152-2003.
Biography Ahmed Faheem Zobaa graduated from and received the M.Sc. and Ph.D. degrees in Electrical Power & Machines, from Faculty of Engineering, Cairo University, Giza, Egypt, in 1992, 1997 and 2002. He was an Instructor in the Department of Electrical Power & Machines, at Faculty of Engineering, Cairo University from 1992 to 1997. He was a Teaching Assistant in the Department of Electrical Power & Machines, at Faculty of Engineering, Cairo University from 1997 to 2002. He is currently an Assistant Professor in the Department of Electrical Power & Machines, at Faculty of Engineering, Cairo University. Dr. Zobaa has been a member of the IEEE Power Engineering/Industry Applications/Industrial Electronics/Power Electronics Societies, and the International Solar Energy Society. On the technical side, he is an Editorial Board member for International Journal of Power and Energy Systems, and Electric Power Components & Systems Journal. He regularly reviews papers for most IEEE Transactions in his area.. He is author or co-author of many refereed Journal and Conference papers. Areas of research include harmonics, compensation of reactive power, power quality, photovoltaics, wind energy, education and distance learning.