Jul 2, 1991 - Abstract - An active power line conditioner (APLC) is a type of active filter that compensates for power system waveform distortion. This paper ...
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Minimizing Network Harmonic Voltage Distortion with an Active Power Line Conditioner W. M. Grady, Senior Member The University of Texas at Austin Austin, TX 78712
M. J. Samotyj, Member Electric Power Research Institute Palo Alto, CA 94303
A. H. Noyola, Student Member The University of Texas at Austin Austin, TX 78712
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Keywords Harmonic Distortion, Active Power Line Conditioner, Distortion Minimization Abstract - An active power line conditioner (APLC) is a type of active filter that compensates for power system waveform distortion. This paper describes a new procedure for computing the APLC harmonic injection currents needed to minimize voltage distortion throughout a network. The procedure is based upon nonlinear optimization theory.
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INTRODUCTION At present, most harmonic problems are attributable to large individual sources of harmonics, such as adjustable speed drives and rectifiers. These problems are usually found on distribution feeders or industrial power systems, and conventional solutions include network reconfiguration, capacitor switching, and passive filtering. As the dispersion of many small power electronic loads continues, background harmonic levels will gradually increase. Conventional solutions for individual source problems will likely become less practical and less effective. Therefore, there will be an increasing need for active filters that can adjust to widely varying harmonic levels and network impedances. Ideally, these active filters should monitor and minimize voltage distortion throughout an entire network, rather than at an individual bus.
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151 -
Current-Type APLC
Figure 1: Voltage-Type and Current-Type APLCs
A candidate active filter that has been described in the literature is the active power line conditioner (APLC) [ l ] - [9]. It has the potential to reduce harmonic distortion, voltage spikes, transients, and flicker by injecting equal-but-opposite distortion. APLCs employ power electronic devices to rapidly switch dc sources, synthesizing the chopped injection current waveform needed to compensate for existing voltage or current distortion.
Other authors have already studied the problem of synthesizing frequency domain current injection waveforms with power electronic switches [3], [6], [lo], [ 111. The equations developed in this paper can be used to specify the waveforms for which synthesis can be performed.
The two types of inverters employed by APLCs are voltage-type and current-type, as shown in Figure 1. The dc storage element (capacitor or inductor) receives its power from the ac system, either through the APLC switching action or through a separate charging circuit.
The chief contribution of this paper lies in developing a simple procedure for finding the Fourier series of the optimum APLC injection current waveform. The procedure is intended to apply in any of the following situations: 1) one single-phase APLC in a single-phase network; 2) one single-phase APLC in a three-phase network; or 3) one three-phase APLC in a balanced three-phase network.
APLCs can be programmed to operate in either time or frequency domains. Both operating domains are described in detail in [9]. Thus far, APLCs have been applied to individual bus problems only, and their potential for improving overall power quality throughout a network has not been previously addressed. The objective of this paper is to develop and illustrate a procedure for calculating the APLC injection currents needed to minimize voltage harmonic distortion throughout a power network. The procedure is intended for use with APLC frequency domain correction in networks that are experiencing periodic harmonic distortion. The injection currents are determined using nonlinear optimization theory.
THE SINGLE HARMONIC PROBLEM The first situation to be considered is where a single harmonic component (harmonic h) of voltage distortion is to be minimized using an APLC. The APLC is connected to bus m of a K-bus power system, as shown in Figure 2. Distorting loads are assumed to exist in the network, producing voltage distortion throughout. It is assumed that the APLC is provided with measurements of voltage distortion throughout the network. It is also assumed that column m of the network impedance matrix is known (either measured or calculated).
Objective Function 91 WM 113-1 PWRD A paper recommended and approved by the IEEE Transmission and Distribution Committee of the IEEE Power Engineering Society for presentation at the IEEE/PES 1991 Winter Meeting, New York, New York, February 3, 1991. Manuscript submitted August 31, 1990; made available for printing December 18, 1990.
The desired objective function for the single harmonic problem can now be formulated. Ideally, the APLC injection current should minimize the sum of voltage magnitudes for harmonic h throughout the network. This sum can be. written as
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and expanding and collecting current terms yields
Now, defining
Ah rn
Sampled Data
5 [ V2 L;:ld
=
k= 1
+ V2:;Ald ] ,
-
Figure 2: APLC Connected to Bus m of a Power System
(9)
where subscripts old and new correspond to bus voltages before and after APLC injection current, and where superscript h refers to harmonic h. Voltage changes at busk due to APLC injection current at bus m can be written in terms of elements of the network impedance ma^ as
the final form of the objective function for the single harmonic h problem becomes
Unconstrained Minimization From an optimization viewpoint, it is more convenient to work with the sum of squared voltage magnitudes rather than the sum of voltage magnitudes. This leads to objective function f
[
I=
5 I Vk,",,
k= 1
I
2
K
=
C
k= 1
I
vk,old
+ AXh
I
(3)
>
The unconstrained minimum of (10) will be found using nonlinear optimization theory. The first-order necessary conditions for minimization are obtained by setting the partial derivatives of (10) to zero, yielding af
[
I;~, I ; ~
atr
where 1;
=Q'+jti,
';,old
= d:;l'
AV:
= AV?
+
hi jvk:old
] = BL+2Dke=0,
'
+ jAV,".i,
Solving yields
and where subscripts r and i refer to real and imaginary components, respectively. Expressing (3) in rectangular form yields f
[ ,':I
I:i
1=
5 [(V;;:,,
k= 1
+ AV:
+ (V2:ld + AV:"
and expanding the right-hand term of (2) yields
Substituting the result into (4)gives
)'],
(4)
The second-order conditions for minimizing (10) yield second partial derivatives
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Therefore, the Hessian matrix H is
By taking the second partial derivatives of the Lagrangian function, it can be shown that H for the constrained problem is positive definite;
[ '3 2i;].
H =
Since D:
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The solution of (14) (18) is more easily understood through geometric interpretation, as follows.
is, according to (91,always positive, H is positive
definite. Therefore, the objective function for the single harmonic case is strictly convex with an unconstrained global minimum at
-
Geometric Interpretation Examining the quadratic objective function given by (lo), and recognizing that D: is positive, contours of equal f
[
e,
I;i
] must be circles whose axes are real
and imaginary injection currents
*
and .I:'
These circles are
shown in Figure 3 and are centered at point Q.
Constrained Minimization The unconstrained solution, given by (11) and (12). will likely exceed the APLC current rating in most practical situations. Therefore, the problem of constrained minimization, where the current is limited to a maximum value, will now be considered for the single harmonic case. The APLC current constraint can be expressed as
e,
g[
I~']=I2~+12~i_I2~"
