IEEE Transactions on Power Delivery, Vol.
7, No. 3, July 1992
1379
The Application of Network Objective Functions for Actively Minimizing the Impact of Voltage Harmonics in Power Systems W. M. Grady. Senior Member The university of Texas at AusM Austin,TX 78712
M. J. Samotyj. Member Elecaic Power Research Institute Pal0 Alto, CA 94303
-
A. H. Noyola, Student Member The University of Texas at Austin Austin, TX 78712
Keyords Harmonic Voltage Distortion, Distortion Minunuation, Active Filtering, Telephone Influence Factor, Power Quality.
NETWORK OBJECTIVE FUNCTIONS
[email protected] .- The impact of voltage harmonics on a power system CanbemlnlXWd by using active filters to inject distortion-cancelling currents. However, a network objective function must be specified before the optimum filter injection currents can be &tennined. This paper illustrates the application of a distortion-minimizingprocedure with each of the following four network correction strategies: total harmonic voltage distortion, telephone influence factor, motor loadloss function, and single-bus sine wave COrreCtion.
The problem that was formulated and solved in [2] was that of ~ in a minimizing the sum of squared harmonic voltages (in p e unit) network by injecting APLC nonsinusoidd currents at a given bus m. The corresponding objective function has the form
Sum of Squared Harmonic Voltages
2 h=2 k=l
where
INTRODUCTION Harmonic distortion is increasing in distribution power systems due to the proliferation of nonlinear distorting loads. Two methods are commonly used to hold distortion levels in check preventative measures such as encouraging the use of low-distortion loads, and mitigation measures such as passive or active filtering. Passive filtering has been the most frequently applied mitigation technique because it is relatively inexpensive and can be achieved by adding series chokes to power factor correction capacitom. Active power line conditioners ( A P E S )are relatively new filteaing devices that reduce distortion by injecting equal-but-opposite distortioncancellingcurrents [13. Unlike passive filters. APLCs can be precisely controlled to reduce distortion levels at a single bus or across an entire power system [2]. The objective of an APLC can be as simple as minimizing one voltage harmonic at one bus, or as complicated as minimizing the sum of squared voltage harmonics across a network. The authors previously developed an optimization procedure for minimizing the sum of squared harmonic voltages in networks with one APLC (either one balanced three-phase APLC, or one singlephase APLC) [2]. The procedure was illustrated using an example 12.5 kV distribution system. This paper extends the previously developed optimization procedure to include the followingdistortion correction strategies: 1. Total Harmonic Distortion (THD),
h k m: I,:
harmonicorder, bus number, APLCbusnumber, APLC injection current,
IVL I: voltage magnitude at bus k for harmonic h. The minimization problem was solved in [2] for unconstrained and constrainedAPLC current cases. Weighting constants can be added to (1) to generalize it for use with any of the four objective functions illustrated in this paper. The generalized form becomes the following weighted sum of squares objective function:
where ~ ( h ) : weighting constant for harmonic h,
P(k):
weighting constant for bus k.
---
Note that (1) is identical to (2) when W(h) = 1 for h = 2,3,4. , H, and when P(k) = 1 for k = 1, 2, 3, , K. The manner in which weighting constants and P can be incorporated into the minimization procedurepresented in [2] is given in the Appendix.
-
w
2. Telephone Influence Factor 0,
Total Harmonic Distortion
3. Motor Load Loss Function (MLL),
The total harmonic distortion (THD) of voltage at any bus k is defined as
.
4. Single-Bus Sine Wave Correction.
The contributions of this paper include a comparison of results using each strategy as well as examples showing that the location of an APLC is important in its ability to minimize network distortion. As with any active device, an APLC can have a detrimental effect on overall network distortion if it is not properly utilized.
(3)
THD can be incorporated into the minimization procedure in [2] by considering a network function that equals the sum of squared THD,'s, or
91 Sl4 400-2 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 Summer Meeting, San Diego, California, July 28 August 1, 1991. Manuscript submitted February 1, 1991; made available for printing July 1, 1991.
r
I
..
1 2
-
H
K
=h=2e k=lc -I V: O885-89?7~3.00y180' with respect to fundamental. This effect could be exploited in our algorithm (with some difficulty) by modifying objective functions (S), (Al), and (A2) to include separate terms for the squared differences between 5th harmonic voltage phase angles and 180'. Relative weightings for squared magnitudes and squared angular differences would be required. Considering the fact that harmonic voltage phase angles vary considerably and somewhat randomly, one possible correction strategy might be to force 5th harmonic voltage magnitudes to be as small as possible while simultaneously driving their phase angles as close to 180' as possible. Concerning item 1 in Dr. El-Amin's discussion, the optimization procedure assumes that harmonic impedances for the system are known. These impedances can be calculated using harmonics simulation programs such as [13], or they can be measured by observing the relationship between APLC injection currents and bus ; , voltages. In either case, (Al) has a derivative with respect to 1 providing that the impedance elements are finite.
If the impedances are unknown, then the relationships between harmonic voltages and APLC currents are unknown, and thus it is impossible to determine the best location for an APLC. Concerning item 2, it is true that improperly-controlled APLCs can excite system resonances in the same way that distorting loads excite resonances. Therefore, it is important to monitor a sufficient number of busses in a network to guarantee that resonances are detected. While we have not included constraints for individual busses in our optimization procedure, weighting factors P(k) in (A2) can be increased at problem busses until individual distortions are reduced to tolerable levels. Again, we would like to thank the discussers for their helpful comments and suggestions. Manuscript r e c e i v e d November 8 , 1991.
