Optimizing Power Flows in Lossy Polyphase Systems: Effects of Source Impedance

Transcription

1 Optimizing Power Flows in Lossy Polyphase Systems: Effects of Source Impedance HANOCH LEVARI, ALEKSANDAR M. STANKOVIĆ Department of Electrical and Computer Engineering, Northeastern University, Boston, USA The paper formulates the problem of optimizing power flows in polyphase systems with significant source (line) impedance. The problem is stated in a Hilbert space, and then solved explicitly for the case of a linear line and load. Properties of the solution are illustrated on the example of an asymmetrical threephase induction motor supplied with unbalanced nonsinusoidal voltages. While the optimal compensator current depends on explicit knowledge of numerous network and load parameters, the paper also explores an alternative closetooptimal solution which generalizes the Fryze compensator and relies only on knowledge of load voltage and current, thus being easier to implement in practice. 1 Introduction The role of compensation in power system efficiency optimization is to reduce the power delivered to the (equivalent) source (or line ) impedance, so that almost all the source power is delivered to the load (Fig. 1). A classical result by Fryze [1] states that when the voltage drop in the (equivalent) line is negligible in comparison with the load voltage, i.e., when then the smallest possible source (line) current (in rms) is where is the original real (average) power delivered to the load without compensation, and denotes the rms value of a waveform. Source Impedance Compensator Load Recently we have put compensation (with negligible voltage drop) in a convenient geometric (Hilbert space) setting, and determined optimal solutions under unequal line resistances and bandwidth limitation on the compensator current [2, 3, 4]; we summarize briefly some of these results in Sec. 3. However, when the source impedance becomes significant (namely, a few percent of the load impedance, or higher), the traditional Fryze current is no longer the smallest (by rms) line current that supplies the same real power to the load as the original load current. In that case every adjustment in the compensator current results in a change of voltage load, which in turn, requires further adjustments in compensator current. Thus, we need to formulate an optimization problem that explicitly accounts for the voltage drop in the line. In this paper we present such a formulation and provide an explicit expression for the optimal source current. We use an example to demonstrate the improvement achieved with our optimized compensator in a case with approximately 15% voltage drop in the line. It turns out that the optimized compensator current depends on many (possibly uncertain) parameters, such as line impedances and load characteristics over multiple frequencies. Thus, it is very desirable to obtain (closetooptimal) solutions that do not require so many parameters. We present in Sec. 4 an approach that generalizes the Fryze compensator and achieves very good results while relying only on knowledge of load voltage and current. 2 Problem description Figure 1: Load compensation in a power delivery system. Consider an phase system, i.e., a system with conductors ( wires ) in which the first are referenced either to a common ground, or to the st ( neutral ) conductor. Then we can define the dimensional voltage and current (row) vectors,, where

2 all currents have reference directions toward the load (Fig. 1). We use the Hilbert space terminology of [2] to formulate our objectives and derive our results. Thus and are row vectors representing load voltage and current, which we view as elements in a Hilbert space of phase, periodic, squareintegrable waveforms, with the inner product defined by where the superscript denotes transposition. For instance, in this terminology the rms value of the polyphase voltage is expressed as. The instantaneous power flow into the load is and its harmonic (i.e., Fourier) coefficients, are. Squareintegrable, periodic waveforms can be represented by a summable Fourier series, viz., where the row vector Fourier coefficients by (1a) are given (1b) Since is real, it follows from the conjugate symmetry property of the Fourier transform that, where the asterisk () denotes elementwise complex conjugation. We opt here to use onesided phasor arrays, which are obtained by stacking the vector Fourier coefficients into a long row vector, viz.,!!! " (2) so that #$%&' where the superscript & indicates conjugate transpose, and, % are the phasor arrays representing the waveforms,, respectively. Our longterm goal is the characterization of an optimal compensator current (Fig. 1) subject to constraints on the frequency content of as well as on the power flow into the compensator. We consider, in general, three distinct measures of optimality: ( Ohmic loss in the phase wires of the polyphase source line ()*,). This loss is a function of the polyphase line current and the real part of the source impedance. ( Ohmic loss in the neutral wire of the polyphase This loss is a function of the current and the resistance in the neutral wire. source line (,./)). ( Cost of the capacitor required for temporary energy storage during the cycle in the compensator. It is proportional to the peak instantaneous energy flow into the compensator (0/1). Thus, the selection of the best compensator current is a multicriterion optimization problem, with constraints imposed by physical limitations of the compensator circuitry, as explained below. The wellknown compensators of Fryze and Akagi Nabae, as well as the recently introduced family of PHM compensators [2], focus on a single cost function ()*,), and a single constraint (power flow into the compensator). Furthermore, these compensation schemes are optimized under the assumption of a negligible voltage drop across the source impedance (in comparison with the load voltage). We provide below a brief summary of these compensation methods, all of which are intended for nearlyideal lines, namely lines in which the voltage drop across the source impedance can be ignored. We then reformulate the singlecost/singleconstraint compensation problem to take into account the effects of significant source impedance. Next, we describe a closedform optimal solution to this problem in Sec. 3, and a simplified nearoptimal solution, based on a generalization of the Fryze approach, in Sec Nearlyideal line previous results The Fryze compensator minimizes only the line loss )*, (while ignoring 0/1 and,./)), subject to the constraint 2, which prevents average power flow into the compensator. The optimal compensator current is, where is the smallest current (by rms) that supplies the same dc component of as the load current. However, the energy storage cost 0/1 can be quite high for the Fryze compensator. An alternative approach to compensation is provided by the AkagiNabae instantaneous compensator. It minimizes the line loss )*, subject to the constraint 2, so that there is no instantaneous power flow into the compensator and, as a result, 0/1 2. The optimal compensator current is /, where / is the smallest current (by rms) that supplies the same as the load current. The price paid for achieving 0/1 2 with the instantaneous compensator is a possibly much higher ohmic line loss )*,. Thus, a basic problem is to obtain a desired tradeoff between power loss and hardware cost. The two extreme choices are the Fryze compensator, which has the lowest power loss, and the instantaneous compensator, which has the lowest capacitor cost (0/1 2 ). The family of PHMcompensators, which was introduced in [2], spans the range between those two extremes. The optimal com

3 pensator current is, where is the smallest current (by rms) that supplies the same power harmonics (up to the th harmonic) as the load current. In other words, the first harmonics of the power flow into the PHM compensator vanish (so at these frequencies the load requirement is met by the source). In 2 particular, corresponds to the Fryze compensator, while the instantaneous compensator is obtained for. Thus the costperformance tradeoff can be controlled by varying the index, as illustrated in [3]. All three methods of compensation Fryze, Akagi Nabae and PHM (which includes the former two as special cases) can be modified to incorporate an explicit bandwidth limitation on the compensator current [4]. When applied to the AkagiNabae compensator, this modification generalizes the results in [5] (which are restricted to sinusoidal waveforms and balanced three phase voltage) to arbitrary harmonic content, and arbitrary polyphase sources and loads. 2.2 Lossy line problem formulation We now turn to consider the effect of an equivalent (Thevenin) source impedance that gives rise to a nonnegligible voltage drop. We describe this case to show the complexities involved in optimization of energy systems involving long transmission lines. In this case the voltage across the load becomes dependent on the compensating current, as well as the source impedance and the characteristics of the load. In order to determine the polyphase load voltage and load current we need to consider the circuit equations $ ' and the load currentvoltage relation (3) % $ ' (4) $ ' Here is an operator that describes the voltagecurrent % relation of the source impedance, while $ ' denotes an operator that maps the voltage applied across the load % into the resulting current flow through the load. While $ ' can, in general, $ ' be nonlinear, it is reasonable to assume that is a linear timeinvariant operator. In the absence of a compensator, the load voltage and current satisfy the nonlinear equation $% $ '' (5) which involves the composition of the operators % $ ' and $ '. This equation has a unique solution, for every periodic squareintegrable %, under some mild constraints on $ ' $ ' and [6, Theorem 4]. Because now, we need to specify in general two fundamental constraints for the compensation problem: (i) the average power supplied to the load remains unaltered as we vary the compensating current, and (ii) no average power flows into the compensator. Additional constraints, such as a bandwidth limitation on the compensator current, may also be required. The optimal solution of this multiplecriterion, multiplyconstrained nonlinear optimization is still an open research problem. To facilitate a closedform solution, and to enhance our understanding of this problem, we choose to focus here on a singlecriterion/singleconstraint version, and assume a linear load. Consequently, the currentvoltage relation (4) can be conveniently expressed in the frequency domain, viz., % (6) where (resp. ) is the phasor array corresponding to the polyphase % waveform (resp. ), as defined in (2), and is a complexvalued admittance matrix representing the linear operator % $ ' in the frequency domain. The circuit equations (3) can also be translated into the frequency domain, viz., (7) where is a complexvalued diagonal matrix representing the source impedance. The power loss in this impedance is )*, & #$ ' (8a) & where the superscript denotes conjugate transposition. Similarly, the real power flowing out of the compensator is # $ &' (8b) The singlecriterion/singleconstraint optimization problem that we consider here is a frequencydomain equivalent of the timedomain formulation that we discussed and solved in [7]. In this version the objective is to minimize the line loss )*, &, subject to the (single) constraint # &' 2 $ (8c) by adjusting the compensated source current equivalently, the compensator current. The onetoone correspondence between these two currents is evident from the relation % % which follows directly from (6)(7). When the matrix % is nonsingular, this relation establishes a onetoone mapping between and. or,

4 3 Lossy line: optimal solution The singlecriterion/singleconstraint version of the compensation problem that was formulated in Sec. 2 translates into quadraticallyconstrained quadraticcost optimization. Indeed, from (6)(8) we deduce that )*, & (9a) & & & (9b) where is the power that would have been delivered to the load in the absence of a source impedance, viz., & % %& (9c) and is the phasor array associated with the source voltage. Notice that the matrix is complex valued, except when % is diagonal, which occurs for a linear timeinvariant load without (magnetic) phase coupling. When phase currents are coupled, as is the case of an electric motor, the matrix is block diagonal, with each block representing a single harmonic (as in the following example). Finally, is a normalized current phasor array, viz., where (9d) is a diagonal matrix consisting of the (positive) square roots of the diagonal elements of. Also & & (9e) (9f) Since both the line and the load are assumed passive, we have 2 and 2, so that the Hermitian matrix is also strictly positivedefinite. 3.1 Optimal compensator The line current that minimizes )*, for a prespecified value of can be determined, for instance, via the Lagrange multiplier method. It is given by the expression (10a) where the Lagrange multiplier is given by the & unique positive solution of the equation & (10b) In particular, if we opt to have 2, then is determined by solving the equation & & which is the frequency domain equivalent of [7, eq. (13b)]. Since the lefthandside of (10b) is a monotonically decreasing function of, this equation has a unique solution, which is relatively easy to determine numerically. We illustrate the utility of the optimal compensator (10) via a practically relevant example of an induction machine supplied by unbalanced voltages containing harmonics (fifth in our case). Loads of this type are very common in power systems, as it is often estimated that they consume up to a half of the total energy that is generated and transmitted in an electric power system. 3.2 Example a threephase unbalanced induction motor We consider the case of a threephase induction machine rated at 25, supplied by from an unbalanced source (variations of roughly 10% in magnitude and phase, with no zero sequence) that contains the fundamental and the fifth harmonic. For simplicity, we assume that phases are not coupled in the line an extension in this regard is straightforward. Numerical values of these parameters for the fundamental harmonic are taken from [8]. Rotorrelated parameters are frequency dependent due to skin effect, and they are estimated for the 5th harmonic using [9]. The induction motor admittance matrix (in Ohms) is computed using the method outlined in [10]. Standard models for the fundamental and the fifth harmonics are used, and the only neglected effects are those of magnetic saturation. The numerical values for this example are!!! 22!2!!!! " 2! diag % % 2 2 % with the blocks!"#$ %!##$&!"'( "!')( & "! % " % #! % % & "!!"'( % " % #"!')(! % % &!"#$ %!##$&!"'( "!')( & & "! % " % #! % % &!"#$ %!##$&, "! ## "!.#' & "!" % % $ & % "!"(#$ " #& "!"(#$ " #& "! ## "!.#' & % "! % $ & % "!" % % "! % $ & "!"(#$ "! " #& "! ## "!.#' & Notice that we have included only the first and fifth harmonic information in % (and, ), since the remaining diagonal blocks of % are all zero. * *

5 In the absence of a compensator, the power loss in the source impedance is, and the power delivered to the load is 2 2. Also, the power that would be delivered to the load in the absence of source impedance is 2. With the optimal compensator in place (adjusted to obtain 2 ), the power loss in the source impedance is reduced to, while the power delivered to the load is increased to. This is quite realistic, as successful compensation increases the load voltages, which in turn increases the load power. The overall efficiency (from the voltage source to the load) increases from 89.2% to 92.9%. In utility practice, this would lead to a transformer tap adjustment, which would lower the load voltage (and power), and further reduce the line losses, as currents would decrease as well. Perhaps the main use of power flow optimization is to serve as an input to engineering design, as it establishes the limits of achievable performance. The role of a design engineer is then to decide if the energy savings justify the inclusion of the compensator. 4 Lossy line: nearoptimal solution The optimal solution (10) relies on explicit knowledge of numerous network and load parameters, namely the elements of the phasor arrays associated with the source voltage ( ), source impedance ( ), as well as the % elements of the load admittance matrix. When such parameters are difficult to estimate accurately and/or tend to vary in time, the utility of the optimal expression (10) can become severely compromised. One alternative is to develop suboptimal compensator schemes that rely on a relatively small number of design parameters. We propose below one such efficient nearoptimal compensation method. We consider a generalization of the Fryze compensation rule: (i) the compensated line current is a linear combination of the load voltage and its Hilbert transform, viz., (11a) where, are realvalued scaling coefficients, and (ii) the voltage is the load voltage in the presence of the compensator. In the frequency domain this translates into (11b) We shall call this compensation method quadrature Fryze because the Hilbert transform imparts a 2 phase delay to. The utility of the proposed compensation scheme is demonstrated by Figs. 23, which display the )*, vs. tradeoff curve for both the optimal compen sator and several versions of the quadrature Fryze compensator. It is evident from the plots that the latter 2 is reasonably close to the optimum even when (dashed P line (W) P (W) comp Figure 2: Tradeoff between line losses and compensator power output, showing optimal compensator (solid), quadrature Fryze with 2 (dashed) and quadrature Fryze with arg! uncompensated load. P line (W) (dashdot). Circle indicates P comp (W) Figure 3: Tradeoff between line losses and compensator power output, showing optimal compensator (solid), quadrature Fryze with 2 (dashed) and quadrature Fryze with arg (dashdot). Circle indicates uncompensated load. line in both figures). Moreover, adjusting the value of (or, equivalently, the value of arg ) allows further reduction of the performance gap between the quadrature Fryze compensator and the optimal one. This gap is a monotone function of the voltage drop across the source impedance as the effect of source impedance increases, the difference between using 2, as compared with an optimal choice for, becomes more noticeable.

6 The two (realvalued) degrees of freedom in (11a) can be used to satisfy two independent constraints, such as setting the values of (say, to zero) and )/. For instance, we observe that )/ # &' $ # which determines the value of #. 5 Concluding remarks The paper begins by observing that classical solutions for optimized load compensation neglect the effects of source (line) impedance on the voltage drop between the source and the load. When this impedance is significant, the compensation problem becomes much more complicated, as adjustments in the compensator currents result in changes in load voltage, possibly necessitating iterative adjustments. In this paper we formulate the compensation problem in the general case using a Hilbert space framework, and explicitly solve in the frequency domain for the optimal source current in the case of linear line and load. We illustrate properties of the solution on the example of an asymmetrical threephase induction motor supplied with unbalanced nonsinusoidal voltages. The optimized compensator current depends on explicit knowledge of numerous network and load parameters, such as source voltages, line impedances and load characteristics over multiple frequencies. This naturally serves as a motivation to explore closetooptimal solutions that do not require so many parameters. We present an approach that generalizes the Fryze compensator and achieves very good results while relying only on knowledge of load voltage and current, which may be obtained during operation. REFERENCES 1. S. Fryze: Wirk, Blind, und Scheinleistung in elctrischen Stromkreisen mit nichtsinusformigem Verlauf von Strom und Spannung, Elektrotech. Z., Vol. 53, No. 25, June 1932, pp H. LevAri, A.M. Stanković: Hilbert Space Techniques for Modeling and Compensation of Reactive Power in Energy Processing Systems, IEEE Trans. Circuits & Systems, Vol. 50, No. 4, Apr. 2003, pp H. LevAri, A.M. Stanković, K. Xu, M.M. Perišić: Hilbert Space Techniques for Evaluating Tradeoffs in Reactive Power Compensation, Proceedings of the IEEE International Symposium on Circuits and Systems, Vancouver, Canada, Vol. V, May 2004, pp H. LevAri, A.M. Stanković: Hilbert Space Techniques for Reactive Power Compensation with Limited Current Bandwidth, Proceedings of the IEEE International Symposium on Circuits and Systems, Vancouver, Canada, Vol. V, May 2004, pp J.L. Willems, D. Aeyels: New Decomposition for 3 Phase Currents in Power Systems, IEE Proceedings C, Vol. 140, No. 4, July 1993, pp I.W. Sandberg: On Truncation Techniques in the Approximate Analysis of Periodically TimeVarying Nonlinear Networks, IEEE Transactions on Circuit Theory, Vol. 11, No. 2, June 1964, pp H. LevAri, A.M. Stanković: Optimization of Power Flows in Polyphase Systems with Significant Source Impedance, 37th North American Power Symposium, Ames, IA, Oct W.H. Kersting: Causes and Effects of Unbalanced Voltages Serving an Induction Motor, Rural Electric Power Conference, 2000, pp. B3/1B3/8. 9. T. Kataoka, Y. Kandatsu, T. Akasaka: Measurement of Equivalent Circuit Parameters of Inverter Fed Induction Motors, IEEE Transactions on Magnetics, Vol. MAG23, No. 5, Sept. 1987, pp W.H. Kersting, W.H. Phillips: Phase Frame Analysis of the Effects of Voltage Unbalance on Induction Machines, IEEE Transactions on Industry Applications, Vol. 33, No. 2, March/April 1997, pp