of the Accuracy of the LTI System Model
Zbigniew Staroszczyk
Warsaw University of Technology Warsaw, Poland stazby@iem.pw.edu.pl
Abstract—The paper presents experimental verification of the quality of LTI power grid model valid for the frequency range of 8 kHz. The single PCC was investigated so the paper is the local, University grid investigations oriented, however concerns two general important problems: one is the absolute accuracy of real grid impedance measurements to which no direct reference data are accessible, the second is the validity of the LTI model, which is used for description of the complex structure composed of nonlinear and switched power installations. The LTI models form very interesting and useful approximation of the complex reality and have meaningful time and frequency domain interpretation.
When used for power grids they deal with their Norton/Thevenin equivalent models. Two components of such a model: impedance and system side sources describe the grid and are discussed in the paper. Rich experimental data were referenced to the structural model of the grid so the paper gives evaluative information on the quality of ad-choc grid modeling. Interesting, incremental method for the estimation of absolute impedance errors was proposed and used for improvement of the implemented DSP.
Time varying nature of the investigated grid impedance was presented, and system errors with implementation of LTI model to such a time-varying PCC description are discussed.
Keywords-component; PCC impedance ,LTI grid model, LPTV grid model, grid identification, grid modeling
I. INTRODUCTION
The impedance is commonly used for power grid current/voltage relations description in the user accessible PCC when system harmonics are considered. It delivers physical compact information on the power system which can be treated as a black box. The “black box” power system description is necessary if there is a lack or limited knowledge
on the power grid structure and its loads. In such situations the impedance identification / measurement experiment is a principal source of the information on the grid [1],[2],[5].
Impedance and voltage observations in the PCC deliver the linear model of the grid which has plenty of useful implementations [5],[6],[7].
The power grid modeling is essential for virtual experiments in which dangerous and untypical power system states have to be investigated. Even for standard situations the experiment on the real grid is tedious, costly and, in controlled and limited degree, unsafe to the system.
In power grid behavior modeling the mathematical or structural approaches can be used.
• With mathematical approach the frequency (linear case) [2],[3],[4],[5] or time domain (linear and time variant case) [8],[9],[10] models of the whole grid, treated as a unity, are used. The data to both models originate from experiments.
The physical background of the model assures its correctness, and easy to perform validation of its quality, as there is a physical reference to model generated data – the real grid and measuring/signal processing instrumentation for its observation.
• With a structural approach the grid model is built from verified partial models of power grid components, connected in the network with a structure of the real grid.
For that modeling the structure and components data of the real grid are required, but there is a risk that the model can be totally incorrect for real system description if some settings of its parameters, or assumed structure are wrong.
That menace creates continuous development of better and more flexible multiparametric models of the power system components, which are implemented as libraries in software
978-1-4244-5172-2/09/$26.00 ©2009 IEEE
Impedance in Voltage-Current Relations Description of the Power System PCC - Experimental Investigations
Institute of the Theory of Electrical Engineering, Information and Measurement Systems,
simulation packages (SimPowerSystems, ATP/EMTP, DIgSILENT, PSpice). Bad initial setting of such, often too complex components models and limited knowledge of the system structure is the principal source of the structural model inaccuracy for the real grid description.
The mixed method of grid modeling in which the same grid is treated with both approaches is presented in the paper. Even for relatively simple structure of the University grid there exist significant discrepancy between is structural end experimental model. Additionally in real systems time varying properties are observed. They deteriorate usefulness of the LTI modeling and confirm the importance of measurements made on a real grids if reliable and accurate information on power system is required.
II. PROBLEM, METODS AND INSTRUMENTATION
The paper is the University grid oriented, in which experiments on the real power grid laboratory PCC, are presented. The investigated PCC is supplied from three phase 1000 kVA, 15.75/0.4 kV DYng transformer. Due to secondary Yng connection three single phase coupled subsystems were accessible for experiments. As they have similar properties, in all presented experiments the PCC of the same, selected randomly phase, was used to assure comparability of observations.
Fig. 1 Simplified single phase structural model of the investigated grid – single phase scaled three phase load levels (P, Q)
The quick and initial information on the grid is obtained with the use of its simplified model (Fig 1, 2) and GIHA (Grid Impedance/Harmonics Analyzer) observations (Fig. 3, 4) [4].
More advanced information, presented in the paper was obtained with the DSP of rough signals delivered by GIHA to external computer. The signals observation frequency band, due to limitations of GIHA, is restricted to 8 kHz.
The structural model of the grid from Fig. 1 can be treated analytically (classical circuit theory, or with the Impedance Measurement tool built in into Simulink). The structure of the grid was taken from University grid documentation, the parameters of all system elements (supplying lines, transformer, capacitor bank) were calculated basing on catalogue data.
In the first experiment the laboratory PCC loaded with capacitor C=40 µF was tested. The simulated system was
slightly loaded (10% of the peak power) to approximate real system loading conditions (see Fig. 1 parameters settings). The loading capacitor creates parallel resonance with supplying lines inductance. That resonance indirectly delivers estimative information on the system side PCC equivalent impedance [4],[9].
For such testing conditions in the calculated (Fig. 3) and measured (Fig 2) impedance three PCC resonances are observed, coming from the paths marked with 1 (parallel:
transformer inductance lt – reactive power compensating capacitors Qc), 2 (serial: compensating capacitors Qc – supplying lines inductance lez, llz) and 3 (parallel: supplying lines inductance lez, llz – PCC loading capacitor C413) in Fig.
1.
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0 2 4 6
Mag. (ohms)
0 1000 2000 3000 4000 5000 6000 7000 8000
-100 -50 0 50 100
Phase (degrees)
Frequency (Hz) 1
2 transformer/ 3 Cbank parall. res.
EE suppl. lines/
Cbank serial res.
Lab. Cload/EE suppl. lines parallel resonance
Fig. 2 Loaded PCC (40μF) impedance plots – simulated structural model of the grid from Fig. 1
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0 0.5 1 1.5 2 2.5
magn. [Ohm]
0 1000 2000 3000 4000 5000 6000 7000 8000
-200 -100 0 100
frequency [Hz]
phase [deg]
Fig. 3 Loaded PCC (40μF) impedance plots – real system observations
There is small discrepancy between the structural model and physical system data (resonance frequencies/dampings) with better suit of level (magnitude) then phase characteristics.
Fortunately the general properties of the real grid are preserved in its structural model. It is worth to mention that GIHA impedance data originate from PCC current impulse excitation and advanced DSP of PCC voltage and current
signals [3],[4]. The nature of the GIHA impedance (Fig. 3) is quite different from the nature of analytical/structural impedance (Fig. 2) of the Fig. 1 virtual grid. Nevertheless the starting (documentation and catalogue based) model of the grid is not so bad.
As it has been mentioned, the GIHA and Simulink model data form the staring information on the real grid. The more advanced data are obtained from load reconfiguration experiments followed with the more advanced, off-line DSP of internal GIHA registered signals. The GIHA itself has built-in advanced DSP and user-frindly GUI (Fig. 4) allowing for quick experiments on a real power system. The GIHA DSP is impedance and harmonics oriented. Except of impedance the device is able to trace selected power system harmonics and present them on logarithmic level, classical phase scale, phase diagrams. The PCC harmonics presentation is enriched with system side voltage sources phasor calculation and presentation in which the PCC voltage and current harmonics and impedance information is combined into very useful information on power system PCC side generated harmonics.
That way the polluting/polluter problem is solved, without the need for disconnection of the polluter from the grid [9].
All presented functions of GIHA, supplemented with additional DSP and numerous experiments served to verify the quality of the impedance description of the real grid. In the paper the case study of the University laboratory PCC is presented. The discussion concerns the quality of the real power grid description with the use of LTI tools: impedance and Thevenin sources. The question was – how reliable is impedance information on the real power grid, how well it explains voltage/current relations in PCC and what is its accuracy and drawbacks for power system description.
III. INITIAL EXPERIMENTS
.
The experiments started with the investigation of unloaded PCC.
Fig. 4 Screen of the grid impedance/harmonics analyzer (GIHA) – unloaded PCC observations
Such a PCC was described in previous publications oriented to development of DSP algorithms which were implemented in
GIHA [4].Unloaded PCC represents the real grid properties in a clean form, not affected by the customer loads. The observed harmonics voltages are equal to system side Thevenin harmonics sources, and can serve as the reference to loaded states observations [2][3][5],[9].
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magn. [Ohm]
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-50 0 50 100 150
phase [deg]
frequency [Hz]
Fig. 5 Impedance details of the unloaded PCC – GIHA recorded signals
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Mag. (ohms)
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Phase (degrees)
Frequency (Hz)
Fig. 6 Impedance of the unloaded PCC – simulated model data
The screen of GIHA in Fig. 4 presents the impedance plot of the unloaded PCC and voltage harmonics phasors observed each 5s in a minute time period (take care to the logarithmic level scale of the polar plot). The original harmonics presentation was reedited – to make the picture more clear harmonics positions were clustered and additionally numbered. The left bottom corner of the original harmonics label points out the mean position of phasor location.
The permanent harmonics migration is a typical property of real power systems. For the presented study the relatively stable are only positions of high level, low order harmonics (3- 9). Some of them are calm (3-9, 21, 27, 29), some
“wondering” (11, 13, 23) – that kind of power system nonstationariness is the main problem for signal processing algorithms [6],[10]. We can assume that a single experiment on the grid takes a minute, and is aimed to deliver information on the instant state of the grid. For such a long experiment the
grid is nonstationary, so the observed grid changes influence (deteriorate) that instant state grid descriptor.
In Fig. 5 and 6 the impedance of unloaded PCC of the real and simulated system is presented. There are no doubts that experimental data are more reliable, as the starting parameters for the structural model are relatively imprecise (unknown configuration of customer loads, unknown state of reactive power compensating capacitor switched banks etc). If we accept the measurements as the basic source of the truth information on the grid, the open question is what is the quality of that measured PCC impedance.
The system structured model can be improved by tuning its parameters. That improvement requires repeated passes:
experiment >model correction >simulation where the essential is the verifying experiment on the real grid with it reliable measurements. That procedure is not the aim of the paper, so is not presented here. There was no effort made for improvement the quality of the structural model of the grid.
The paper focus is on the reliability of real system observations.
IV. GRID MEASURED MODEL QUALITY EVALUATION
The experimental model is the observed/measured impedance based model of the grid (Fig 3, 5). To verify the measured impedance quality the alternative, verified grid observation tools or the absolute power grid physical model (grid standard) should be used. Such costly solutions were not accessible for experiments.
0 0.5 1 1.5 2 2.5 3 3.5 4
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
real (Z) [Ohm]
imag (Z) [Ohm]
grid
grid + R (2.7)
Fig. 7 Impedance trajectory of the unloaded orginal and modified PCC, restricted observation frequency range 0 – 4 kHz.
Two procedures were used instead.
• The first is incremental. The background for it forms the assumption that if the unknown impedance is well determined, so are its controlled and known modifications.
Two observations of impedance for natural and modified grid configurations are then compared and referenced.
• The second relies on the fact, that if the true system impedance is known, it allows for proper determination of internal system voltage drops on PCC supplying lines if PCC loading currents are known. That way the system side
voltage sources can be determined from loaded state PPC voltage/current signals observation. The discrepancy between observed (on unloaded PCC) and calculated system side voltage sources serve here as a measure of impedance errors.
A. Incremental method
To illustrate the method the simplest modification of the grid with serial resistor (2.7 Ω) was made. In Fig 7, 8 and 9 three different representations of original and new, modified grid impedance, are shown.
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0 0.5 1 1.5 2 2.5 3 3.5
frequency [Hz]
abs (Z) [Ohm]
grid grid + R (2.7)
R calculated
Fig. 8 Impedance level (magnitude) of the unloaded, original and modified PCC, magnitude plot of differential impedance (Rcalculated).
In polar plot (Fig. 7) the impedance trajectory for original and modified PCC in the frequency range 4 kHz is presented.
Because of serial resistor, the modified PCC drawing should be shifted right 2.7 Ω. The observed shift is rather 2.5 Ω, depends on frequency, and contains imaginary component.
0 1000 2000 3000 4000 5000 6000 7000 8000
-80 -60 -40 -20 0 20 40 60 80
frequency [Hz]
phase [deg]
grid
grid + R (2.7)
R calculated
Fig. 9 Phase characteristics of the unloaded, original and modified PCC, phase plot of the differential impedance (Rcalculated).
Frequency dependent plots (real/imag, magnitude/phase) give more information on the nature of observed errors. The most informative are magnitude/phase plots shown in Fig. 8 and 9.
enriched with the differential impedance (Rcalculated)
presentation. As it was mentioned its phase should be zero, as it origins from resistor, but is not - see Fig. 9.
The linear phase shift observed for the calculated resistor is caused by multichannel non simultaneous sampling of GIHA.
For limit frequency of 8 kHz the shift is -57 deg ( 1 rd, Fig. 8), so the corrective factor to GIHA impedance observations should be 2.7/2.5 exp(j*f/8000) for frequency given in Hz.
With that correction Fig. 3 and Fig. 5 observed impedance patterns get phase properties closer to those of a structural model.
With the incremental method absolute corrections to the impedance observations can be made, as the absolute value of the grid modifying component is known and has his absolute reflection in impedance data.
B. Voltage sources identification method
That method touches physical properties of the impedance and backgrounds of linear model of the grid. Three grid observations are compared: two of them are experimental (measured), one is calculated. Here there is not direct impedance observation (as it was in the case A), but voltage harmonics phasor observations for the open (unloaded) PCC, and the PCC loaded with known (measured) distorted currents.
The spectral information of PCC voltage and current signals serves to find system side harmonics – equivalent system side Thevenin voltage sources.
Fig. 10 Unloaded and loaded PCC voltage harmonics observations, incremental vectors marked with lines
In Fig. 10 harmonics voltage signal phasors observed in the PCC for unloaded and loaded conditions are presented. The unloaded harmonics are marked with a black dots as they form Thevenin voltage sources of the investigated grid model and can serve as power grid LTI model reference data. As the PCC load the thyristor controlled rectifier (with a pure resistive load) is used. Distorted load currents generate voltage drops
on internal impedance of the grid (and PCC), so the loaded voltage PCC harmonics get new positions (Fig 10).
Fig. 11 Calculated and observed PPC system side harmonics sources
The lines in Fig 10 show how big the voltage drop is on marked harmonics so give the knowledge on grid stiffness for different frequencies. One can observe that there is no marked reference for higher order harmonics in Fig. 10 (n>20). In reality it is, but in the middle of the plot, as for unloaded PCC state those harmonics levels are very low (<20 mV) – take care to the logarithmic level scale of the plot.
In Fig 11 on the background of the unloaded PCC reference harmonics the real-time calculated Thevenin sources are shown. The phasors positions are calculated from loaded PCC harmonics observations (Fig 10) and the impedance information. There is almost ideal correspondence between calculated and observed system side sources. Those observations confirm that the linear model of the system is valid, as the same results (Fig 11) are obtained for different loadings of the PCC (changes of peak current levels and firing angles), except of situation presented in Fig. 12.
It was observed that a good quality of Thevenin sources estimation is not assured for situations when the load current rapidly changes on tops of power system voltage levels. For such “top level” loadings there is a discrepancy between GIHA calculated and observed sources on medium frequency harmonics (Fig. 12, n=5-17). That is not the case of PCC nonlinearity, as situation does not change if different level/the same shape loading currents are used. That is a case of system time-variance in which power system nonlinear components working points follow the system voltage changes making the grid time-varying with doubled fundamental frequency [8],[9],[10]. The impedance measurement procedure built in into GIHA is based on PCC shortcuts in the regions of system voltage zero-crossings [3], [4]. For the case of observed errors (Fig.12) the load excitation is shifted 5 ms and deals with a different “top voltage” state of the time-varying grid.
Fig. 12 Erroneous sources identification for time varying grid
Two different PCC impedance patterns observed for zero- crossing and top voltage located impulse excitations, presented in Fig. 13) explain Fig. 11 and 12 discrepancy.
Fig. 13 Two impedance patterns of a real grid: zero-crossing (top), and peak voltage regions (bottom) excitations
The physical provenance of the observed, time-varying change of impedance is currently not well explained. Some damping of the resonance results, without doubts, from higher system loadings for top level voltages. Total resonance cancellation (Fig. 13 bottom) is however difficult to explain and needs future investigations.
V. CONCLUSIONS
In the paper the quality evaluation of LTI model of the real power grid was made. It was pointed out that with the use of
controlled reconfiguration of the PCC it is possible to evaluate the quality of impedance determination of the power grid in real situations even if the impedance is unknown and can not be measured with the use of alternative methods. In the presented case (section IVA discussion) not only errors of the phase determination were detected but the source of them was found and the method of their correction was proposed. The presented measuring data accompanied with parallel simulations inform the reader how accurate is the LTI model of the power grid obtained with the DSP built in into grid virtual impedance/harmonics analyzer (GIHA).
The section IVB discussion touches the problem of non-LTI power grid behavior, which can not be solved with the corrections of DSP, but requires time varying approach to the grid description. All presented data in Fig 10-13 were double checked with the alternative system excitation source not described here [9],[10], so the power grid time variance is the fact, existing at least in the PCC accessible for presented experiments.
The main value of paper is in its experimental orientation. All presented, discussed and solved problems concern real, not simulated only, power grid. The observed grid time-variance is of great importance for power system description, not discussed in the current, simulation-mainly oriented, literature.
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