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Load modeling at electric power distribution substations using dynamic load parameters estimation
Electrical Power and Energy Systems 26 (2004) 805–811 www.elsevier.com/locate/ijepes
Load modeling at electric power distribution substations using dynamic load parameters estimation Lia Toledo Moreira Mota, Alexandre Assis Mota* UNICAMP-FEEC-DSEE, CEP 13081-970, Campinas, Sa˜o Paulo, Brazil Received 6 March 2002; revised 5 April 2004; accepted 8 July 2004
Abstract This paper is focused on electric power distribution substations load modeling using dynamic load parameters estimation. The load parameters are estimated using two models: the exponential and the ZIP load models. Since the load bus voltage and parameters are known one can determine the active and reactive power injections of this bus and include these pseudo-measurements in the state estimation in order to improve observability and estimation accuracy. The dynamic parameter estimation is developed using the weighted least squares method in a recursive form and the tests are carried out based on actual measurements. It is shown that the estimated parameters (for both load models) at a distribution substation are valid, since the obtained active and reactive power residuals are very close to zero. q 2004 Elsevier Ltd. All rights reserved. Keywords: Load modeling; Parameter estimation; Electric power systems
1. Introduction Load modeling has a significant influence on power systems operation, simulation and analysis. It is shown, for example, that the performance of the state estimation function, which plays a critical role in electric power systems operations, is affected by the power network parameters, including load parameters, and that the presence of parameter errors can lead to unreliable state estimation results [1,2]. Usually, the power system loads are modeled as constants. However, this kind of model is inadequate for some studies like power system dynamic studies and voltage collapse studies. Moreover, the unbalance between the power generation and the power demand, very common nowadays, reduces the systems security margins and increases the risks to safe operation. In order to reduce these risks, system studies have to be developed with better * Corresponding author. Tel.: C55-019-37883708; fax: C55-01932891395. E-mail address: [email protected] (A.A. Mota). 0142-0615/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2004.07.002
models for the systems components including better load models [3]. This paper is focused on electric power distribution substations load modeling with the estimation of the load parameters that represent these substations. Besides, this work deals with the parameter estimation problem in a dynamic way allowing a data bank construction (with the results of the prediction step) that could be used in the security analysis problem and that provides an increase of the redundancy related to the parameters to be estimated. Since the load bus voltage and parameters are known one can determine the active and reactive power injections of this bus and include these pseudo-measurements in the state estimation in order to improve both observability and estimation accuracy [10,11,13,14]. In this work, load parameters are estimated using two different models: the exponential load model and the ZIP (or polynomial) load model, both described in Section 2. The dynamic load parameters estimation is developed using the weighted least squares (WLS) method in a recursive form, which is presented in Section 3.
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L.T.M. Mota, A.A. Mota / Electrical Power and Energy Systems 26 (2004) 805–811
The tests and results of a substation dynamic parameter estimation are shown in Section 4.
2. Load models A load model is a mathematical representation of the relation between the active or reactive power (or the current injection) in a load bus and the complex voltage of this same bus [4]. Load modeling has been carried out based on two different approaches. The first approach measures the voltage and frequency sensitivity of the active and reactive powers at substations [5]. The second approach constructs a composite load model for a given substation, based on the mix of load classes served by this substation. In this paper, the load modeling problem is treated using the first approach. The most commonly load models found in the literature [9] are described as follows: Constant impedance: In the constant impedance model, the active and reactive power injections at a given load bus vary directly with the square of the nodal voltage magnitude. This model is also called constant admitance model: P Z f ðV 2 Þ
(1)
where P is the active power injection and V is the voltage magnitude at the load bus. Constant current: In this model, the active and reactive power injections at a given load bus vary directly with the nodal voltage magnitude: P Z f ðVÞ
(2)
Constant power: Here, the power of the load bus is assumed to be constant and does not vary with the nodal voltage magnitude: PZk
(3)
where k is a constant. In these three load models (constant impedance, constant current and constant power) the reactive power injection can also be modeled using the Eqs. (1)–(3). Frequency dependent model: In this case, the active and reactive power injections of the load bus are related to the bus voltage frequency through an equation as follows. Factor Z ½1 C af $ðf K f0 Þ;
(4)
where: † af is the model sensitivity parameter; † f is the nodal voltage frequency; † f0 is the nominal frequency. In this work, other commonly used load models are implemented: the exponential and the ZIP load models. These models are described in Sections 2.1 and 2.2 [6,7].
2.1. ZIP (polynomial) load model In the ZIP load model, the active and reactive power injections comprise three components: a constant impedance (Z), a constant current (I) and a constant power injection (P): 2 V V C c$ (5) PL Z P0 $ a C b$ V0 V0 2 V V QL Z Q0 $ d C e$ Cf$ V0 V0
(6)
where a, b, c, d, e and f are the load parameters to be estimated, PL is the active power injection, QL is the reactive power injection, and P0, Q0 and V0 are the nominal values of the active and reactive power injections and the voltage magnitude, respectively. 2.2. Exponential load model In the exponential load model [12], the active and reactive power injections of the load bus are related to the bus voltage through an exponential function: a V P L Z P0 $ (7) V0 QL Z Q0 $
V V0
b (8)
where PL, QL, P0, Q0 and V0 were described in the ZIP load model and a and b are the parameters to be estimated.
3. Dynamic load parameters estimation Differently of the static estimation, the dynamic estimation requires system information from past scans of measurements, like for example, the estimated augmented state vector of the last scan. It is an augmented state vector because the load parameters are also included in the conventional state vector. Fig. 1 illustrates the data from the last scan of measurement (tiK1) required for actual parameters estimation (scan ti). In this figure, z represents the measurement vector, Rz is the covariance matrix of the measurements errors, x is the augmented state vector and Rx is the covariance matrix of the state (and parameters) errors. The dynamic estimation of load parameters is implemented using the WLS method in a recursive form. For each scan of measurements one can obtain one estimated parameter vector with the estimated parameters of the adopted load model (exponential or ZIP). The algorithm used for the WLS (recursive) implementation is described as follows. In this algorithm the superior indices
L.T.M. Mota, A.A. Mota / Electrical Power and Energy Systems 26 (2004) 805–811
807
and Rx ðtiC1 Þ Z ðGn ðti ÞÞK1 :
(14)
^ i Þ is the estimated augmented state vector at where xðt scan ti. It is important to notice that the recursivity of the WLS method is incorporated through this last step (step 7 or prediction step).
4. Tests and results Fig. 1. Information required for the parameter estimation of the scan ti.
(n) are related to the iterations of the process for the scan of measurement ti [7]. Step 1. Initialize the iteration counter. Step 2. Compute the gain matrix Gy ðti Þ : T
Gn ðti Þ Z Hn ðti Þ$ðRz ðti ÞÞK1 $Hn ðti Þ
(9)
where H is the system augmented Jacobian matrix. H is an augmented matrix because the load parameters are also included in the state vector; Step 3. Update the Dzn ðti Þ vector: Dzn ðti Þ Z zðti Þ K z^nK1 ðti Þ;
(10)
nK1
where z^ ðti Þ represents the estimated measurement vector that is computed using the estimated state vector of the last iteration. Step 4. Update the Dxn ðti Þ vector: y
y
yT
y
Dx ðti Þ Z ðG ðti ÞÞ $ðH ðti Þ$ðRz ðti ÞÞ $Dz ðti ÞÞ K1
K1
(11)
Step 5. Update the estimated state vector: xn ðti Þ Z xnK1 ðti Þ C Dxn ðti Þ
(12)
Step 6. Convergence test: † if Dxn ðti Þ! tolerance or if the number of iterations (n) has reached the adopted maximum number (nmax), then the estimated state vector of the tith scan is given by the xn ðti Þ vector. As the estimated vector is an augmented state vector it includes the estimated load parameters for the scan of measurement ti. The iterative process is over. Proceed to step 7; † if Dxn ðti ÞR tolerance and if n!nmax, update the iteration counter (nZnC1) and the augmented Jacobian matrix, based on the updated estimated state vector, and go back to step 2.
4.1. The adopted system The system used for the tests was adapted from [8] and is described in Fig. 2, which shows that the load bus has one active power, one reactive power and one voltage measurements. 4.2. Observability problem A system is considered to be observable if there are sufficient measurements and if these measurements are adequately distributed to obtain an unique solution for the state estimation problem [8]. In order to obtain this unique solution the Jacobian matrix (H) must have a full rank or the gain matrix (GZHT$H) must be nonsingular what guarantees the system observability. For the exponential load model, the Jacobian matrix is: 2
vP 6 vV 6 6 H Z 6 vQ 6 4 vV 1
vP va 0 0
3 0 7 7 7 vQ 7 7 vb 5 0
In this case, the H matrix has full rank (rank 3) and the gain matrix (G) is nonsingular, what guarantees the system observability. For the ZIP load model,
Step 7. Prediction step: Update the state vector and its covariance matrix (Rx) for the next scan of measurements: ^ iÞ xðtiC1 Þ Z xðt
(13)
(15)
Fig. 2. System used for the tests.
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L.T.M. Mota, A.A. Mota / Electrical Power and Energy Systems 26 (2004) 805–811
the Jacobian matrix is: 2 vP vP vP vP 6 vV 6 H Z 6 vQ 4 vV 1
va
vb
vc
0
0
0
0
0
0
0
0
vQ vd 0
vQ ve 0
0
3
7 7 vQ 7 5 vf 0
(16)
The H matrix of the ZIP model has not a full rank (rank 3) and the gain matrix (G) is singular, and so the system is not observable. In order to make the system observable for the application of the ZIP model, pseudo-measurements are incorporated to the measurement vector (z). In this work, these pseudo-measurements are the estimated parameters (a, b, c, d, e and f) of the last scan (tiK1). Thus, the augmented measurement vector, with the additional measurements (pseudomeasurements) is: T zZ P Q V a b c d e f (17) And the augmented Jacobian matrix is: 3 2 vP vP vP vP 0 0 0 7 6 vV va vb vc 7 6 vQ vQ vQ 7 6 vQ 7 6 0 0 0 6 vV vd ve vf 7 7 6 0 0 0 0 0 0 7 6 1 7 6 6 0 1 0 0 0 0 0 7 7 6 7 H Z6 7 6 0 0 1 0 0 0 0 7 6 7 6 7 6 0 0 0 1 0 0 0 7 6 7 6 6 0 0 0 0 1 0 0 7 7 6 7 6 0 0 0 0 1 0 5 4 0 0
0
0
0
0
0
Fig. 4. Reactive power injection of Pinhal substation—1:00 a.m. of 05/Jun/1994 to 1:00 a.m. of 06/Jun/1994.
nonsingular and so the system can now be considered as observable. 4.3. Actual data
(18)
1
One can notice that the new Jacobian matrix has full rank (rank 7) and the new gain matrix (HT$H) is
Fig. 3. Active power injection of Pinhal substation—1:00 a.m. of 05/Jun/1994 to 1:00 a.m. of 06/Jun/1994.
The simulations were based on actual data from the Pinhal substation of CPFL (Companhia Paulista de Forc¸a e Luz) related to voltage magnitude, active and reactive power injections measurements, collected in periods of 1 h from 1:00 a.m. of 05/Jun/1994 to 1:00 a.m. of 06/Jun/1994. The measurements of active and reactive power injections are illustrated in Figs. 3 and 4. Figs. 5 and 6 show the evolution of the estimated parameters a and b for the exponential load model. Figs. 7–9 show that the error between the actual measurement and the estimated measurement (in the case of voltage and active and reactive power injections), or the residual, is close to zero, what demonstrates that the estimated parameters are valid for the measurements adopted and for the exponential
Fig. 5. Parameter a evolution—exponential load model.
L.T.M. Mota, A.A. Mota / Electrical Power and Energy Systems 26 (2004) 805–811
Fig. 6. Parameter b evolution—exponential load model.
Fig. 7. Residual of the active power—exponential load model.
Fig. 8. Residual of the reactive power—exponential load model.
809
Fig. 9. Residual of the voltage magnitude—exponential load model.
load model. Similar results were obtained using the ZIP load model. Figs. 10–15 show the evolution of the estimated parameters a, b, c, d, e and f for the ZIP load model. In this case the errors between the actual and the estimated measurements are also very small (less than 1!10K4), what demonstrates that these estimated parameters are valid for the ZIP load model. One can notice the interesting behaviour of the parameters evolution curves obtained for the ZIP load model. The active power parameters (a, b and c) depicted in Figs. 10–12 have almost the same form in the adopted estimation interval. A similar behaviour is presented by the reactive power parameters (d, e and f) as shown in Figs. 13–15. This is due to the absence of restrictions in the proposed parameters related to the physical limits of the load variables under estimation. These variables correspond to maximum and minimum values of constant active and reactive power demand and constant current under steady-state conditions. The determination of these limits is
Fig. 10. Parameter a evolution—ZIP load model.
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L.T.M. Mota, A.A. Mota / Electrical Power and Energy Systems 26 (2004) 805–811
Fig. 11. Parameter b evolution—ZIP load model.
Fig. 14. Parameter e evolution—ZIP load model.
Fig. 12. Parameter c evolution—ZIP load model.
Fig. 15. Parameter f evolution—ZIP load model.
related to the knowledge of the electric characteristics of the equipments distributed along the feeders. However, the main focus of this work is in the estimation methods and thus, these limits were not considered in the problem formulation. So, the parameters proposed for the ZIP model have total freedom to evolve through the state space. Since actual measurements were used to perform the tests, the voltage magnitude data entries were close to 1 p.u. (controlled power system steady-state condition). Thus the voltage dependent terms (linear and quadratic) in Eqs. (5) and (6) became very close to the unity and the load behaviour is reflected in a similar way in the estimated parameters during the WLS recursive algorithm solution.
5. Conclusions
Fig. 13. Parameter d evolution—ZIP load model.
This work dealt with the problem of the load parameters estimation in electric power systems using two different
L.T.M. Mota, A.A. Mota / Electrical Power and Energy Systems 26 (2004) 805–811
load models: the exponential load model and the ZIP load model. The load parameters estimation was treated in a dynamic way, using data from the system past and providing an increase of the redundancy related to the parameters to be estimated. The algorithm for the accomplishment of this estimation process was implemented based on a recursive version of WLS method. Actual data (actual measurements) from a CPFL substation were used in the tests. It was shown that non-observable systems can become observable by the inclusion of pseudo-measurements to the measurement vector. That was the case of the ZIP load model. First, the application of this model was not feasible because the gain matrix (G) was singular. However, after the inclusion of the pseudo-measurements (in this case the parameters estimated at the last scan) to the measurement vector, this matrix became nonsingular and so the parameters of the ZIP load model could be estimated. The results demonstrated the efficiency of the dynamic estimation. In this case, the estimated parameters could be considered as valid, since the errors between the actual and the estimated measurements (residuals) were sufficiently close to zero. These results were obtained with both the exponential and the ZIP load models. Since these load parameters can be previously estimated, one can use them in order to determine, for example, the active and the reactive power injections of a load bus, and include these pseudomeasurements in the state estimation in order to improve observability and estimation accuracy.
Acknowledgements The authors gratefully acknowledge Prof. Alcir Monticelli (in memoriam), for his valuable orientation, corrections and contributions during the development of this work,
811
and also the support of FAPESP (Fundac¸a˜o de Amparo a` Pesquisa do Estado de Sa˜o Paulo) under the grants 98/14400-9 and 00/11164-4.
References [1] Alsac¸ O, Vempati N, Stott B, Monticelli A. Generalized state estimation. IEEE Trans Power Syst 1998;13:1069–75. [2] Debs AS. Estimation of steady-state power system model parameters. IEEE Power Apparatus Syst 1974;1260–8. [3] Zarco P, Expo´sito AG. Power system parameter estimation: a survey. IEEE Trans Power Syst 2000;15:216–21. [4] Mota LTM. Estimac¸a˜o Dinaˆmica de Paraˆmetros da Carga em Sistemas de Energia Ele´trica. M.Sc. Thesis. 2001. [5] Coker ML, Kgasoane H. Load modeling. IEEE Africon 1999. [6] Ju P, Handschin E, Karlsson D. Nonlinear dynamic load modeling: model and parameter estimation. IEEE Trans Power Syst 1996;1689–97. [7] IEEE Task Force on load representation for dynamic performance. Load representation for dynamic performance analysis. IEEE Trans Power Syst 1993;8(2):472–82. [8] Monticelli A. In: State estimation in electric power systems—a generalized approach. Massachusetts: Kluwer’s Power Electronics and Power Systems Series; 1999. [9] Kao W, Lin C, Huang C, Chen Y, Chiou C. Comparison of simulated power system dynamics applying various load models with actual recorded data. IEEE Trans Power Syst 1994;248–54. [10] Monticelli A. Electric power system state estimation—invited paper. IEEE Proc 2000. [11] Liu WE, Lim S. Parameter error identification and estimation in power system state estimation. IEEE Trans Power Syst 1995;10: 200–9. [12] Milanovic JV. On unreliability of exponential load models. Electric Power Syst Res 1999;1–9. [13] Rousseaux P, Van Cutsem Th, Dy Liacco TE. Whither dynamic state estimation? Int J Electrical Power Energy Syst 1990;104–16. [14] Silva AML, Do Coutto Filho MB, Cantera JMC. An efficient dynamic state estimator algorithm including bad data processing. IEEE Trans Power Syst 1987;2(4).
Load modeling at electric power distribution substations using dynamic load parameters estimation Lia Toledo Moreira Mota, Alexandre Assis Mota* UNICAMP-FEEC-DSEE, CEP 13081-970, Campinas, Sa˜o Paulo, Brazil Received 6 March 2002; revised 5 April 2004; accepted 8 July 2004
Abstract This paper is focused on electric power distribution substations load modeling using dynamic load parameters estimation. The load parameters are estimated using two models: the exponential and the ZIP load models. Since the load bus voltage and parameters are known one can determine the active and reactive power injections of this bus and include these pseudo-measurements in the state estimation in order to improve observability and estimation accuracy. The dynamic parameter estimation is developed using the weighted least squares method in a recursive form and the tests are carried out based on actual measurements. It is shown that the estimated parameters (for both load models) at a distribution substation are valid, since the obtained active and reactive power residuals are very close to zero. q 2004 Elsevier Ltd. All rights reserved. Keywords: Load modeling; Parameter estimation; Electric power systems
1. Introduction Load modeling has a significant influence on power systems operation, simulation and analysis. It is shown, for example, that the performance of the state estimation function, which plays a critical role in electric power systems operations, is affected by the power network parameters, including load parameters, and that the presence of parameter errors can lead to unreliable state estimation results [1,2]. Usually, the power system loads are modeled as constants. However, this kind of model is inadequate for some studies like power system dynamic studies and voltage collapse studies. Moreover, the unbalance between the power generation and the power demand, very common nowadays, reduces the systems security margins and increases the risks to safe operation. In order to reduce these risks, system studies have to be developed with better * Corresponding author. Tel.: C55-019-37883708; fax: C55-01932891395. E-mail address: [email protected] (A.A. Mota). 0142-0615/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2004.07.002
models for the systems components including better load models [3]. This paper is focused on electric power distribution substations load modeling with the estimation of the load parameters that represent these substations. Besides, this work deals with the parameter estimation problem in a dynamic way allowing a data bank construction (with the results of the prediction step) that could be used in the security analysis problem and that provides an increase of the redundancy related to the parameters to be estimated. Since the load bus voltage and parameters are known one can determine the active and reactive power injections of this bus and include these pseudo-measurements in the state estimation in order to improve both observability and estimation accuracy [10,11,13,14]. In this work, load parameters are estimated using two different models: the exponential load model and the ZIP (or polynomial) load model, both described in Section 2. The dynamic load parameters estimation is developed using the weighted least squares (WLS) method in a recursive form, which is presented in Section 3.
806
L.T.M. Mota, A.A. Mota / Electrical Power and Energy Systems 26 (2004) 805–811
The tests and results of a substation dynamic parameter estimation are shown in Section 4.
2. Load models A load model is a mathematical representation of the relation between the active or reactive power (or the current injection) in a load bus and the complex voltage of this same bus [4]. Load modeling has been carried out based on two different approaches. The first approach measures the voltage and frequency sensitivity of the active and reactive powers at substations [5]. The second approach constructs a composite load model for a given substation, based on the mix of load classes served by this substation. In this paper, the load modeling problem is treated using the first approach. The most commonly load models found in the literature [9] are described as follows: Constant impedance: In the constant impedance model, the active and reactive power injections at a given load bus vary directly with the square of the nodal voltage magnitude. This model is also called constant admitance model: P Z f ðV 2 Þ
(1)
where P is the active power injection and V is the voltage magnitude at the load bus. Constant current: In this model, the active and reactive power injections at a given load bus vary directly with the nodal voltage magnitude: P Z f ðVÞ
(2)
Constant power: Here, the power of the load bus is assumed to be constant and does not vary with the nodal voltage magnitude: PZk
(3)
where k is a constant. In these three load models (constant impedance, constant current and constant power) the reactive power injection can also be modeled using the Eqs. (1)–(3). Frequency dependent model: In this case, the active and reactive power injections of the load bus are related to the bus voltage frequency through an equation as follows. Factor Z ½1 C af $ðf K f0 Þ;
(4)
where: † af is the model sensitivity parameter; † f is the nodal voltage frequency; † f0 is the nominal frequency. In this work, other commonly used load models are implemented: the exponential and the ZIP load models. These models are described in Sections 2.1 and 2.2 [6,7].
2.1. ZIP (polynomial) load model In the ZIP load model, the active and reactive power injections comprise three components: a constant impedance (Z), a constant current (I) and a constant power injection (P): 2 V V C c$ (5) PL Z P0 $ a C b$ V0 V0 2 V V QL Z Q0 $ d C e$ Cf$ V0 V0
(6)
where a, b, c, d, e and f are the load parameters to be estimated, PL is the active power injection, QL is the reactive power injection, and P0, Q0 and V0 are the nominal values of the active and reactive power injections and the voltage magnitude, respectively. 2.2. Exponential load model In the exponential load model [12], the active and reactive power injections of the load bus are related to the bus voltage through an exponential function: a V P L Z P0 $ (7) V0 QL Z Q0 $
V V0
b (8)
where PL, QL, P0, Q0 and V0 were described in the ZIP load model and a and b are the parameters to be estimated.
3. Dynamic load parameters estimation Differently of the static estimation, the dynamic estimation requires system information from past scans of measurements, like for example, the estimated augmented state vector of the last scan. It is an augmented state vector because the load parameters are also included in the conventional state vector. Fig. 1 illustrates the data from the last scan of measurement (tiK1) required for actual parameters estimation (scan ti). In this figure, z represents the measurement vector, Rz is the covariance matrix of the measurements errors, x is the augmented state vector and Rx is the covariance matrix of the state (and parameters) errors. The dynamic estimation of load parameters is implemented using the WLS method in a recursive form. For each scan of measurements one can obtain one estimated parameter vector with the estimated parameters of the adopted load model (exponential or ZIP). The algorithm used for the WLS (recursive) implementation is described as follows. In this algorithm the superior indices
L.T.M. Mota, A.A. Mota / Electrical Power and Energy Systems 26 (2004) 805–811
807
and Rx ðtiC1 Þ Z ðGn ðti ÞÞK1 :
(14)
^ i Þ is the estimated augmented state vector at where xðt scan ti. It is important to notice that the recursivity of the WLS method is incorporated through this last step (step 7 or prediction step).
4. Tests and results Fig. 1. Information required for the parameter estimation of the scan ti.
(n) are related to the iterations of the process for the scan of measurement ti [7]. Step 1. Initialize the iteration counter. Step 2. Compute the gain matrix Gy ðti Þ : T
Gn ðti Þ Z Hn ðti Þ$ðRz ðti ÞÞK1 $Hn ðti Þ
(9)
where H is the system augmented Jacobian matrix. H is an augmented matrix because the load parameters are also included in the state vector; Step 3. Update the Dzn ðti Þ vector: Dzn ðti Þ Z zðti Þ K z^nK1 ðti Þ;
(10)
nK1
where z^ ðti Þ represents the estimated measurement vector that is computed using the estimated state vector of the last iteration. Step 4. Update the Dxn ðti Þ vector: y
y
yT
y
Dx ðti Þ Z ðG ðti ÞÞ $ðH ðti Þ$ðRz ðti ÞÞ $Dz ðti ÞÞ K1
K1
(11)
Step 5. Update the estimated state vector: xn ðti Þ Z xnK1 ðti Þ C Dxn ðti Þ
(12)
Step 6. Convergence test: † if Dxn ðti Þ! tolerance or if the number of iterations (n) has reached the adopted maximum number (nmax), then the estimated state vector of the tith scan is given by the xn ðti Þ vector. As the estimated vector is an augmented state vector it includes the estimated load parameters for the scan of measurement ti. The iterative process is over. Proceed to step 7; † if Dxn ðti ÞR tolerance and if n!nmax, update the iteration counter (nZnC1) and the augmented Jacobian matrix, based on the updated estimated state vector, and go back to step 2.
4.1. The adopted system The system used for the tests was adapted from [8] and is described in Fig. 2, which shows that the load bus has one active power, one reactive power and one voltage measurements. 4.2. Observability problem A system is considered to be observable if there are sufficient measurements and if these measurements are adequately distributed to obtain an unique solution for the state estimation problem [8]. In order to obtain this unique solution the Jacobian matrix (H) must have a full rank or the gain matrix (GZHT$H) must be nonsingular what guarantees the system observability. For the exponential load model, the Jacobian matrix is: 2
vP 6 vV 6 6 H Z 6 vQ 6 4 vV 1
vP va 0 0
3 0 7 7 7 vQ 7 7 vb 5 0
In this case, the H matrix has full rank (rank 3) and the gain matrix (G) is nonsingular, what guarantees the system observability. For the ZIP load model,
Step 7. Prediction step: Update the state vector and its covariance matrix (Rx) for the next scan of measurements: ^ iÞ xðtiC1 Þ Z xðt
(13)
(15)
Fig. 2. System used for the tests.
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the Jacobian matrix is: 2 vP vP vP vP 6 vV 6 H Z 6 vQ 4 vV 1
va
vb
vc
0
0
0
0
0
0
0
0
vQ vd 0
vQ ve 0
0
3
7 7 vQ 7 5 vf 0
(16)
The H matrix of the ZIP model has not a full rank (rank 3) and the gain matrix (G) is singular, and so the system is not observable. In order to make the system observable for the application of the ZIP model, pseudo-measurements are incorporated to the measurement vector (z). In this work, these pseudo-measurements are the estimated parameters (a, b, c, d, e and f) of the last scan (tiK1). Thus, the augmented measurement vector, with the additional measurements (pseudomeasurements) is: T zZ P Q V a b c d e f (17) And the augmented Jacobian matrix is: 3 2 vP vP vP vP 0 0 0 7 6 vV va vb vc 7 6 vQ vQ vQ 7 6 vQ 7 6 0 0 0 6 vV vd ve vf 7 7 6 0 0 0 0 0 0 7 6 1 7 6 6 0 1 0 0 0 0 0 7 7 6 7 H Z6 7 6 0 0 1 0 0 0 0 7 6 7 6 7 6 0 0 0 1 0 0 0 7 6 7 6 6 0 0 0 0 1 0 0 7 7 6 7 6 0 0 0 0 1 0 5 4 0 0
0
0
0
0
0
Fig. 4. Reactive power injection of Pinhal substation—1:00 a.m. of 05/Jun/1994 to 1:00 a.m. of 06/Jun/1994.
nonsingular and so the system can now be considered as observable. 4.3. Actual data
(18)
1
One can notice that the new Jacobian matrix has full rank (rank 7) and the new gain matrix (HT$H) is
Fig. 3. Active power injection of Pinhal substation—1:00 a.m. of 05/Jun/1994 to 1:00 a.m. of 06/Jun/1994.
The simulations were based on actual data from the Pinhal substation of CPFL (Companhia Paulista de Forc¸a e Luz) related to voltage magnitude, active and reactive power injections measurements, collected in periods of 1 h from 1:00 a.m. of 05/Jun/1994 to 1:00 a.m. of 06/Jun/1994. The measurements of active and reactive power injections are illustrated in Figs. 3 and 4. Figs. 5 and 6 show the evolution of the estimated parameters a and b for the exponential load model. Figs. 7–9 show that the error between the actual measurement and the estimated measurement (in the case of voltage and active and reactive power injections), or the residual, is close to zero, what demonstrates that the estimated parameters are valid for the measurements adopted and for the exponential
Fig. 5. Parameter a evolution—exponential load model.
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Fig. 6. Parameter b evolution—exponential load model.
Fig. 7. Residual of the active power—exponential load model.
Fig. 8. Residual of the reactive power—exponential load model.
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Fig. 9. Residual of the voltage magnitude—exponential load model.
load model. Similar results were obtained using the ZIP load model. Figs. 10–15 show the evolution of the estimated parameters a, b, c, d, e and f for the ZIP load model. In this case the errors between the actual and the estimated measurements are also very small (less than 1!10K4), what demonstrates that these estimated parameters are valid for the ZIP load model. One can notice the interesting behaviour of the parameters evolution curves obtained for the ZIP load model. The active power parameters (a, b and c) depicted in Figs. 10–12 have almost the same form in the adopted estimation interval. A similar behaviour is presented by the reactive power parameters (d, e and f) as shown in Figs. 13–15. This is due to the absence of restrictions in the proposed parameters related to the physical limits of the load variables under estimation. These variables correspond to maximum and minimum values of constant active and reactive power demand and constant current under steady-state conditions. The determination of these limits is
Fig. 10. Parameter a evolution—ZIP load model.
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Fig. 11. Parameter b evolution—ZIP load model.
Fig. 14. Parameter e evolution—ZIP load model.
Fig. 12. Parameter c evolution—ZIP load model.
Fig. 15. Parameter f evolution—ZIP load model.
related to the knowledge of the electric characteristics of the equipments distributed along the feeders. However, the main focus of this work is in the estimation methods and thus, these limits were not considered in the problem formulation. So, the parameters proposed for the ZIP model have total freedom to evolve through the state space. Since actual measurements were used to perform the tests, the voltage magnitude data entries were close to 1 p.u. (controlled power system steady-state condition). Thus the voltage dependent terms (linear and quadratic) in Eqs. (5) and (6) became very close to the unity and the load behaviour is reflected in a similar way in the estimated parameters during the WLS recursive algorithm solution.
5. Conclusions
Fig. 13. Parameter d evolution—ZIP load model.
This work dealt with the problem of the load parameters estimation in electric power systems using two different
L.T.M. Mota, A.A. Mota / Electrical Power and Energy Systems 26 (2004) 805–811
load models: the exponential load model and the ZIP load model. The load parameters estimation was treated in a dynamic way, using data from the system past and providing an increase of the redundancy related to the parameters to be estimated. The algorithm for the accomplishment of this estimation process was implemented based on a recursive version of WLS method. Actual data (actual measurements) from a CPFL substation were used in the tests. It was shown that non-observable systems can become observable by the inclusion of pseudo-measurements to the measurement vector. That was the case of the ZIP load model. First, the application of this model was not feasible because the gain matrix (G) was singular. However, after the inclusion of the pseudo-measurements (in this case the parameters estimated at the last scan) to the measurement vector, this matrix became nonsingular and so the parameters of the ZIP load model could be estimated. The results demonstrated the efficiency of the dynamic estimation. In this case, the estimated parameters could be considered as valid, since the errors between the actual and the estimated measurements (residuals) were sufficiently close to zero. These results were obtained with both the exponential and the ZIP load models. Since these load parameters can be previously estimated, one can use them in order to determine, for example, the active and the reactive power injections of a load bus, and include these pseudomeasurements in the state estimation in order to improve observability and estimation accuracy.
Acknowledgements The authors gratefully acknowledge Prof. Alcir Monticelli (in memoriam), for his valuable orientation, corrections and contributions during the development of this work,
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and also the support of FAPESP (Fundac¸a˜o de Amparo a` Pesquisa do Estado de Sa˜o Paulo) under the grants 98/14400-9 and 00/11164-4.
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