State Estimation and Power Flow Analysis of Power Systems
JOURNAL OF COMPUTERS, VOL. 7, NO. 3, MARCH 2012
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State Estimation and Power Flow Analysis of Power Systems Jiaxiong Chen University of Kentucky, Lexington, Kentucky 40508 U.S.A. Email: [email protected]
Yuan Liao University of Kentucky, Lexington, Kentucky 40508 U.S.A. Email: [email protected]
Abstract— State estimation and power flow analysis are important tools for analysis, operation and planning of a power system. In this paper, a new state estimation method based on the extended weighted least squares (WLS) method for considering both measurement errors and model inaccuracy is presented. Two bus, three bus, and IEEE 14 bus test cases are employed to evaluate the accuracy of the method. The comparison results show that the extended WLS method may outperform traditional WLS approach when the model is not accurate. In addition, this paper investigates a method based on Z matrix to implement power flow in a transmission system with multiple types of loads (e.g. constant PQ, constant impedance and constant current magnitude loads or mixed loads). The load flow results demonstrate that the method is effective and easy to implement when composite load types exist in the system. Our studies also show that it may be possible that multiple solutions exist for a power flow problem. Index Terms— State estimation, Power flow analysis, Weighted least squares method
I. INTRODUCTION Weighted least squares (WLS) state estimation (SE) approach has been an important tool for determining the optimal estimate of power system states [1]. The power network model used in the function relating the measurements to the states in WLS SE is implicitly assumed to be exact. However, the real model may contain two types of errors: parametric and nonparametric errors [2]. Parametric errors are errors that occur in circuit parameters such as the resistance and reactance of the transmission lines. Non-parametric errors could occur when utilizing a DC load flow model instead of a full AC model, or using approximate positive sequence impedance for representing untransposed lines, etc. Various approaches have been proposed to account for inaccuracy in network parameters, for example, by augmenting the state vector with addition of the suspicious parameters and augmenting the Jacobian matrix at the same time, or by utilizing Kalman filtering theory [3]. However, due to limited measurements, these approaches may not always be possible to identify all © 2012 ACADEMY PUBLISHER doi:10.4304/jcp.7.3.685-691
suspicious parameters or accurately estimate these parameters in order to alleviate non-parametric errors. Various approaches for distinguishing measurement errors from model errors in the context of “errors-invariable” model are discussed in paper [4]. The author of [5] proposes an approach called extended least squares for estimating parameters of pseudo-linear models. This approach is firstly applied to power system state estimation in paper [2] for considering both potential measurement errors and model errors. This paper further investigates its characteristics and evaluates its effectiveness. Both simple power system cases and a larger power system IEEE 14 bus test case are utilized. The power flow study (also known as load-flow study) has been widely applied to power systems for system planning, operating and control. Over the past decades, researchers have made an enormous amount of efforts to develop numerical power flow calculation approaches. The paper in [6] gives a review and comparison of power flow calculation methods, including Y-matrix, Z-matrix, and Newton methods. The authors of [7] develop a method based on Z matrix which converges more reliably than Y-matrix methods. In all the above methods, the type of loads is usually considered as constant PQ load. However, in reality, there are other types of loads, such as constant impedance load, constant current magnitude load, and mixed two and more types of loads. This paper will examine the effectiveness of Z matrix based power flow method to account for multiple types of loads in the transmission system. II. WLS METHODS A. Traditional WLS Method The system measurement equation is as follows [1]: 𝑧 = ℎ(𝑥) + 𝑒
(1)
where: z is the (m x 1) measurement vector; x is the an (n x 1) state vector to be estimated; h is a vector of nonlinear functions that relate the states to the measurements; and e is an (m x 1) measurement error vector. It is necessary that m ≥ n and the Jacobian matrix of ℎ(𝑥) has rank n.
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The optimal state estimate vector x may be determined by minimizing the sum of weighted squares of residuals Min 𝐹(𝑥) = [𝑧 − ℎ(𝑥)]𝑇 𝑅 −1 [𝑧 − ℎ(𝑥)]
(2)
where, R is the measurement covariance matrix. (2) is linearized using the Taylor series expansion, retaining the first two terms and ignoring higher-order terms. This leads to a linear WLS problem having the solution ∆𝑥 = (𝐻𝑇 𝑅−1 𝐻)−1 𝐻𝑇 𝑅−1 ∆𝑧
(3)
𝑥 = (𝐻𝑇 𝑅−1 𝐻)−1 𝐻𝑇 𝑅 −1 𝑧
(4)
where, H is the Jacobian matrix of h(x). The iterative approach is applied to obtain the state update until the absolute value of the difference of the states between successive iterations is less than the tolerance value 𝜀, typically, 𝜀 is set to 1e-4. The function h(x) can be linear or nonlinear. H is a constant matrix (m × n) in the case of linear function, and it needs only one iteration to converge. Alternatively, the solution to (2) can be obtained as follows:
B. The New Extended WLS Method
(5)
where, 𝑈 is the (m × 1) zero vector, m is total number of measurements, 𝜀 is the (m × 1) model mismatch error vector, and 𝑒 is the (m × 1) measurement error vector. Define 𝜀 𝑈 𝑧𝑛𝑒𝑤 = � � , 𝐹𝑛𝑒𝑤 (𝑋𝑛𝑒𝑤 ) = �ℎ(𝑥) − 𝑧̅ � , 𝜂𝑛𝑒𝑤 = � �. 𝑒 𝑧 𝑧̅ Therefore we obtain
𝑧𝑛𝑒𝑤 = 𝐹𝑛𝑒𝑤 (𝑋𝑛𝑒𝑤 ) + 𝜂𝑛𝑒𝑤
(6)
𝑓(𝑥) = [𝑧̅ − ℎ(𝑥)]𝑇 𝑊𝜀 [𝑧̅ − ℎ(𝑥)] + [𝑧 − 𝑧̅]𝑇 𝑊𝑒 [𝑧 − 𝑧̅]
(7)
The cost function, consisting of weighting matrix 𝑊𝜀 due to model mismatches, and weighting matrix 𝑊𝑒 due to measurement error, is defined as follows [2]: It is note that 𝐹𝑛𝑒𝑤 (𝑋𝑛𝑒𝑤 ) = 𝐻𝑛𝑒𝑤 𝑋𝑛𝑒𝑤 .
𝐻 −𝐼 where, 𝐻𝑛𝑒𝑤 = � � , 𝐼 is an (m × m) identity matrix, 𝑂 𝐼 and 𝑂 is a (m × 𝑛) zero matrix.
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This section presents case studies for evaluating the state estimation methods presented in Section II. A. Case Study Utilizing a Three-Bus System Based on DC Load Flow Model Figure 1 shows a sample system with branch impedances labeled in per unit. The correct value of 𝑅12 is 0.25𝑗, however, for some reason, the model database shows 0.3𝑗. The reason for this mistake might be the model database was not updated accordingly after the series compensation capacitor on the line was switched off. Table 1 gives the measurement data. All the meter readings are in per unit based on 100 MW base. The bus voltage angle 𝛿1 at bus 1, 𝛿2 at bus 2 and line flows at meter locations need to be estimated.
Figure 1. A three-bus sample power system
𝑋 Let the new state vector be 𝑋𝑛𝑒𝑤 = � � , with 𝑧̅ being 𝑧̅ the true value or best estimate of the measurement z. Hence, equation (1) is expanded as follows [2]. 𝜀 𝑈 � � = �ℎ(𝑥) − 𝑧̅ � + � � 𝑒 𝑧 𝑧̅
III. CASE STUDIES OF STATE ESTIMATION
TABLE 1. MEASUREMENT DATA FOR SYSTEM IN FIGURE 1 Quantity measured Real power from bus 1 to 2 Real power from bus 1 to 3 Real power from bus 3 to 2
Metering point M1
Error variance 0.0001
Meter reading 0.72
M2
0.0001
0.09
M3
0.0001
0.65
According to DC state estimation model, power flow equations can be expressed as follows: 𝑝12 = (𝛿1 − 𝛿2 )/𝑅12 𝑝13 = 2.5𝛿1 𝑝32 = −5𝛿2
(8) (9) (10)
where, 𝑝12 , 𝑝13 and 𝑝32 represent real power flow from bus 1 to 2, from bus 1 to 3, and from bus 3 to 2, respectively. The 𝐻𝑛𝑒𝑤 , obtained using 0.25j and 0.3j can be expressed as follows, respectively. 𝐻𝑛𝑒𝑤_0.25
4 ⎡2.5 ⎢ =⎢ 0 ⎢0 ⎢0 ⎣0
−4 −1 0 0 0 −5 1 0 0 0 0 0
0 0 −1 0 ⎤ ⎥ 0 −1⎥ 0 0⎥ 1 0⎥ 0 1⎦
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𝐻𝑛𝑒𝑤_0.3
1/0.3 −1/0.3 −1 0 ⎡ 0 −1 2.5 0 ⎢ 0 0 −5 =⎢ 0 1 0 0 ⎢ 0 ⎢ 0 0 1 0 ⎣ 0 0 0 0
0 0⎤ ⎥ −1⎥ 0 ⎥ 0 ⎥ 1 ⎦
687
TABLE 4. MEASUREMENT DATA FOR SYSTEM IN FIGURE 2
(12)
The error variances for model mismatches are assumed to take the same values in the extended WLS method. Table 2 shows the estimated results by using the correct 𝑅12 . The estimated voltage angles are the same for the two methods, while the estimated power flows by the extended WLS method are much closer to the meter readings than the traditional WLS method. The estimated voltage angles and power flows by using the mistaken 𝑅12 are shown in Table 3. The traditional WLS approach results in large errors in the estimate for power flows, while the extended WLS approach achieves more accurate results. TABLE 2. ESTIMATED VOLTAGE ANGLES AND POWER FLOW USING 𝑹𝟏𝟐 = 0.25j Traditional WLS Extended WLS
𝛿1 (radians)
𝛿2 (radians)
𝑝12 ���� (p.u)
𝑝13 ���� (p.u)
𝑝 ���� 32 (p.u)
0.0445
-0.1321
0.7067
0.1113
0.6607
0.0445
-0.1321
0.7133
0.1007
0.6553
TABLE 3. ESTIMATED VOLTAGE ANGLES AND POWER FLOW USING 𝑹𝟏𝟐 = 0.3j Traditional WLS Extended WLS
𝛿1 (radians)
𝛿2 (radians)
𝑝12 ���� (p.u)
𝑝13 ���� (p.u)
𝑝 ���� 32 (p.u)
0.0636
-0.1369
0.6683
0.1590
0.6845
0.0636
-0.1369
0.6941
0.1245
0.6672
B. Case Study Utilizing a Two-Bus System Based on AC Load Flow Model This section presents a case study based on AC load flow measurement Jacobian 𝐻𝐴 . Figure 2 shows a twobus sample power system, with branch impedances labeled in per unit. The correct value of the line impedance is 0.4𝑗, however somehow, the data model database shows a value of 0.5𝑗.
Figure 2. A Two-bus Sample Power System
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Quantity measured
Metering point
Error variance
Meter reading
Voltage magnitude at bus 1 Voltage magnitude at bus 2 Real power from bus 1 to 2 Real power received at bus 2 from bus 1 Reactive power from bus 1 to 2
M1 M2 M1 M2
0.0001 0.0004 0.0001 0.0004
1.02 0.93 1.97 -1.97
M1
0.0009
1.08
The states that need to be estimated are voltage magnitudes at bus 1 and bus 2 and voltage angle at bus 2. Table 4 gives the measurement data in per unit. Based on Figure 2 and Table 4, the system equations can be expressed as follows: ℎ1 = 𝑉1 ℎ2 = 𝑉2 ℎ3 = −𝑉1 𝑉2 sin𝛿/𝑅12 ℎ4 = 𝑉1 𝑉2 sin𝛿/𝑅12 ℎ5 = 𝑉1 2 /𝑅12 − 𝑉1 𝑉2 cos𝛿/𝑅12
where, 𝑅12 is line impedance between bus 1 and 2. 𝑉1 is voltage magnitude at bus 1. 𝑉2 is voltage magnitude at bus 2. 𝛿 is the voltage angle at bus 2.
(13) (14) (15) (16) (17)
Functions ℎ1 , ℎ2 , ℎ3 , ℎ4 and ℎ5 represent the bus 1 voltage magnitude, bus 2 voltage magnitude, real power from bus 1 to 2 measured at bus 1, real power received at bus 2 from bus 1 measured at bus 2, and reactive power from bus 1 to 2 measured at bus 1, respectively. Table 5 and 6 show the estimation results obtained by using 𝑅12 = 0.4𝑗 and 𝑅12 = 0.5𝑗, respectively. The variances for the model mismatches are set to 1e-6, 1e-6, 1e-1, 1e-1 and 1e-1 corresponding to equations (13), (14), (15), (16) and (17), respectively. The results yielded by the extended WLS are much closer to the meter readings than the traditional approach. Table 7 illustrates the effects of the variances of model mismatches on the estimation results by using the incorrect line impedance for different variances. From the last row of the table, it is seen that when the variances are properly chosen, more accurate estimate can be obtained by the extended WLS approach. Note that if the error variances for model mismatches are set to be small enough in both DC and AC load flow based studies, then both approaches will result in the same results.
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TABLE 5. ESTIMATED VOLTAGES AND POWER FLOW USING 𝑹𝟏𝟐 = 0.4j FOR THE TWO-BUS SYSTEM
Traditional WLS Extended WLS
Voltage Magnitude 𝑉�1 (p.u)
1.0175 1.0196
Voltage Magnitude 𝑉�2 (p.u)
Voltage Angle of bus 2 (radians)
0.9696 0.9343
-0.9236 -0.8622
Real power from bus 1 to 2 (p.u)
Real power received at bus 2(p.u)
1.9675 1.9698
-1.9675 -1.9694
Reactive power form bus 1 to 2 (p.u) 1.1011 1.0825
TABLE 6. ESTIMATED VOLTAGES AND POWER FLOW USING 𝑹𝟏𝟐 = 0.5j FOR THE TWO-BUS SYSTEM
Traditional WLS Extended WLS
Voltage Magnitude 𝑉�1 (p.u)
1.0253 1.0201
Voltage Magnitude 𝑉�2 (p.u)
Voltage Angle of bus 2 (radians)
1.0632 0.9342
-1.1158 -1.1450
Real power from bus 1 to 2 (p.u)
Real power received at bus 2(p.u)
1.9583 1.9698
-1.9583 -1.9691
Reactive power form bus 1 to 2 (p.u) 1.1442 1.0819
TABLE 7. ESTIMATED VOLTAGES AND POWER FLOW USING 𝑹𝟏𝟐 = 0.5j FOR THE TWO-BUS SYSTEM
Variances for model mismatches
Voltage Magnitude 𝑉�1 (p.u)
1/[1e6 1e6 1e2 1e2 1e2] 1/[1e6 1e6 1e1 1e1 1e1] 1/[1e6 1e6 1 1 1]
Voltage Magnitude 𝑉�2 (p.u)
1.0209
0.9626
Voltage Angle of bus 2 (radians) -1.1389
Real power from bus 1 to 2 (p.u)
Real power received at bus 2(p.u)
Reactive power form bus 1 to 2 (p.u)
1.9682
-1.9629
1.0950
1.0201
0.9342
-1.1450
1.9698
-1.9691
1.0819
1.0200
0.9304
-1.1457
1.9700
-1.9699
1.0802
C. Case Study Utilizing IEEE 14 Bus System Based on AC Load Flow Model The application of both traditional WLS and the extended WLS approaches for the state estimation of the IEEE 14 bus test case will be presented in this section [8]. Figure 3 shows the test case. Newton Raphson load flow method can be applied to obtain load flow results of the system [9]. The measurement data set chosen from load flow results must satisfy network observability [10]. In the state estimation, the chosen measurement data set for the formulation of state-estimation Jacobian 𝐻𝑥 are voltage magnitudes, real and reactive power injections, real and reactive power flows [11]. The variances of measurement errors for the above three kinds of measurement data are set to 9e-5, 1e-4 and 64e-6. To simplify the calculation, the variances for model mismatches are all set to 1.0. Two tests are implemented: first, all the branch impedances are correct, the results are shown in Table 8. Second, the branch impedance from bus 4 to bus 7 𝑍47 is mistakenly changed to 0.3j, the results are shown in table 9. The true values of voltage magnitude and angle are from Newton Raphson load flow results. To compare the state estimate accuracy of both traditional WLS and the extended WLS approaches, mean absolute percentage error (MAPE) is introduced as follows: 1
𝐴𝑡 −𝐹𝑡
𝑀𝐴𝑃𝐸 = ∑𝑛𝑡=1 � 𝑛
𝐴𝑡
� × 100%
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(18)
Figure 3. IEEE 14 Bus Test case [8]
where, 𝐴𝑡 is the actual value and 𝐹𝑡 is the calculated value. A smaller value of MAPE indicates a more accurate state estimation result. The MAPE values are shown in the last row of both Tables 8 and 9. As can be seen from Tables 8 and 9, the MAPE values of the extended WLS approach are smaller than those of the traditional WLS approach, indicating that the extended WLS approach can obtain more accurate state estimation results.
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TABLE 8. FIRST CONDITION: ALL THE BRANCH IMPEDANCES ARE CORRECT. True Value Bus ID
Traditional WLS
|𝑉| (p.u)
𝜃 (degree)
1
1.060
0
1.041
0
1.045
0
2
1.045
-4.99
1.026
-5.166
1.029
-5.138
3
1.010
-12.75
0.988
-13.241
0.992
-13.163
4
1.013
-10.24
0.994
-10.658
0.997
-10.591
5
1.017
-8.76
0.998
-9.122
1.001
-9.064
6
1.070
-14.45
1.053
-15.012
1.055
-14.923
7
1.046
-13.24
1.027
-13.764
1.030
-13.680
8
1.080
-13.24
1.063
-13.763
1.065
-13.680
9
1.031
-14.82
1.012
-15.405
1.015
-15.313
10
1.030
-15.04
1.012
-15.627
1.014
-15.534
11
1.046
-14.86
1.028
-15.440
1.031
-15.349
12
1.053
-15.30
1.036
-15.891
1.038
-15.798
13
1.047
-15.33
1.029
-15.927
1.031
-15.833
14
1.019
-16.08
1.001
-16.699
1.004
-16.600
1.78%
3.62%
1.51%
3.05%
M%
|𝑉| (p.u)
𝜃 (degree)
Extended WLS |𝑉| (p.u)
𝜃 (degree)
TABLE 9. SECOND CONDITION: 𝒁𝟒𝟕 = 0.3j
True Value Bus ID
Traditional WLS
|𝑉| (p.u)
𝜃 (degree)
1
1.06
0
1.043
0
1.046
0
2
1.045
-4.99
1.027
-5.154
1.030
-5.127
3
1.010
-12.75
0.989
-13.210
0.993
-13.133
4
1.013
-10.24
0.996
-10.632
0.998
-10.565
5
1.017
-8.76
0.999
-9.104
1.002
-9.046
6
1.070
-14.45
1.055
-15.077
1.058
-14.999
7
1.046
-13.24
1.038
-14.669
1.040
-14.580
8
1.080
-13.24
1.074
-14.621
1.076
-14.540
9
1.031
-14.82
1.023
-16.204
1.025
-16.099
10
1.030
-15.04
1.020
-16.246
1.023
-16.159
11
1.046
-14.86
1.033
-15.699
1.035
-15.641
12
1.053
-15.3
1.038
-15.953
1.041
-15.871
13
1.047
-15.33
1.031
-15.988
1.034
-15.906
14
1.019
-16.08
1.008
-17.162
1.010
-17.062
1.34%
5.61%
1.08%
5.06%
M%
|𝑉| (p.u)
𝜃 (degree)
Extended WLS
|𝑉| (p.u)
𝜃 (degree)
IV. POWER FLOW STUDY IN A SYSTEM WITH MULTIPLE TYPES OF LOADS This section presents a power flow method based on Z matrix to implement power flow in the transmission system with multiple types of loads. For a power system
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with n nodes, under specified generating and loading conditions, the network equation is given by the matrix equation [7] 𝑌𝑉 = 𝐼
(19)
𝑌 ′ 𝑉′ = 𝐼 ′ 𝐼 ′ = 𝐼𝑘 − 𝑌𝑘1 𝑉1
(20) (21)
𝑉′ = 𝑍′𝐼′
(22)
where, 𝑌 is n × n bus admittance matrix, 𝑉 is n × 1 vector of node voltages, and 𝐼 is n × 1 vector of current injections. Let bus 1 be designated as the slack bus. The above general set of equations includes the equation for the known slack-bus voltage, which is eliminated by modifying 𝑌 and 𝐼 . The matrix 𝑌 is modified by removing the row and column corresponding to the slack bus, and the second to the nth row of matrix I is modified by subtracting from any current element 𝐼𝑘 the product 𝑌𝑘1 𝑉1 , where 𝑌𝑘1 represents the element of the kth row and first column of matrix 𝑌 , and 𝑉1 is the constant slack-bus voltage. Denoting the modified bus admittance matrix and current vector as 𝑌 ′ and 𝐼 ′ , respectively. The equations for the above modifications are [7]
where, k ranges from 2 to n. 𝑉′ is voltage vector from bus 2 to bus n. The node voltages from the second to the nth node are obtained by the product 𝑍′𝐼′ where, bus impedance matrix 𝑍′ is obtained from matrix 𝑌 ′ . Then, an iterative technique can be applied to obtain the node voltages. During each iteration, the current injection is calculated using the most recent voltage vector. In the transmission system, the loads are usually considered as constant PQ loads. However, there are other types of loads, such as constant impedance load, constant current magnitude load and mixed two and more types of loads at a bus. The load modeling for the above loads are expressed as follows: For a constant impedance load, the injection current into the bus is determined as [12] 𝐼𝑙 = −
𝑉(𝑃𝑏 +𝑗𝑄𝑏 )∗ |𝑉𝑏 |2
(23)
where, 𝑃𝑏 and 𝑄𝑏 are nominal real and reactive power of the load, 𝑉 is the voltage of the bus. 𝑉𝑏 is the nominal voltage. For a constant PQ load [12], 𝑃𝑏 +𝑗𝑄𝑏 ∗ ) 𝑉
𝐼𝑙 = −(
For a constant current magnitude load [12],
(24)
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𝐼𝑙 = −
|𝑃𝑏 +𝑗𝑄𝑏 | |𝑉𝑏 |
∗
𝑉
(25)
|𝑉|
For the mixed two and more types of loads, i.e. mixed loads have a constant PQ load and a constant impedance load, each load accounts for 50% of total load of the bus, 𝐼𝑙 = −0.5 ∗ (
𝑃𝑏 +𝑗𝑄𝑏 ∗ ) 𝑉
− 0.5 ∗
𝑉(𝑃𝑏 +𝑗𝑄𝑏 )∗
(26)
|𝑉𝑏 |2
A modified IEEE 14 bus test case is utilized for case study. The loads of bus 4 and 10 are changed to constant impedance load, the load of bus 13 is changed to constant current magnitude load, and the load of bus 9 is changed to mixed loads with 50% constant PQ load and 50% constant impedance load. Table 10 shows the power flow results and the convergence criterion is a tolerance value of 1e-4 for voltage magnitudes. TABLE 10. POWER FLOW RESULTS FOR THE MODIFIED IEEE 14 BUS TEST CASE
Bus ID 1
Voltage Magnitude (p.u.) 1.06
Voltage Angle (Degree) 0
which may be convenient for performing studies spanning a horizon where the load changes according to specified load shape modifiers. It is noted that using the LU factorization technique may determine the solution to (20) more efficiently without the need of obtaining the Z matrix. An interesting observation has been found that the power flow problem may have multiple valid solutions. As an example, we have used the traditional Newton Raphson and the Z-matrix methods to solve the original IEEE 14 bus test case. The results obtained by the two methods are shown in Table 11. The convergence criterion is a tolerance of 1e-4 for voltage magnitude. Both solutions are valid since both satisfy the KCL law, but it can be seen that they are slightly different. TABLE 11. POWER FLOW RESULTS OBTAINED BY Z-MATRIX METHOD AND NEWTON RAPHSON METHOD Z-matrix Method Bus ID
Newton Raphson Method |𝑉| 𝜃 (p.u) (degree) 1.06 0
1
|𝑉| (p.u) 1.06
𝜃 (degree) 0
2
1.0465
-5.0049
1.0450
-4.9891
2
1.0480
-4.9464
3
1.0100
-12.7178
1.0100
-12.7492
3
1.0100
-12.5456
4
1.0159
-10.2610
1.0132
-10.2420
4
1.0200
-10.1148
5
1.0191
-8.7804
1.0166
-8.7601
5
1.0232
-8.6985
6
1.0755
-14.4212
1.0700
-14.4469
6
1.0882
-14.3504
7
1.0525
-13.2319
1.0457
-13.2368
7
1.0600
-13.0442
8
1.0912
-13.2318
1.0800
-13.2368
8
1.0985
-13.0442
9
1.0366
-14.7975
1.0305
-14.8201
9
1.0461
-14.5802
10
1.0359
-15.0099
1.0299
-15.0360
-14.8050
11
1.0519
-14.8312
1.0461
-14.8581
12
1.0589
-15.2629
1.0533
-15.2973
13
1.0523
-15.2974
1.0466
-15.3313
14
1.0253
-16.0325
1.0193
-16.0717
10
1.0462
11
1.0634
-14.6886
12
1.0739
-15.2360
13
1.0689
-15.3784
14
1.0381
-15.9211
Iteration
Iteration
14
7
15
V. FUTURE RESEARCH To test the power flow results, Kirchhoff’s current law (KCL) is applied to each bus of the modified IEEE 14 bus test case; that is, at any bus, the sum of branch currents injected into that bus should be equal to the load currents flowing out of that bus. Load currents can be calculated from equations (23) – (26). Utilizing voltages of each bus of the system as shown in Table 10, the sum of the branch currents injected into the i-th bus can be obtained from the equation: 𝐼𝑖 = ∑𝑛𝑘=1 𝑌𝑖𝑘 𝑉𝑘
(27)
The test results show that the power flow results in Table 10 are correct. This method has the following benefits. It is easy to consider all types of loads, since the loads can be readily represented by injection currents. The bus admittance/impedance matrix, once formulated, doesn’t need to be changed if the network remains unchanged, © 2012 ACADEMY PUBLISHER
In state estimation, the selection of error variances for measurements and model mismatches may have significant effects on the state estimation accuracy. Error variances for the measurements can be chosen according to meter measurement characteristic, which is usually found in meter’s manual. However, it is more difficult to find error variances for model mismatches. Thus, one future work is to find suitable methods to determine the error variances for model mismatches for enhanced state estimation accuracy. VI. CONCLUSION This paper presents a new state estimation algorithm based on the extended WLS method for accounting for both model errors and measurement errors. For comparison purposes, both the traditional WLS method
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and the new method are evaluated by performing case studies. The estimated results evince that the extended WLS method may outperform the traditional approach when the power network model used is not accurate. The extended WLS method is best suited for cases with known variances for model errors. In addition, this paper also examines a power flow method based on Z matrix for transmission systems with multiple types of loads, and the test results demonstrate that the method is effective and correct. The benefit of this method is that it is straightforward to model composite types of loads in the analysis. Studies also show that multiple valid solutions may exist for a power flow problem. REFERENCES [1] A. Monticelli, State Estimation in Electric Power Systems: a Generalized Approach, New York: Kluwer, 1999. [2] Y. Liao, “State estimation algorithm considering effects of model”, SoutheastCon, 2007. Proceedings. IEEE. pp. 450453. [3] P. Zarco and A. Gomez, “Power system parameter estimation: A survey”, IEEE Transactions on Power systems, vol. 15, no. 1, Feb. 2000, pp. 216-222. [4] W. A. Fuller, Measurement Error Models. New York: Wiley, 1987. [5] A. Yeredor, The extended least squares criterion: minimization algorithms and applications, IEEE Transactions on signal processing, vol. 49, no. 1, January 2001, pp. 74-86. [6] B. Stott, “Review of Load-Flow Calculation Methods”, IEEE Transactions on Power Delivery, Vol. 62, no. 7, Jul 1974, pp. 916 – 929. [7] A. Brameller and J. K Denmead, “Some improved methods of digital network analysis, Proc. Znst. Elec. Eng., vol. 109A, pp. 109- 116, Feb. 1962. [8] http://www.ee.washington.edu/research/pstca/, Power System Test Case Archive. [9] W.F. Tinney and C.E. Hart, “Power flow solution by Newton’s method", IEEE Transactions on Power Apparatus and Systems, Vol PAS-86, No. 11, 1967, pp. 1449 – 1460. [10] A. Monticelli and F. F. Wu, “Network Observability: Theory”, IEEE Trans. On Power Apparatus and Systems, vol. PAS-104, no. 5, pp. 1042-1048, May 1985. [11] J. Grainger and W. Stevenson, Power System Analysis, New York: McGraw-Hill, 1994. [12] Yuan Liao, Lecture Notes of Smart Grid class, Department of Electrical and Computer Engineering, University of Kentucky, 2011.
© 2012 ACADEMY PUBLISHER
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Jiaxiong Chen was born in Fuqing, China, on September 7, 1985. He received his B.S. degree from Department of Electrical Engineering, Hunan University, Changsha, China, in 2008. In 2010, he received his M.S. degree from Department of Electrical and Computer Engineering, University of Kentucky, Lexington, KY, USA. He is currently pursuing his PH.D. degree in electrical engineering at University of Kentucky. His research field is power system state estimation. Yuan Liao (S’98, M’00, SM’05) is currently an Associate Professor at the department of electrical and computer engineering at the University of Kentucky. From August 2000 to August 2005, he was with the ABB Corporate Research Center as an R&D Consulting Engineer and then Principal R&D Consulting Engineer. His research interests include protection, power quality analysis, large-scale resource scheduling optimization and EMS/SCADA design.
685
State Estimation and Power Flow Analysis of Power Systems Jiaxiong Chen University of Kentucky, Lexington, Kentucky 40508 U.S.A. Email: [email protected]
Yuan Liao University of Kentucky, Lexington, Kentucky 40508 U.S.A. Email: [email protected]
Abstract— State estimation and power flow analysis are important tools for analysis, operation and planning of a power system. In this paper, a new state estimation method based on the extended weighted least squares (WLS) method for considering both measurement errors and model inaccuracy is presented. Two bus, three bus, and IEEE 14 bus test cases are employed to evaluate the accuracy of the method. The comparison results show that the extended WLS method may outperform traditional WLS approach when the model is not accurate. In addition, this paper investigates a method based on Z matrix to implement power flow in a transmission system with multiple types of loads (e.g. constant PQ, constant impedance and constant current magnitude loads or mixed loads). The load flow results demonstrate that the method is effective and easy to implement when composite load types exist in the system. Our studies also show that it may be possible that multiple solutions exist for a power flow problem. Index Terms— State estimation, Power flow analysis, Weighted least squares method
I. INTRODUCTION Weighted least squares (WLS) state estimation (SE) approach has been an important tool for determining the optimal estimate of power system states [1]. The power network model used in the function relating the measurements to the states in WLS SE is implicitly assumed to be exact. However, the real model may contain two types of errors: parametric and nonparametric errors [2]. Parametric errors are errors that occur in circuit parameters such as the resistance and reactance of the transmission lines. Non-parametric errors could occur when utilizing a DC load flow model instead of a full AC model, or using approximate positive sequence impedance for representing untransposed lines, etc. Various approaches have been proposed to account for inaccuracy in network parameters, for example, by augmenting the state vector with addition of the suspicious parameters and augmenting the Jacobian matrix at the same time, or by utilizing Kalman filtering theory [3]. However, due to limited measurements, these approaches may not always be possible to identify all © 2012 ACADEMY PUBLISHER doi:10.4304/jcp.7.3.685-691
suspicious parameters or accurately estimate these parameters in order to alleviate non-parametric errors. Various approaches for distinguishing measurement errors from model errors in the context of “errors-invariable” model are discussed in paper [4]. The author of [5] proposes an approach called extended least squares for estimating parameters of pseudo-linear models. This approach is firstly applied to power system state estimation in paper [2] for considering both potential measurement errors and model errors. This paper further investigates its characteristics and evaluates its effectiveness. Both simple power system cases and a larger power system IEEE 14 bus test case are utilized. The power flow study (also known as load-flow study) has been widely applied to power systems for system planning, operating and control. Over the past decades, researchers have made an enormous amount of efforts to develop numerical power flow calculation approaches. The paper in [6] gives a review and comparison of power flow calculation methods, including Y-matrix, Z-matrix, and Newton methods. The authors of [7] develop a method based on Z matrix which converges more reliably than Y-matrix methods. In all the above methods, the type of loads is usually considered as constant PQ load. However, in reality, there are other types of loads, such as constant impedance load, constant current magnitude load, and mixed two and more types of loads. This paper will examine the effectiveness of Z matrix based power flow method to account for multiple types of loads in the transmission system. II. WLS METHODS A. Traditional WLS Method The system measurement equation is as follows [1]: 𝑧 = ℎ(𝑥) + 𝑒
(1)
where: z is the (m x 1) measurement vector; x is the an (n x 1) state vector to be estimated; h is a vector of nonlinear functions that relate the states to the measurements; and e is an (m x 1) measurement error vector. It is necessary that m ≥ n and the Jacobian matrix of ℎ(𝑥) has rank n.
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The optimal state estimate vector x may be determined by minimizing the sum of weighted squares of residuals Min 𝐹(𝑥) = [𝑧 − ℎ(𝑥)]𝑇 𝑅 −1 [𝑧 − ℎ(𝑥)]
(2)
where, R is the measurement covariance matrix. (2) is linearized using the Taylor series expansion, retaining the first two terms and ignoring higher-order terms. This leads to a linear WLS problem having the solution ∆𝑥 = (𝐻𝑇 𝑅−1 𝐻)−1 𝐻𝑇 𝑅−1 ∆𝑧
(3)
𝑥 = (𝐻𝑇 𝑅−1 𝐻)−1 𝐻𝑇 𝑅 −1 𝑧
(4)
where, H is the Jacobian matrix of h(x). The iterative approach is applied to obtain the state update until the absolute value of the difference of the states between successive iterations is less than the tolerance value 𝜀, typically, 𝜀 is set to 1e-4. The function h(x) can be linear or nonlinear. H is a constant matrix (m × n) in the case of linear function, and it needs only one iteration to converge. Alternatively, the solution to (2) can be obtained as follows:
B. The New Extended WLS Method
(5)
where, 𝑈 is the (m × 1) zero vector, m is total number of measurements, 𝜀 is the (m × 1) model mismatch error vector, and 𝑒 is the (m × 1) measurement error vector. Define 𝜀 𝑈 𝑧𝑛𝑒𝑤 = � � , 𝐹𝑛𝑒𝑤 (𝑋𝑛𝑒𝑤 ) = �ℎ(𝑥) − 𝑧̅ � , 𝜂𝑛𝑒𝑤 = � �. 𝑒 𝑧 𝑧̅ Therefore we obtain
𝑧𝑛𝑒𝑤 = 𝐹𝑛𝑒𝑤 (𝑋𝑛𝑒𝑤 ) + 𝜂𝑛𝑒𝑤
(6)
𝑓(𝑥) = [𝑧̅ − ℎ(𝑥)]𝑇 𝑊𝜀 [𝑧̅ − ℎ(𝑥)] + [𝑧 − 𝑧̅]𝑇 𝑊𝑒 [𝑧 − 𝑧̅]
(7)
The cost function, consisting of weighting matrix 𝑊𝜀 due to model mismatches, and weighting matrix 𝑊𝑒 due to measurement error, is defined as follows [2]: It is note that 𝐹𝑛𝑒𝑤 (𝑋𝑛𝑒𝑤 ) = 𝐻𝑛𝑒𝑤 𝑋𝑛𝑒𝑤 .
𝐻 −𝐼 where, 𝐻𝑛𝑒𝑤 = � � , 𝐼 is an (m × m) identity matrix, 𝑂 𝐼 and 𝑂 is a (m × 𝑛) zero matrix.
© 2012 ACADEMY PUBLISHER
This section presents case studies for evaluating the state estimation methods presented in Section II. A. Case Study Utilizing a Three-Bus System Based on DC Load Flow Model Figure 1 shows a sample system with branch impedances labeled in per unit. The correct value of 𝑅12 is 0.25𝑗, however, for some reason, the model database shows 0.3𝑗. The reason for this mistake might be the model database was not updated accordingly after the series compensation capacitor on the line was switched off. Table 1 gives the measurement data. All the meter readings are in per unit based on 100 MW base. The bus voltage angle 𝛿1 at bus 1, 𝛿2 at bus 2 and line flows at meter locations need to be estimated.
Figure 1. A three-bus sample power system
𝑋 Let the new state vector be 𝑋𝑛𝑒𝑤 = � � , with 𝑧̅ being 𝑧̅ the true value or best estimate of the measurement z. Hence, equation (1) is expanded as follows [2]. 𝜀 𝑈 � � = �ℎ(𝑥) − 𝑧̅ � + � � 𝑒 𝑧 𝑧̅
III. CASE STUDIES OF STATE ESTIMATION
TABLE 1. MEASUREMENT DATA FOR SYSTEM IN FIGURE 1 Quantity measured Real power from bus 1 to 2 Real power from bus 1 to 3 Real power from bus 3 to 2
Metering point M1
Error variance 0.0001
Meter reading 0.72
M2
0.0001
0.09
M3
0.0001
0.65
According to DC state estimation model, power flow equations can be expressed as follows: 𝑝12 = (𝛿1 − 𝛿2 )/𝑅12 𝑝13 = 2.5𝛿1 𝑝32 = −5𝛿2
(8) (9) (10)
where, 𝑝12 , 𝑝13 and 𝑝32 represent real power flow from bus 1 to 2, from bus 1 to 3, and from bus 3 to 2, respectively. The 𝐻𝑛𝑒𝑤 , obtained using 0.25j and 0.3j can be expressed as follows, respectively. 𝐻𝑛𝑒𝑤_0.25
4 ⎡2.5 ⎢ =⎢ 0 ⎢0 ⎢0 ⎣0
−4 −1 0 0 0 −5 1 0 0 0 0 0
0 0 −1 0 ⎤ ⎥ 0 −1⎥ 0 0⎥ 1 0⎥ 0 1⎦
(11)
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𝐻𝑛𝑒𝑤_0.3
1/0.3 −1/0.3 −1 0 ⎡ 0 −1 2.5 0 ⎢ 0 0 −5 =⎢ 0 1 0 0 ⎢ 0 ⎢ 0 0 1 0 ⎣ 0 0 0 0
0 0⎤ ⎥ −1⎥ 0 ⎥ 0 ⎥ 1 ⎦
687
TABLE 4. MEASUREMENT DATA FOR SYSTEM IN FIGURE 2
(12)
The error variances for model mismatches are assumed to take the same values in the extended WLS method. Table 2 shows the estimated results by using the correct 𝑅12 . The estimated voltage angles are the same for the two methods, while the estimated power flows by the extended WLS method are much closer to the meter readings than the traditional WLS method. The estimated voltage angles and power flows by using the mistaken 𝑅12 are shown in Table 3. The traditional WLS approach results in large errors in the estimate for power flows, while the extended WLS approach achieves more accurate results. TABLE 2. ESTIMATED VOLTAGE ANGLES AND POWER FLOW USING 𝑹𝟏𝟐 = 0.25j Traditional WLS Extended WLS
𝛿1 (radians)
𝛿2 (radians)
𝑝12 ���� (p.u)
𝑝13 ���� (p.u)
𝑝 ���� 32 (p.u)
0.0445
-0.1321
0.7067
0.1113
0.6607
0.0445
-0.1321
0.7133
0.1007
0.6553
TABLE 3. ESTIMATED VOLTAGE ANGLES AND POWER FLOW USING 𝑹𝟏𝟐 = 0.3j Traditional WLS Extended WLS
𝛿1 (radians)
𝛿2 (radians)
𝑝12 ���� (p.u)
𝑝13 ���� (p.u)
𝑝 ���� 32 (p.u)
0.0636
-0.1369
0.6683
0.1590
0.6845
0.0636
-0.1369
0.6941
0.1245
0.6672
B. Case Study Utilizing a Two-Bus System Based on AC Load Flow Model This section presents a case study based on AC load flow measurement Jacobian 𝐻𝐴 . Figure 2 shows a twobus sample power system, with branch impedances labeled in per unit. The correct value of the line impedance is 0.4𝑗, however somehow, the data model database shows a value of 0.5𝑗.
Figure 2. A Two-bus Sample Power System
© 2012 ACADEMY PUBLISHER
Quantity measured
Metering point
Error variance
Meter reading
Voltage magnitude at bus 1 Voltage magnitude at bus 2 Real power from bus 1 to 2 Real power received at bus 2 from bus 1 Reactive power from bus 1 to 2
M1 M2 M1 M2
0.0001 0.0004 0.0001 0.0004
1.02 0.93 1.97 -1.97
M1
0.0009
1.08
The states that need to be estimated are voltage magnitudes at bus 1 and bus 2 and voltage angle at bus 2. Table 4 gives the measurement data in per unit. Based on Figure 2 and Table 4, the system equations can be expressed as follows: ℎ1 = 𝑉1 ℎ2 = 𝑉2 ℎ3 = −𝑉1 𝑉2 sin𝛿/𝑅12 ℎ4 = 𝑉1 𝑉2 sin𝛿/𝑅12 ℎ5 = 𝑉1 2 /𝑅12 − 𝑉1 𝑉2 cos𝛿/𝑅12
where, 𝑅12 is line impedance between bus 1 and 2. 𝑉1 is voltage magnitude at bus 1. 𝑉2 is voltage magnitude at bus 2. 𝛿 is the voltage angle at bus 2.
(13) (14) (15) (16) (17)
Functions ℎ1 , ℎ2 , ℎ3 , ℎ4 and ℎ5 represent the bus 1 voltage magnitude, bus 2 voltage magnitude, real power from bus 1 to 2 measured at bus 1, real power received at bus 2 from bus 1 measured at bus 2, and reactive power from bus 1 to 2 measured at bus 1, respectively. Table 5 and 6 show the estimation results obtained by using 𝑅12 = 0.4𝑗 and 𝑅12 = 0.5𝑗, respectively. The variances for the model mismatches are set to 1e-6, 1e-6, 1e-1, 1e-1 and 1e-1 corresponding to equations (13), (14), (15), (16) and (17), respectively. The results yielded by the extended WLS are much closer to the meter readings than the traditional approach. Table 7 illustrates the effects of the variances of model mismatches on the estimation results by using the incorrect line impedance for different variances. From the last row of the table, it is seen that when the variances are properly chosen, more accurate estimate can be obtained by the extended WLS approach. Note that if the error variances for model mismatches are set to be small enough in both DC and AC load flow based studies, then both approaches will result in the same results.
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TABLE 5. ESTIMATED VOLTAGES AND POWER FLOW USING 𝑹𝟏𝟐 = 0.4j FOR THE TWO-BUS SYSTEM
Traditional WLS Extended WLS
Voltage Magnitude 𝑉�1 (p.u)
1.0175 1.0196
Voltage Magnitude 𝑉�2 (p.u)
Voltage Angle of bus 2 (radians)
0.9696 0.9343
-0.9236 -0.8622
Real power from bus 1 to 2 (p.u)
Real power received at bus 2(p.u)
1.9675 1.9698
-1.9675 -1.9694
Reactive power form bus 1 to 2 (p.u) 1.1011 1.0825
TABLE 6. ESTIMATED VOLTAGES AND POWER FLOW USING 𝑹𝟏𝟐 = 0.5j FOR THE TWO-BUS SYSTEM
Traditional WLS Extended WLS
Voltage Magnitude 𝑉�1 (p.u)
1.0253 1.0201
Voltage Magnitude 𝑉�2 (p.u)
Voltage Angle of bus 2 (radians)
1.0632 0.9342
-1.1158 -1.1450
Real power from bus 1 to 2 (p.u)
Real power received at bus 2(p.u)
1.9583 1.9698
-1.9583 -1.9691
Reactive power form bus 1 to 2 (p.u) 1.1442 1.0819
TABLE 7. ESTIMATED VOLTAGES AND POWER FLOW USING 𝑹𝟏𝟐 = 0.5j FOR THE TWO-BUS SYSTEM
Variances for model mismatches
Voltage Magnitude 𝑉�1 (p.u)
1/[1e6 1e6 1e2 1e2 1e2] 1/[1e6 1e6 1e1 1e1 1e1] 1/[1e6 1e6 1 1 1]
Voltage Magnitude 𝑉�2 (p.u)
1.0209
0.9626
Voltage Angle of bus 2 (radians) -1.1389
Real power from bus 1 to 2 (p.u)
Real power received at bus 2(p.u)
Reactive power form bus 1 to 2 (p.u)
1.9682
-1.9629
1.0950
1.0201
0.9342
-1.1450
1.9698
-1.9691
1.0819
1.0200
0.9304
-1.1457
1.9700
-1.9699
1.0802
C. Case Study Utilizing IEEE 14 Bus System Based on AC Load Flow Model The application of both traditional WLS and the extended WLS approaches for the state estimation of the IEEE 14 bus test case will be presented in this section [8]. Figure 3 shows the test case. Newton Raphson load flow method can be applied to obtain load flow results of the system [9]. The measurement data set chosen from load flow results must satisfy network observability [10]. In the state estimation, the chosen measurement data set for the formulation of state-estimation Jacobian 𝐻𝑥 are voltage magnitudes, real and reactive power injections, real and reactive power flows [11]. The variances of measurement errors for the above three kinds of measurement data are set to 9e-5, 1e-4 and 64e-6. To simplify the calculation, the variances for model mismatches are all set to 1.0. Two tests are implemented: first, all the branch impedances are correct, the results are shown in Table 8. Second, the branch impedance from bus 4 to bus 7 𝑍47 is mistakenly changed to 0.3j, the results are shown in table 9. The true values of voltage magnitude and angle are from Newton Raphson load flow results. To compare the state estimate accuracy of both traditional WLS and the extended WLS approaches, mean absolute percentage error (MAPE) is introduced as follows: 1
𝐴𝑡 −𝐹𝑡
𝑀𝐴𝑃𝐸 = ∑𝑛𝑡=1 � 𝑛
𝐴𝑡
� × 100%
© 2012 ACADEMY PUBLISHER
(18)
Figure 3. IEEE 14 Bus Test case [8]
where, 𝐴𝑡 is the actual value and 𝐹𝑡 is the calculated value. A smaller value of MAPE indicates a more accurate state estimation result. The MAPE values are shown in the last row of both Tables 8 and 9. As can be seen from Tables 8 and 9, the MAPE values of the extended WLS approach are smaller than those of the traditional WLS approach, indicating that the extended WLS approach can obtain more accurate state estimation results.
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TABLE 8. FIRST CONDITION: ALL THE BRANCH IMPEDANCES ARE CORRECT. True Value Bus ID
Traditional WLS
|𝑉| (p.u)
𝜃 (degree)
1
1.060
0
1.041
0
1.045
0
2
1.045
-4.99
1.026
-5.166
1.029
-5.138
3
1.010
-12.75
0.988
-13.241
0.992
-13.163
4
1.013
-10.24
0.994
-10.658
0.997
-10.591
5
1.017
-8.76
0.998
-9.122
1.001
-9.064
6
1.070
-14.45
1.053
-15.012
1.055
-14.923
7
1.046
-13.24
1.027
-13.764
1.030
-13.680
8
1.080
-13.24
1.063
-13.763
1.065
-13.680
9
1.031
-14.82
1.012
-15.405
1.015
-15.313
10
1.030
-15.04
1.012
-15.627
1.014
-15.534
11
1.046
-14.86
1.028
-15.440
1.031
-15.349
12
1.053
-15.30
1.036
-15.891
1.038
-15.798
13
1.047
-15.33
1.029
-15.927
1.031
-15.833
14
1.019
-16.08
1.001
-16.699
1.004
-16.600
1.78%
3.62%
1.51%
3.05%
M%
|𝑉| (p.u)
𝜃 (degree)
Extended WLS |𝑉| (p.u)
𝜃 (degree)
TABLE 9. SECOND CONDITION: 𝒁𝟒𝟕 = 0.3j
True Value Bus ID
Traditional WLS
|𝑉| (p.u)
𝜃 (degree)
1
1.06
0
1.043
0
1.046
0
2
1.045
-4.99
1.027
-5.154
1.030
-5.127
3
1.010
-12.75
0.989
-13.210
0.993
-13.133
4
1.013
-10.24
0.996
-10.632
0.998
-10.565
5
1.017
-8.76
0.999
-9.104
1.002
-9.046
6
1.070
-14.45
1.055
-15.077
1.058
-14.999
7
1.046
-13.24
1.038
-14.669
1.040
-14.580
8
1.080
-13.24
1.074
-14.621
1.076
-14.540
9
1.031
-14.82
1.023
-16.204
1.025
-16.099
10
1.030
-15.04
1.020
-16.246
1.023
-16.159
11
1.046
-14.86
1.033
-15.699
1.035
-15.641
12
1.053
-15.3
1.038
-15.953
1.041
-15.871
13
1.047
-15.33
1.031
-15.988
1.034
-15.906
14
1.019
-16.08
1.008
-17.162
1.010
-17.062
1.34%
5.61%
1.08%
5.06%
M%
|𝑉| (p.u)
𝜃 (degree)
Extended WLS
|𝑉| (p.u)
𝜃 (degree)
IV. POWER FLOW STUDY IN A SYSTEM WITH MULTIPLE TYPES OF LOADS This section presents a power flow method based on Z matrix to implement power flow in the transmission system with multiple types of loads. For a power system
© 2012 ACADEMY PUBLISHER
with n nodes, under specified generating and loading conditions, the network equation is given by the matrix equation [7] 𝑌𝑉 = 𝐼
(19)
𝑌 ′ 𝑉′ = 𝐼 ′ 𝐼 ′ = 𝐼𝑘 − 𝑌𝑘1 𝑉1
(20) (21)
𝑉′ = 𝑍′𝐼′
(22)
where, 𝑌 is n × n bus admittance matrix, 𝑉 is n × 1 vector of node voltages, and 𝐼 is n × 1 vector of current injections. Let bus 1 be designated as the slack bus. The above general set of equations includes the equation for the known slack-bus voltage, which is eliminated by modifying 𝑌 and 𝐼 . The matrix 𝑌 is modified by removing the row and column corresponding to the slack bus, and the second to the nth row of matrix I is modified by subtracting from any current element 𝐼𝑘 the product 𝑌𝑘1 𝑉1 , where 𝑌𝑘1 represents the element of the kth row and first column of matrix 𝑌 , and 𝑉1 is the constant slack-bus voltage. Denoting the modified bus admittance matrix and current vector as 𝑌 ′ and 𝐼 ′ , respectively. The equations for the above modifications are [7]
where, k ranges from 2 to n. 𝑉′ is voltage vector from bus 2 to bus n. The node voltages from the second to the nth node are obtained by the product 𝑍′𝐼′ where, bus impedance matrix 𝑍′ is obtained from matrix 𝑌 ′ . Then, an iterative technique can be applied to obtain the node voltages. During each iteration, the current injection is calculated using the most recent voltage vector. In the transmission system, the loads are usually considered as constant PQ loads. However, there are other types of loads, such as constant impedance load, constant current magnitude load and mixed two and more types of loads at a bus. The load modeling for the above loads are expressed as follows: For a constant impedance load, the injection current into the bus is determined as [12] 𝐼𝑙 = −
𝑉(𝑃𝑏 +𝑗𝑄𝑏 )∗ |𝑉𝑏 |2
(23)
where, 𝑃𝑏 and 𝑄𝑏 are nominal real and reactive power of the load, 𝑉 is the voltage of the bus. 𝑉𝑏 is the nominal voltage. For a constant PQ load [12], 𝑃𝑏 +𝑗𝑄𝑏 ∗ ) 𝑉
𝐼𝑙 = −(
For a constant current magnitude load [12],
(24)
690
JOURNAL OF COMPUTERS, VOL. 7, NO. 3, MARCH 2012
𝐼𝑙 = −
|𝑃𝑏 +𝑗𝑄𝑏 | |𝑉𝑏 |
∗
𝑉
(25)
|𝑉|
For the mixed two and more types of loads, i.e. mixed loads have a constant PQ load and a constant impedance load, each load accounts for 50% of total load of the bus, 𝐼𝑙 = −0.5 ∗ (
𝑃𝑏 +𝑗𝑄𝑏 ∗ ) 𝑉
− 0.5 ∗
𝑉(𝑃𝑏 +𝑗𝑄𝑏 )∗
(26)
|𝑉𝑏 |2
A modified IEEE 14 bus test case is utilized for case study. The loads of bus 4 and 10 are changed to constant impedance load, the load of bus 13 is changed to constant current magnitude load, and the load of bus 9 is changed to mixed loads with 50% constant PQ load and 50% constant impedance load. Table 10 shows the power flow results and the convergence criterion is a tolerance value of 1e-4 for voltage magnitudes. TABLE 10. POWER FLOW RESULTS FOR THE MODIFIED IEEE 14 BUS TEST CASE
Bus ID 1
Voltage Magnitude (p.u.) 1.06
Voltage Angle (Degree) 0
which may be convenient for performing studies spanning a horizon where the load changes according to specified load shape modifiers. It is noted that using the LU factorization technique may determine the solution to (20) more efficiently without the need of obtaining the Z matrix. An interesting observation has been found that the power flow problem may have multiple valid solutions. As an example, we have used the traditional Newton Raphson and the Z-matrix methods to solve the original IEEE 14 bus test case. The results obtained by the two methods are shown in Table 11. The convergence criterion is a tolerance of 1e-4 for voltage magnitude. Both solutions are valid since both satisfy the KCL law, but it can be seen that they are slightly different. TABLE 11. POWER FLOW RESULTS OBTAINED BY Z-MATRIX METHOD AND NEWTON RAPHSON METHOD Z-matrix Method Bus ID
Newton Raphson Method |𝑉| 𝜃 (p.u) (degree) 1.06 0
1
|𝑉| (p.u) 1.06
𝜃 (degree) 0
2
1.0465
-5.0049
1.0450
-4.9891
2
1.0480
-4.9464
3
1.0100
-12.7178
1.0100
-12.7492
3
1.0100
-12.5456
4
1.0159
-10.2610
1.0132
-10.2420
4
1.0200
-10.1148
5
1.0191
-8.7804
1.0166
-8.7601
5
1.0232
-8.6985
6
1.0755
-14.4212
1.0700
-14.4469
6
1.0882
-14.3504
7
1.0525
-13.2319
1.0457
-13.2368
7
1.0600
-13.0442
8
1.0912
-13.2318
1.0800
-13.2368
8
1.0985
-13.0442
9
1.0366
-14.7975
1.0305
-14.8201
9
1.0461
-14.5802
10
1.0359
-15.0099
1.0299
-15.0360
-14.8050
11
1.0519
-14.8312
1.0461
-14.8581
12
1.0589
-15.2629
1.0533
-15.2973
13
1.0523
-15.2974
1.0466
-15.3313
14
1.0253
-16.0325
1.0193
-16.0717
10
1.0462
11
1.0634
-14.6886
12
1.0739
-15.2360
13
1.0689
-15.3784
14
1.0381
-15.9211
Iteration
Iteration
14
7
15
V. FUTURE RESEARCH To test the power flow results, Kirchhoff’s current law (KCL) is applied to each bus of the modified IEEE 14 bus test case; that is, at any bus, the sum of branch currents injected into that bus should be equal to the load currents flowing out of that bus. Load currents can be calculated from equations (23) – (26). Utilizing voltages of each bus of the system as shown in Table 10, the sum of the branch currents injected into the i-th bus can be obtained from the equation: 𝐼𝑖 = ∑𝑛𝑘=1 𝑌𝑖𝑘 𝑉𝑘
(27)
The test results show that the power flow results in Table 10 are correct. This method has the following benefits. It is easy to consider all types of loads, since the loads can be readily represented by injection currents. The bus admittance/impedance matrix, once formulated, doesn’t need to be changed if the network remains unchanged, © 2012 ACADEMY PUBLISHER
In state estimation, the selection of error variances for measurements and model mismatches may have significant effects on the state estimation accuracy. Error variances for the measurements can be chosen according to meter measurement characteristic, which is usually found in meter’s manual. However, it is more difficult to find error variances for model mismatches. Thus, one future work is to find suitable methods to determine the error variances for model mismatches for enhanced state estimation accuracy. VI. CONCLUSION This paper presents a new state estimation algorithm based on the extended WLS method for accounting for both model errors and measurement errors. For comparison purposes, both the traditional WLS method
JOURNAL OF COMPUTERS, VOL. 7, NO. 3, MARCH 2012
and the new method are evaluated by performing case studies. The estimated results evince that the extended WLS method may outperform the traditional approach when the power network model used is not accurate. The extended WLS method is best suited for cases with known variances for model errors. In addition, this paper also examines a power flow method based on Z matrix for transmission systems with multiple types of loads, and the test results demonstrate that the method is effective and correct. The benefit of this method is that it is straightforward to model composite types of loads in the analysis. Studies also show that multiple valid solutions may exist for a power flow problem. REFERENCES [1] A. Monticelli, State Estimation in Electric Power Systems: a Generalized Approach, New York: Kluwer, 1999. [2] Y. Liao, “State estimation algorithm considering effects of model”, SoutheastCon, 2007. Proceedings. IEEE. pp. 450453. [3] P. Zarco and A. Gomez, “Power system parameter estimation: A survey”, IEEE Transactions on Power systems, vol. 15, no. 1, Feb. 2000, pp. 216-222. [4] W. A. Fuller, Measurement Error Models. New York: Wiley, 1987. [5] A. Yeredor, The extended least squares criterion: minimization algorithms and applications, IEEE Transactions on signal processing, vol. 49, no. 1, January 2001, pp. 74-86. [6] B. Stott, “Review of Load-Flow Calculation Methods”, IEEE Transactions on Power Delivery, Vol. 62, no. 7, Jul 1974, pp. 916 – 929. [7] A. Brameller and J. K Denmead, “Some improved methods of digital network analysis, Proc. Znst. Elec. Eng., vol. 109A, pp. 109- 116, Feb. 1962. [8] http://www.ee.washington.edu/research/pstca/, Power System Test Case Archive. [9] W.F. Tinney and C.E. Hart, “Power flow solution by Newton’s method", IEEE Transactions on Power Apparatus and Systems, Vol PAS-86, No. 11, 1967, pp. 1449 – 1460. [10] A. Monticelli and F. F. Wu, “Network Observability: Theory”, IEEE Trans. On Power Apparatus and Systems, vol. PAS-104, no. 5, pp. 1042-1048, May 1985. [11] J. Grainger and W. Stevenson, Power System Analysis, New York: McGraw-Hill, 1994. [12] Yuan Liao, Lecture Notes of Smart Grid class, Department of Electrical and Computer Engineering, University of Kentucky, 2011.
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Jiaxiong Chen was born in Fuqing, China, on September 7, 1985. He received his B.S. degree from Department of Electrical Engineering, Hunan University, Changsha, China, in 2008. In 2010, he received his M.S. degree from Department of Electrical and Computer Engineering, University of Kentucky, Lexington, KY, USA. He is currently pursuing his PH.D. degree in electrical engineering at University of Kentucky. His research field is power system state estimation. Yuan Liao (S’98, M’00, SM’05) is currently an Associate Professor at the department of electrical and computer engineering at the University of Kentucky. From August 2000 to August 2005, he was with the ABB Corporate Research Center as an R&D Consulting Engineer and then Principal R&D Consulting Engineer. His research interests include protection, power quality analysis, large-scale resource scheduling optimization and EMS/SCADA design.
