Arab J Sci Eng DOI 10.1007/s13369-015-1698-6
RESEARCH ARTICLE - ELECTRICAL ENGINEERING
State Space Least Mean Fourth Algorithm for Dynamic State Estimation in Power Systems Arif Ahmed1,2 · Muhammad Moinuddin1,2
· Ubaid M. Al-Saggaf1,2
Received: 20 October 2014 / Accepted: 17 May 2015 © King Fahd University of Petroleum & Minerals 2015
Abstract Power system dynamic state estimation (DSE) has always been a critical problem in studying power systems. One of the essential parts of power systems are synchronous machines. In this work, we dealt with the problem of DSE of a synchronous machine by introducing a novel state space-based least mean fourth (SSLMF) algorithm. The rationale behind the proposed algorithm is the fact that a power system may encounter non-Gaussian disturbances/state errors and the least mean fourth algorithm is proven to be better in such environments. Moreover, we have also introduced a normalized version of the proposed algorithm, namely state space normalized least mean fourth (SSNLMF) algorithm to deal with the stability issue under Gaussian disturbances. Another motivation for developing the SSLMF algorithm is its simplicity as compared to other model-based nonlinear filtering algorithms such as Kalman filter, extended Kalman filter (EKF). Moreover, we also investigate the performance of the recently introduced state space least mean square (SSLMS). Performance of the SSLMF and the SSLMS is compared with existing EKF in both Gaussian and non-Gaussian noise environments. Extensive simulation results are presented which show superiority
B
Muhammad Moinuddin
[email protected] Arif Ahmed
[email protected] Ubaid M. Al-Saggaf
[email protected]
1
The Electrical and Computer Engineering Department, King Abdul Aziz University, Jeddah 21859, Saudi Arabia
2
Center of Excellence in Intelligent Engineering Systems (CEIES), King Abdul Aziz University, Jeddah 21859, Saudi Arabia
of the proposed algorithms, and hence, it verifies our rationale behind the work. Keywords SSLMF algorithm · SSLMS algorithm · Power system state estimation · Synchronous generator dynamic state estimation · Least mean fourth · Least mean square
1 Introduction Power systems have evolved into very complex networks having capabilities of bidirectional power flow, distributed generation and much more. To ensure proper performance and operation of a power system, its monitoring, control and optimization are essential which require proper estimation of the necessary states. This has enabled state estimation (SE) to become a very important part of the energy management systems (EMS) [1]. SE assists in performing contingency analysis, security assessment, bad data detection, optimization and real-time modelling of the grid [2–6]. Another vital development is the phasor measurement unit (PMU) which has highly contributed to power system SE [1,7,8]. These synchronized devices take data samples at specific rates which aid in having an overview of the power system dynamics. Among the vast literature available on power system SE, many deal with the SE at the synchronous machine level [9–16]; however, this field is always open to better, accurate, robust and lesser computationally complex estimation techniques. To perform different analyses by the utility for large-scale power system, it is necessary to have a precise model of the power system (such as switch statuses, line parameters and their constraints) aided by precise representation of states. However, the aim of this work is to deal with only the DSE of the synchronous generator using our pro-
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posed algorithms as it is essential in analyses, monitoring and control of the power system. In the domain of DSE of power systems, remarkable work has been done by Debs and Larson [17], where Kalman filtering was applied for the estimation. Since then, a lot of adaptive filtering algorithms have been applied to the problem of DSE of power systems [9–14]. For synchronous generator SE, [18] presented dynamic estimation technique using Kalman filtering for the second-order model where performance was investigated with respect to sampling rate and noise level. In [13], a hybrid observer scheme for estimation was used using third-order synchronous generator model and estimation of the third-order synchronous generator states was performed using square-root unscented Kalman filter (SQ-UKF) in [14]. In [11], a fourth-order synchronous generator model was used for the estimation using EKF and its variant. Ghahremani and Kamwa further proposed in [12], an UKF-based algorithm which has an upper hand compared to the EKF because of its ability to estimate even with bad initial guess. More recently, [19] proposed a particle filtering approach to estimate the states of a synchronous generator in a multi-machine system having non-Gaussian noise. The work was further complimented with [20], where an extended particle filter was proposed to estimate the states of a fourth-order nonlinear synchronous generator which was shown to be more accurate and robust in the presence of non-Gaussian noise. However, all of these model-based estimation techniques either have very high computational complexity [21] or need very precise initial guess of states to be estimated [11,21,22]. Recently, in [23], it was demonstrated that the SSLMS algorithm exhibits superior tracking performance compared to traditional least mean square (LMS) and recursive least squares (RLS) algorithms while having lesser computational complexity compared to existing model-based algorithms. This inspired us to develop a simple and efficient state spacebased least mean fourth (SSLMF) algorithm which not only inherits simplicity in its architecture but also promises better performance under non-Gaussian noise environments. Moreover, the existing SSLMS has not been implemented in a nonlinear and particularly power system DSE problem. Therefore, in this work, both the existing SSLMS and the proposed SSLMF algorithms are investigated in power system DSE. This paper has a twofold contribution to the problem of DSE of synchronous generator. Firstly, we proposed a state space least mean fourth (SSLMF) algorithm for the DSE of a fourth-order synchronous generator which can overcome the limitations of the techniques mentioned in previous paragraph. More precisely, we developed a novel SSLMF algorithm by minimizing the mean fourth prediction error via stochastic gradient optimization method. The motivation behind this proposed algorithm lies in the fact that LMF algo-
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rithm performs better in non-Gaussian noise environments [24,25]. This fact along with the observation in the existing literature that disturbance/noise in power system can be non-Gaussian [26–28], provides the basis of our proposed work. Secondly, we also implemented the existing SSLMS algorithm [23] in DSE problem and compared these with the performance of the EKF. All of the investigations were performed in the presence of both Gaussian and non-Gaussian environments such as binary and uniform noise environments [26]. This paper is organized as follows: Sect. 2 reviews the basic SS model of linear and nonlinear systems with the special case of an unforced linear system. Section 3 gives an overview of SSLMS followed by Sect. 4 where the SSLMF algorithm is proposed and developed. An overview of the fourth-order nonlinear synchronous generator model is presented in Sect. 5. While in Sect. 6, the simulation results are presented for three different cases, and for each case, simulations in the presence of three different types of noises were investigated. This is then followed by a discussion of the results, thereby concluding the paper in Sect. 7.
2 State Space Model We begin by defining the general state space model of a linear time-varying system. x[k + 1] = A[k]x[k] + B[k]u[k] + w[k], y[k] = C[k]x[k] + D[k]u[k] + v[k]
(1a) (1b)
where x ∈ n are the process states, y ∈ m are the measured outputs such that m ≤ n. A[k] is the state transition matrix, B[k] is the input matrix, and u[k] is the input vector, where u ∈ p , w ∈ n is the process noise vector and v ∈ m is the measurement noise vector. The matrix C[k] is the output matrix, where dim[C[k]] = m × n, and D[k] is the feed through matrix with dim[D[k]] = m × p. It is assumed that the above system is observable. A special case is the unforced (autonomous) linear time-varying system, represented as x[k + 1] = A[k]x[k] + w[k], y[k] = C[k]x[k] + v[k]
(2a) (2b)
The state space representation for a nonlinear continuous time system is x˙ = f (x, u, w),
(3a)
y = h(x, u, v)
(3b)
where f and h are nonlinear functions and the parameters are as defined before.
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3 Overview of the SSLMS
The estimator equation from Eq. (10) and (12) is hence derived as
Considering the system described by Eq. (2) above, a modelbased adaptive estimation process can be divided into the following two steps. Step 1, the time update which is given by x¯ [k] = A[k − 1]ˆx[k − 1]
(4)
where xˆ [k − 1] is the state estimate for the (k − 1)th instant and x¯ [k] is the predicted state at the kth instant. Step 2, the measurement update which is given by xˆ [k] = x¯ [k] + K[k]ε[k]
(5)
(6)
K[k] = μGCT [k]
y¯ [k] = C[k]¯x[k]
4 Derivation of Proposed SSLMF
This is basically the structure employed in all KF techniques [29–32]. From Eqs. (4), (6) and (7), Eq. (5) can be written as xˆ [k] = [I − K[k]C[k]]A[k − 1]ˆx[k − 1] + K[k]y[k]
(8)
In the case of SSLMS [23], the gain matrix K[k] is derived as follows e[k] = y[k] − yˆ [k] = ε[k] − C[k]δ[k]
(9)
where δ[k] = xˆ [k] − x¯ [k]
(16)
The update Eq. (5) can be thought of as the steepest descentbased adaptation [21,22] and therefore can be generalized with the following update rule: xˆ [k] = x¯ [k] − μ∇J[k]
J[k] = E
m
εi4 [k]
(18)
i=1
(10) where εi [k] is the ith prediction error defined as εi [k] = yi [k] − y¯ i [k]
ε[k] = C[k]δ[k]
Here
(11)
δ[k] is chosen as the minimum norm solution of the above equation which results in (12)
y¯ i [k] = Ci [k]¯x[k]
(13)
(19)
(20)
Ci [k] is the ith row of the output matrix. The minimization of the cost function with respect to the predicted states results in
here K[k] = CT [k](C[k]CT [k])−1
(17)
where J[k] is the cost function to be minimized and ∇J[k] is the gradient. In our proposed SSLMF algorithm, the cost function is the fourth power of the prediction error ε[k].
Assuming C[k] is full rank, xˆ [k] is chosen such that e[k] = 0 which implies the following
δ[k] = CT [k](C[k]CT [k])−1 ε[k]
(15)
here
here y[k] is as mentioned in Eq. (2), and K[k] is the gain matrix and (7)
(14)
This is termed in [23] as the SSNLMS algorithm. However, the above estimator may be unstable and there are no known conditions for stability, and hence, convergence is not guaranteed. For this reason, a G matrix has been introduced in [23] to overcome this problem. In [23], it is claimed that the choice of this matrix G depends on the nature of the problem and one simple approach is to take G to be all zeros except for the first column to have nonzero entries. In our investigation, we will consider the SSLMS algorithm [23] defined as xˆ [k] = x¯ [k] + μGCT [k]ε[k]
where ε[k] = y[k] − y¯ [k]
xˆ [k] = x¯ [k] + CT [k](C[k]CT [k])−1 ε[k]
∇J[k] = −4
m
εi3 [k]CiT [k]
(21)
i=1
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From Eqs. (17) and (21), the estimator equation is thus represented by xˆ [k] = x¯ [k] + μG
m
εi3 [k]CiT [k]
(22)
x = [δ ω E q E d ]T
(25a)
u = [Tm E f d ]
(25b)
T
where
i=1
where μ is the step size to assist quick achievement of the solution and the matrix G was imposed for the condition of controllability which is required due to the dynamics of the system in which the algorithm is being applied. In other words, if the system is not observable, the G matrix ensures observability by taking into account the effect of all the state variables in the estimation process. It is a well-known fact in the literature that LMF algorithm gives faster convergence at the expense of higher steady-state error [33,34] under Gaussian noise environments and it performs better in terms of both convergence and steady-state error as compared to the LMS in non-Gaussian environments which is the main rationale behind our proposal. Another critical issue with the LMF algorithm is its instability in the presence of Gaussian input or any random input with infinite probability density function (PDF) support [35]. To deal with this problem, we also proposed a normalized version of SSLMF which is designed using the concept of a recently introduced normalized LMF [36] variant that is proven to be stable under any circumstances. Consequently, by employing the concept of [36], it can be shown that the stable normalized version of the proposed SSLMF algorithm is given by following recursion xˆ [k] = x¯ [k] +
μG
m
3 T i=1 εi [k]Ci [k] + ||C[k]||4F
(23)
where is a small positive quantity to avoid division by zero scenario and the subscript F represents the Frobenius norm which is defined as ||C[k]|| F =
trace(C∗ [k]C[k])
(24)
The C∗ [k] here represents the conjugate transpose of C[k]. For the case when C[k] is a vector (single output model), the above norm reduces to conventional l2 norm.
δ˙ = ωo ω 1 ˙ = (Tm − Te − Dω) ω J 1 E˙q = (E f d − E q − (xd − xd )id ) Tdo 1 E˙d = (−E d + (xq − xq )iq ) Tqo
(26b) (26c) (26d)
In the above equations, Tm and E f d are the mechanical input torque (pu) and field excitation voltage (pu), respectively. E q and E d are the q-axis and d-axis component of the induced stator voltages, respectively. δ is the mechanical angle (elec.rad) between the terminal voltage Vt (pu) and the internal voltage E q (pu). ω is the rotor speed deviation (pu), whereas ω0 is the rated angular velocity of the rotor (elec.rad/s). For performing the estimation of the fourthorder synchronous generator model, an extensive model of the system derived in [11] and [12] was considered which is represented as x = [δ ω E q E d ]T
(27a)
u = [Tm E f d Vt ]T
(27b)
y = [Pt Q t ]
(27c)
T
δ˙ = ωo ω (28a) Vt 1 ˙ = Tm − E sin δ ω J xd q
1 Vt2 1 (28b) − sin 2δ − Dω + 2 xq xd
E q − Vt cos δ 1 E˙q = E f d − E q − xd − xd Tdo xd
5 Nonlinear Modelling of Fourth-Order Synchronous Generator
E˙d
For the purpose of estimation, a fourth-order synchronous generator model is needed which complies with the SS model of our estimators. Estimation of fourth-order nonlinear synchronous generator has been dealt with before using various techniques and models that are widely available in various literature [11,12,37]. The fourth-order synchronous generator model can be described by the following set of nonlinear equations [37,38]
Pt
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(26a)
Qt
V sin δ
1 t = −E d + xq − xq Tqo xq Vt V2 1 1 = E q sin δ + t − sin 2δ xd 2 xq xd 2 Vt sin2 δ 2 cos δ = E q sin δ − Vt + xd xd xq
(28c) (28d) (28e) (28f)
The parameters are as mentioned before with input Vt , the terminal voltage (pu). Pt and Q t are the terminal active (pu) and reactive power (pu), respectively. The variables and constants
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can be referred to in [11] with the exception of E f d which was considered to be 2.0. (xd ,xq ) and (xd ,xq ) are the d- and q-axis reactances and transient reactances, respectively. It should be noted that for the purpose of estimation, the saturation effect is neglected. These set of nonlinear equations for a fourthorder generator model, derived in [11] and [12], are essential to validate our SS estimator. One of the requirements of the state space least mean algorithms is the availability of the C[k] matrix which is necessary for the computation of the estimated states. The C[k] matrix was, therefore, calculated by taking the Jacobian of the output equation y = [Pt Q t ]T
(29)
where Vt V2 Pt = E q sin δ + t xd 2 Qt =
Vt E sin δ − Vt2 xd q
1 1 − xq xd
sin 2δ
cos2 δ sin2 δ + xd xq
The C matrix thus becomes ⎡ ∂P ∂P ∂P t t t ∂δ ∂ω ∂ E q C = ∂y = ⎣ ∂x
∂ Pt ∂ E d
∂ Qt ∂ Qt ∂ Qt ∂ Qt ∂δ ∂ω ∂ E q ∂ E d
(30a)
(30b)
⎤ ⎦
(31)
where V2 ∂ Pt Vt = E q cos δ + t ∂δ xd 2 ∂ Pt Vt = sin δ ∂ E q xd
∂ Q t −Vt = E q sin δ − 2Vt2 ∂δ xd ∂ Q t Vt = cos δ ∂ E q xd
1 1 − xq xd
1 1 − xq xd
Scenario 1. In this scenario, we investigated the performance of the three algorithms for DSE of the fourthorder nonlinear synchronous generator model for constant inputs. Scenario 2. For this scenario, we used a varying input signal E f d modelled with ramp function while rest of the parameters are as mentioned. Scenario 3. Lastly, two short-circuit faults at the terminal of the synchronous generator were investigated. 6.1 Scenario 1
2 cos 2δ
(32a) (32b)
EKF algorithm. The elements of the G matrix are taken to be all zeros except for the first column to be unity. As we emphasized earlier that the rationale behind the proposed algorithm is to investigate the performance in non-Gaussian environments, we used three different noise environments, namely Gaussian, uniform and binary, in our study. For the simulation of the real system, the state vector considered is x[0] = [0.8 0 0 0]T , whereas the initial estimate for the compared estimators is set to xˆ [0] = [0.5 0 0 0]T . For the purpose of our study, we compared the performance of three algorithms (EKF, SSLMS and our proposed SSLMF). In the exceptional case of Gaussian noise, we included the performance of the proposed normalized SSLMF algorithm in this comparison. In all the results presented, we have shown the average of 100 independent simulation experiments. The following three different scenarios of system dynamics were investigated.
sin δ cos δ
∂ Pt ∂ Qt ∂ Pt ∂ Qt = = = =0 ∂ω ∂ Ed ∂ω ∂ E d
(32c) (32d)
(33)
6 Simulation Results and Discussion We considered the problem of estimating the states of the fourth-order nonlinear synchronous generator described by Eqs. (27), (28), (29) and (30). It is assumed that the states as well as the measurements are noisy. The process and the observation noises are considered as zero mean white noises of variances σw2 = 0.0012 and σv2 = 0.012 , respectively. The system was sampled at 0.01s. The covariance matrices R[k] and Q[k] were considered as diag([0.22 , 0.22 ]) and diag([0.082 , 0.082 , 0.082 , 0.082 ]), respectively, for the
In the first scenario, we simulated the system for constant input and the estimation performance of the discussed algorithms was observed. The rationale was to compare the overall performance of these algorithms in the presence of the three types of noises mentioned earlier. Due to the lack of space, we are reporting the case of uniform noise for this scenario. In the subplots of Fig. 1, we compared the actual states of the synchronous generator with the ones obtained via the three algorithms. The corresponding mean square estimation errors are also reported in Fig. 2. It can be depicted from these two figures that the proposed SSLMF algorithm performs better than the existing EKF and the SSLMS in this scenario. In the results of the output observation errors shown in Fig. 3, the performance of all the algorithms are comparable. The RMSE of the SE presented in Table 1 clearly shows the superiority of the state space least mean algorithms. 6.2 Scenario 2 The objective of this scenario was to investigate the effectiveness of the proposed algorithm for a time-varying input. Specifically, the input E f d was varied from 0 to 5 s with
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Arab J Sci Eng Fig. 1 State x1 , x2 , x3 and x4 in the presence of uniform noise
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(a) Mean Square Error, P (dB)
Fig. 3 Mean square observation error in the presence of uniform noise
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a ramp signal having a slope of +5 %/s. In this case, we are reporting the result for only Gaussian noise environment.1 The zoomed view of the rotor angle and the rotor speed of Fig. 4 tells us that initially the proposed SSNLMF algorithm has a faster convergence but with larger deviation from the true states. However, as time proceeds, the overall mean square estimation error was found to be lesser than the other compared algorithms. Moreover, the SSLMS algorithm was also found to be better than EKF. This fact can also be observed in the plots of the mean square SE errors shown in Fig. 5. These results show the consistency in the performance of the proposed algorithms even in the time-varying input scenario. 6.3 Scenario 3 In this study, we performed two types of fault analysis obtained by applying a short circuit at the terminal of the generator. Here, we are reporting the result for binary noise only. In the first analysis, the short-circuit fault was applied at 5s and cleared at 5.1s so that the system rendered to be stable.
1
Similar behavior was observed in the other noise environments too, but we are reporting only one case for the comparison purpose.
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The plots for the first two states are reported in Fig. 6, while the other two in Fig. 7. Their corresponding mean square estimation errors are displayed in Fig. 8. It can be easily seen from these figures that the proposed SSLMF outperforms the other compared algorithms which further validates the consistency in its performance. In the second analysis, the short-circuit fault was applied at 5 s but cleared comparatively later at 5.3 s which rendered the system unstable. Here too, similar performance can be observed in Figs. 9 and 10. Moreover, it can also be observed that the EKF is not able to track the system in some of the cases due to short-circuit fault. On the other hand, both the SSLMS and the SSLMF perform very well under such scenarios. Finally, the RMSE obtained by averaging 100 independent simulations for all the three cases in the presence of the three noise environments are reported in Table 1. These results clearly show the supremacy of the proposed SSLMF algorithm in all scenarios with different noise environments.
7 Conclusion In this work, we have explored the performance of our newly proposed SSLMF, the existing SSLMS and the EKF algorithms on the well-known nonlinear synchronous gen-
Arab J Sci Eng Fig. 4 State x1 , x2 , x3 and x4 in the presence of Gaussian noise
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Fig. 5 Mean square estimation error in the presence of Gaussian noise
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Arab J Sci Eng Fig. 9 State x1 , x2 , x3 and x4 with short-circuit fault cleared at 5.3 s (binary noise)
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Time (seconds)
(b) Rotor Speed Δω (pu)
0.2
EKF SSLMS True SSLMF
0.15 0.1 0.05 0 −0.05 0
2
4
6
8
10
12
14
16
18
20
Time (seconds)
(c) True EKF SSLMS SSLMF
q
Voltage E, (pu)
1
0.5
0
−0.5 0
2
4
6
8
10
12
14
16
18
20
Time (seconds)
True EKF SSLMS SSLMF
0.5
d
, Voltage E (pu)
(d)
0
−0.5 0
2
4
6
8
10
12
14
16
18
20
Time (seconds)
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Fig. 10 Mean square estimation error in the presence of binary noise
Mean Square Error, Rotor Angle δ (dB)
Arab J Sci Eng
(a) 0 EKF SSLMS SSLMF
−10 −20 −30 −40 −50 −60
0
2
4
6
8
10
12
14
16
18
20
Mean Square Error, Rotor Speed Δω (dB)
Time (seconds)
(b) −20 EKF SSLMS SSLMF
−40
−60
−80
−100
0
2
4
6
8
10
12
14
16
18
20
, q
Mean Square Error, Voltage E (dB)
Time (seconds)
(c) 0
EKF SSLMS SSLMF
−20 −40 −60 −80 −100
0
2
4
6
8
10
12
14
16
18
20
, d
Mean Square Error, Voltage E (dB)
Time (seconds)
(d) 0
EKF SSLMS SSLMF
−20 −40 −60 −80 −100
0
2
4
6
8
10
12
Time (seconds)
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Arab J Sci Eng Table 1 RMSE taken by averaging 100 simulations
Root mean square error
State δ
State ω
State E q
State E d
Observation Pt
Observation Qt
EKF
0.0176
0.0022
0.0155
0.0084
0.0229
0.0277
SSLMS
0.0132
0.0023
0.0067
0.0081
0.0082
0.0153
SSLMF
0.0113
0.0011
0.0050
0.0061
0.0125
0.0127
EKF
0.0170
0.0022
0.0155
0.0077
0.0211
0.0274
SSLMS
0.0132
0.0023
0.0067
0.0078
0.0073
0.0148
SSLMF
0.0105
0.0011
0.0048
0.0056
0.0107
0.0120
EKF
0.0170
0.0022
0.0155
0.0077
0.0212
0.0274
SSLMS
0.0132
0.0023
0.0067
0.0078
0.0073
0.0147
SSLMF
0.0107
0.0011
0.0050
0.0058
0.0106
0.0122
Algorithm First scenario Constant input Gaussian noise
Uniform noise
Binary noise
Second Scenario Varying input, E f d Gaussian noise EKF
0.0176
0.0023
0.0156
0.0084
0.0228
0.0277
SSLMS
0.0131
0.0026
0.0066
0.0081
0.0082
0.0153
SSLMF
0.0113
0.0011
0.0050
0.0061
0.0126
0.0129
Uniform noise EKF
0.0176
0.0023
0.0156
0.0084
0.0229
0.0277
SSLMS
0.0131
0.0026
0.0066
0.0081
0.0082
0.0153
SSLMF
0.0106
0.0011
0.0048
0.0057
0.0112
0.0122
Binary noise EKF
0.0176
0.0023
0.0156
0.0084
0.0229
0.0277
SSLMS
0.0131
0.0026
0.0066
0.0081
0.0082
0.0153
SSLMF
0.0103
0.0010
0.0046
0.0055
0.0106
0.0119
Third scenario Short-circuit fault cleared at 0.1s Gaussian noise EKF
0.0215
0.0027
0.0164
0.0091
0.0293
0.0287
SSLMS
0.0140
0.0025
0.0071
0.0082
0.0073
0.0151
SSLMF
0.0115
0.0010
0.0054
0.0059
0.0114
0.0129
Uniform noise EKF
0.0215
0.0027
0.0164
0.0091
0.0294
0.0287
SSLMS
0.0140
0.0025
0.0070
0.0082
0.0073
0.0151
SSLMF
0.0106
0.0011
0.0049
0.0056
0.0107
0.0121
Binary noise EKF
0.0216
0.0027
0.0164
0.0091
0.0293
0.0287
SSLMS
0.0142
0.0026
0.0071
0.0082
0.0074
0.0151
SSLMF
0.0102
0.0011
0.0046
0.0055
0.0105
0.0118
Short-circuit fault cleared at 0.3 s Gaussian noise EKF
0.4295
0.0024
0.1835
0.2117
0.6100
0.6099
SSLMS
0.0176
0.0020
0.0074
0.0085
0.0140
0.0163
SSLMF
0.0152
0.0017
0.0066
0.0078
0.0181
0.0187
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Arab J Sci Eng Table 1 continued
Root mean square error Algorithm
State δ
State ω
State E q
State E d
Observation Pt
Observation Qt
Uniform noise EKF
0.4294
0.0024
0.1835
0.2116
0.6099
0.6098
SSLMS
0.0178
0.0020
0.0074
0.0086
0.0140
0.0164
SSLMF
0.0181
0.0011
0.0090
0.0094
0.0172
0.0202
Binary noise EKF
0.4294
0.0024
0.1835
0.2116
0.6099
0.6099
SSLMS
0.0181
0.0020
0.0075
0.0086
0.0141
0.0164
SSLMF
0.0190
0.0011
0.0095
0.0099
0.0170
0.0201
erator dynamic parameter estimation problem. To get more insight, we investigated the parameter estimation for three different case studies including short-circuit fault analysis, variable input and constant input scenarios. Our investigations revealed that both the proposed SSLMF and the existing SSLMS algorithms are very effective in estimating the dynamic states of a power system in the presence of different types of noise environments. However, SSLMF has superior performance over the other compared algorithms. We also proposed a stabilized version (SSNLMF) of our proposed SSLMF in order to avoid its divergence problem for any noise with infinite support PDF. A major benefit of SSLMF and SSLMS algorithms over the existing model-based nonlinear algorithms (EKF, UKF, etc.) is in their computational simplicity. Although all of our investigations were based on simulation experiments, the performance of the proposed algorithms is expected to be well in line with real-world scenarios too. This is due to the fact that our investigation is based on a higher-order nonlinear representation of the synchronous machine which is known to closely resemble the real-world synchronous machine. Acknowledgments The authors acknowledge the support provided by the Centre of Excellence in Intelligent Engineering Systems (CEIES), King Abdulaziz University, Jeddah, Saudi Arabia, to carry out this work. The authors also acknowledge the support from Dr. Esmaeil Ghahremani for providing assistance in studying short-circuit analysis.
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