A new method for power system contingency ranking using combination of neural network and data envelopment analysis

Abstract

In this paper, an integrated algorithm has been proposed for ranking contingencies in the deregulated network. The network security and economical indices should be considered when dealing with market environment. Locational marginal price and congestion cost indices are the best signals to completely illustrate the market operation. In this paper, voltage violation, line flow violation, locational marginal price and congestion cost indices have been simultaneously considered to rank the contingencies. This algorithm uses neural networks method to estimate the power system parameters (locational marginal price, bus voltage magnitudes and angles). The efficiency of each of contingencies was calculated using data envelopment analysis and this index was employed for ranking. The efficiency of each contingency shows its severity and indicates that it affects network security and economic indices concurrently. Considering the proposed formulation for data envelopment analysis, the efficiency of a contingency will be higher if the calculated indices for that contingency are higher. More efficiency leads to increased severity of the contingency and shows that the contingency has concurrently more affected network security and economic indices. The proposed algorithm has been tested on IEEE 30-bus test power system. Simulation results show the high efficiency of the algorithm.

Keywords

Contingency ranking deregulated network neural network network security indices data envelopment analysis

1 Introduction

Maintaining power system security in the deregulated and unbundled electricity market is a challenging task for power system engineers [6]. The main goal of contingency ranking is to find related contingencies including accurate contingencies and ranking them according to their severity. Various methods have been developed for estimating the severity of contingency. In [12], a new composite sensitivity analysis framework has been proposed for voltage contingency evaluation and ranking. The proposed formulation considers the voltage stability margin/instability depth of the entire power system as the severity index for voltage contingencies. The proposed method is tested on the New Zealand test system and Iran’s transmission network. Obtained results indicated that the proposed method can highly reduce the computation time. In [13], a method capable of selecting contingencies that lead to voltage insecurities has been proposed. The contingencies are ordered according to their effect on the system operating state. In [2], a new power sensitivity ranking algorithm for voltage collapse contingency ranking has been proposed. This new ranking algorithm considered the future variations in generation dispatch and the short term load demand forecast. In [18], a fast and precise contingency ranking method for the power systems security analysis is presented. The method proposed in [18], considered both the apparent power overloading and voltage violations, simultaneously. In [5], a method as a combinatorial optimization problem and solved by genetic algorithms is proposed to efficiently perform the selection of multiple contingencies. In [17], a contingency assessment method that takes into account the nature of probability distribution of power system operating conditions to get realistic severity and risk estimations of contingencies is proposed. The developed contingency assessment methods are applied on SEO region of French EHV system to estimate severities and rank the selected contingencies based on risk of voltage collapse. In [15], an alternative methodology is proposed for static contingency analyses that only use continuation methods and thus provides an accurate determination of the loading margin. The applicability and effectiveness of the proposed methodology have been investigated on IEEE test systems (14, 57 and 118 buses) and compared with the continuation power flow.

In recent years, fuzzy system applications and artificial intelligence methodology have received increasing attentions in various areas of power systems. In [4], a novel approach has been proposed for contingency ranking based on static security assessment. The proposed method is applied to IEEE 30-bus power system and different cased were examined. In [6], a hybrid fuzzy-neural network is developed for ranking of critical contingency using pre fault load information at selected buses. A multi-output fuzzy-neural network has also been used for contingency ranking. The membership values of the loads have been categorized in lingual groups of low, middle and high and considered as the input for neural network. A fuzzy composite performance index (FCPI), formulated by combining (i) voltage violations, (ii) line flow violations and (iii) voltage stability margin is proposed for composite ranking of contingencies. The performance of the proposed method has been tested on a 69-bus practical Indian power system. In [14], an approach based on radial basis function neural network is developed to estimate bus voltage magnitudes and angles for normal operation as well as for all possible single-line contingencies. This methodology is extended for contingency ranking. The effectiveness of proposed method is demonstrated on two IEEE test power systems. In [7], a supervised learning approach is proposed to fast and accurate power system security assessment and contingency analysis. In [7], feed-forward artificial neural network is employed that uses pattern recognition methodology for security assessment and contingency analysis. In [11], an integrated algorithm was proposed to measure the efficiency of electric companies using data envelopment analysis (DEA) combined with fuzzy c-means clustering (FCM) and principle component analysis (PCA). In [10], a multi-objective congestion management approach has been proposed to minimize transmission congestion management cost and emission. In [10], in the a posteriori stage, a solution is selected by considering power system security. For this purpose, two strategies were proposed: in the first strategy, based on a proposed managerial vision, a combination of data envelopment analysis introduced by Charnes, Cooper, and Rhodes (CCR-DEA), cross-efficiency technique and robustness analysis is deployed to select the most robust super-efficient solution. In the second strategy, first the effective scenarios due to outage of transmission components are identified using CCR-DEA and next, each scenarios’ degree of severity (DOS) is obtained using the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). In [9], a new approach for evaluation of power generation plans was proposed. Different congestion management attributes were considered besides transmission congestion management cost. Also, effective scenarios due to the outage of transmission components have been considered besides the normal operating case. In order to obtain the effective scenarios, an approach based on conjunctive method and a pessimistic approach based on CCR-DEA has been proposed.

In this paper, in order to rank the contingencies, network security indices (voltage violation and line flow violation indices) and economic indices (locational marginal price and congestion cost indices) have been considered simultaneously. This paper is organized as follows. Section 2 presents the performance indices for contingency ranking. Section 3 presents the proposed algorithm. Section 4 details the application and presents the obtained results. Finally, concluding remarks are made in Section 5.

2 Performance indices for contingency ranking

By experience, it is known that the ranking results highly depend on the operating index definition used to measure the severity of a contingency. Also, the choice of parameters in operating index definition depends on the usage of ranking results. As an example, if the ranking results are to be used for voltage stability issue, it is necessary to employ bus voltage and voltage stability margin parameters in index definition. In power transmission network management and planning, the line flow parameter is generally used. While bus voltage angles and critical clearing time are used to define index in transient stability problem. Voltage violations [4 , 14] and line flow violations [4 , 16] have been proposed as indicators of network security. The network security and economical indices should be considered when dealing with market environment. Locational marginal price (LMP) is the best economical signal to completely illustrate the market operation. Using LMP, the power consumers and producers experience real energy price in their location, and hence, LMP plays a significant role in system management.

In this paper, in order to evaluate the severity of important contingencies in the deregulated network, the network security and economic indices are considered together. The definitions related to these indices are as follows.

2.1 Voltage violation index

The most common index for voltage violation is [14]: ${PI}_{v} = \sum_{j = 1}^{nb} w_{j} {| \frac{Δ v_{j}}{k_{j}^{v}} |}^{2 m}$ (1) $K_{j}^{v} = (V_{j}^{Max} - V_{j}^{Min}) / 2$ (2) $Δ V_{j} = (V_{j} - V_{j}^{norm})$ (3)

Where $V_{j}^{Max}$ and $V_{j}^{Min}$ are over-voltage and under-voltage limits, respectively. $V_{j}^{norm}$ is a user specified for bus j and V_j is post-contingency voltage magnitude for bus j. Also, w_j is a weighting factor for bus j, nb is the number of load buses and m is a positive integer to reduce masking effects.

2.2 Line flow violation index

The most common operating index for line flow violation is [14]: ${PI}_{MVA} = \sum_{j = 1}^{nl} w_{i} {(\frac{s_{i}}{s_{ni}})}^{2 m}$ (4)

Where S_i and S_ni denote the apparent load and apparent power overload limits of line i, respectively. w_i is a weighting factor for line i, and nl is the total number of lines, respectively. m is a positive integer number used to avoid masking effect by increasing its value.

2.3 Locational marginal price (LMP) index

By definition, locational marginal price (LMP) index for a bus is the minimum excess production cost needed to feed 1 MW extra load in the bus without violating transmission constraints, and hence it depends on the cost suggested by producers, market rules and transmission constraints. LMP is one of the most important indications of market price which illustrates a lot of information of the power market. One of the measures of competition level of the market based on LMP is to investigate the distribution of LMP. In a fully competitive market, all producers and consumers sell/buy electricity with the same price, meaning that in all buses the price is the same and price profile is completely uniform, and there is no limitation for consumers on buying electricity from any desired producer. In practice, however, due to the transmission constraints and line losses, the LMPs of the buses cannot be the same, but still a more uniform LMP profile indicates more competition in market. Here, this index is used to evaluate contingencies such that a more important contingency is defined to be the one which increases LMPs standard deviation. The distribution of bus prices is calculated as: ${PI}_{LMP} = Std ({LMP}_{i})$ (5)

Where LMP_i and Std represent the LMP of bus i and standard deviation, respectively.

2.4 Congestion cost index

Congestion cost is another economic index based on LMP. Transmission congestion occurs when there is not enough transmission capability to support all requests for transmission services. The congestion cost of the whole system is the summation of all congestion costs of lines. It can be calculated as: ${PI}_{CON} = \sum_{j = 1}^{nl} ({LMP}_{j 1} - {LMP}_{j 2}) \times P_{j}$ (6)

Where LMP_j1 and LMP_j2 are the LMP at the two ends of the line j and P_j is the active power in line j.

3 The proposed algorithm

The algorithm is proposed through the following steps:

A large number of load patterns (active and reactive powers at all buses) are generated randomly.

AC load flows is performed for all load patterns and all the single line outage contingencies and the performance indices are calculated.

Three neutral networks were trained to predict LMP, voltage magnitude and voltage angle.

During testing, the efficiency of each contingency was evaluated using the data envelopment analysis.

The contingencies were ranked based on efficiency values.

3.1 Data envelopment analysis

Data Envelopment Analysis (DEA) is a non-parametric method that calculates the efficiency in a given set of decision-making units (DMUs). DEA measures the efficiency of a DMU with multiple inputs and outputs by ratios of weighted outputs to weighted inputs [1]. Assuming that there are n DMUs, each with m inputs and s outputs, the efficiency scores can be computed as a solution to the following linear programming (LP) problem: $min θ_{o}$ (7) $\begin{matrix} Subject to : \\ θ_{o} x_{io} \geq \sum_{j = 1}^{n} λ_{j} x_{ij}, i = 1, 2, \dots, m \end{matrix}$ (8) $y_{ro} \leq \sum_{j = 1}^{n} λ_{j} y_{rj}, r = 1, 2, \dots, s$ (9) $λ_{j} \geq 0, j = 1, 2, \dots, n$ (10)

Where,

θ_o : Overall efficiency (OE) of DMUs,

x_kj : Amount of input k utilized by DMU i,

y_kj : Amount of output k produced by DMU i,

λ₁, …, λ_n : The weights for the inputs and outputs of the n DMUs.

This model considers Constant Returns to Scale (CRS) and it may be adapted to other returns to scale situations.

Variable Returns to Scale (VRS) if the equality constrained $\sum_{j = 1}^{n} λ_{j} = 1$ is added to equation. The relative efficiency obtained by this method is called technical efficiency (TE). The scale efficiency (SE) can be estimated by dividing OE into TE.

Non Increasing Returns to Scale (NIRS) if the equality constrained $\sum_{j = 1}^{n} λ_{j} \leq 1$ is added to equation. The nature of the scale inefficiencies (i.e., due to increasing or decreasing returns to scale) for a particular DMU can be determined by seeing whether the NIRS score is equal to the TE score. If they are unequal then increasing returns to scale (IRS) exist for the DMU. If they are equal then decreasing returns to scale (DRS) apply [3].

In this paper, the voltage violation index (PI_V) and line flow violation index (PI_MVA) are considered as the inputs, while outputs are the LMP index (PI_LMP) and congestion cost index (PI_CON). The efficiency of each contingency, illustrating its severity, is then calculated based on these values.

3.2 Neutral network

The generic diagram of the radial basis function (RBF) neutral network employed in this paper is shown in Fig. 1 [14]. The RBF model used here is composed of an input array and two layers (one hidden and one output layers). Also, in this network a Gaussian function is employed, which has the highest output when the input variables are closest to the center position and decreases monotonically as the distance from the center increases. Let X_p is the input array with component, x_1p, x_2p, …, x_rp. The output of the ith RBF unit in the hidden layer, which is y_i (X_p) can be calculated using Equation (11).

Fig.1

The diagram of the RBF neutral network [14].

$y_{i} (X_{p}) = exp (- \frac{\sum_{j = 1}^{r} [X_{jp} - {\hat{X}}_{ji}]^{2}}{σ_{i}^{2}})$ (11)

In Equation (11), X_jp is the jth input pattern p and ${\hat{X}}_{ji}$ is the center of the ith RBF unit, U_i, for the jth input variable. Also, σ_i is the width of the ith RBF unit, U_i. The output layer consists of a linear combiner whose output is presented in Equation (12). $O_{mp} = \sum_{i = 1}^{H} W_{mi} y_{i} (X_{p}) + W_{mB}$ (12)

Where H is the number of hidden RBF units, O_mp is the output of the mth node of output layer for pth input pattern, W_mi is the weight between ith RBF unit, U_i, and mth output node, and W_mB relates to the bias in mth output node in the linear output layer.

In this paper, the orthogonal least squares (OLS) algorithm is used to train and build an RBF neutral network. OLS algorithm is a structure identification algorithm and builds a suitable network structure in an intelligent way during learning. It chooses appropriate RBF centers as neurons and trains the patterns one after the other until it reaches a specified error.

In this paper, three RBF neutral networks are designed for normal conditions and every contingency. Each input pattern includes active injection, P, in all buses except the slack bus and reactive injection, Q, in all load buses. Also, the patterns consisting of zero or constant values are excluded from the input patterns. Each input pattern [x] is represented as: $\begin{matrix} [x] = [P_{G 1}, \dots, P_{Gg}, P_{L 1}, \dots, P_{Ln}, \\ Q_{L 1}, \dots, Q_{Ln}] \end{matrix}$ (13)

For the voltage predictor, the output vector [v] includes all the load bus voltage magnitudes: $[v] = [v_{L 1}, \dots, v_{Ln}]$ (14)

The angle predictor builds the voltage angles [θ] in all buses except the slack bus as: $[θ] = [θ_{G 1}, \dots, θ_{Gg}, θ_{L 1}, \dots, θ_{Ln}]$ (15)

Finally, in the LMP predictor, the LMP values in all buses form the output vector [LMP]: $[LMP] = [{LMP}_{sk}, {LMP}_{Gg}, {LMP}_{L 1}, \dots, {LMP}_{Ln}]$ (16)

Where G is the generator bus, g is the number of generators in the power system, L is the load bus, n is the number of load buses and sk is the slack bus in the power system.

4 Simulation test results and discussion

The proposed algorithm is tested on IEEE 30-bus power system. In this paper, only the single-line outages are considered and the RBF neutral networks are designed to estimate the post-contingency LMP, voltage magnitude, bus angles for every possible contingency in the test systems. In normal operating conditions, for each bus, 1000 load patterns were randomly generates by perturbing the load (the loads are considered as fuzzy). Similarly, for each line outages, 1000 patterns of bus injection are generated for 1000 different operating conditions in both systems. Among the 1000 produced patterns for each system, 750 patterns were randomly chosen and saved for training and the remaining 250 ones are marked to be used as test patterns.

4.1 IEEE 30-bus system

In this paper, MATLAB coding is developed to validate the proposed algorithm. There is no explicit indication of how to select weighting factors and value of exponent (m). It is observed from simulation that for m = 4 (the value of exponent), masking effect has been removed 30-bus test power system. All weighting factors are assumed to be equal.

The IEEE 30-bus system consists of 24 PQ buses, 5 PV buses, a slack bus and 41 lines. The three aforementioned neural networks for prediction of angle, LMP and voltage magnitude are designed for this system. Here, the LMP, voltage magnitude and angle predictors predict the LMP for all buses, voltage magnitude for all 24 PQ buses, and voltage angle for all buses except the slack bus, respectively. The inputs are active power for buses 2 to 30 and reactive power for all PQ buses, meaning that there are 44 elements in input patterns. If the power injection of the some buses is zero, they are not considered in the input pattern. The active and reactive powers in bus 1 are not considered as input, since it is the slack bus. Hence, the LMP predictor includes 44 and 30 neurons in the input and output layers respectively, showing the LMP values for all buses. The voltage magnitude predictor consists of 44 neurons in input layer and 24 ones in the output layer, presenting the voltage magnitudes for 24 PQ (load) buses. Similarly, the angle predictor is composed of 44 input neurons and 29 output neurons (standing for total number of 29 PV and PQ buses, except the slack bus). The proposed neural networks are tested for the entire 250 test patterns for each of the 34 outages. The maximum absolute errors in the estimated voltage magnitude and angle in each outage case are presented in Table 1. Table 1 shows that the maximum absolute errors in prediction of voltage magnitude and angle are of the order of 10^–3. From Table 1, the predicted values of voltage magnitude and angle are very close to those obtained by the AC load flows. In this case, the retrained RBF neural networks in 250 test patterns were performed for each 34 feasible outages in this case.

Table 1
Summary of voltage magnitude and angle maximum absolute errors for all test cases

Contingency No. Voltage magnitude Voltage angle

1 2.29E-03 2.40E-03

2 9.13E-03 1.23E-03

3 1.52E-03 1.84E-03

4 8.26E-03 2.40E-03

5 5.38E-03 4.17E-03

6 9.96E-03 4.97E-04

7 7.82E-04 9.03E-03

8 4.43E-03 9.45E-03

9 1.07E-03 4.91E-03

10 9.62E-03 4.89E-03

11 4.63E-05 3.38E-03

12 7.75E-03 9.00E-03

13 8.17E-03 3.69E-03

14 8.69E-03 1.11E-03

15 8.44E-04 7.80E-03

16 4.00E-03 3.90E-03

17 2.60E-03 2.42E-03

18 8.00E-03 4.04E-03

19 4.31E-03 9.65E-04

20 9.11E-03 1.32E-03

21 1.82E-03 9.42E-03

22 2.64E-03 9.56E-03

23 1.46E-03 5.75E-03

24 1.36E-03 5.98E-04

25 8.69E-03 2.35E-03

26 5.80E-03 3.53E-03

27 5.50E-03 8.21E-03

28 1.45E-03 1.54E-04

29 8.53E-03 4.30E-04

30 6.22E-03 1.69E-03

31 3.51E-03 6.49E-03

32 5.13E-03 7.32E-03

33 4.02E-03 6.48E-03

34 7.60E-04 4.51E-03

Contingency No.	Voltage magnitude	Voltage angle
1	2.29E-03	2.40E-03
2	9.13E-03	1.23E-03
3	1.52E-03	1.84E-03
4	8.26E-03	2.40E-03
5	5.38E-03	4.17E-03
6	9.96E-03	4.97E-04
7	7.82E-04	9.03E-03
8	4.43E-03	9.45E-03
9	1.07E-03	4.91E-03
10	9.62E-03	4.89E-03
11	4.63E-05	3.38E-03
12	7.75E-03	9.00E-03
13	8.17E-03	3.69E-03
14	8.69E-03	1.11E-03
15	8.44E-04	7.80E-03
16	4.00E-03	3.90E-03
17	2.60E-03	2.42E-03
18	8.00E-03	4.04E-03
19	4.31E-03	9.65E-04
20	9.11E-03	1.32E-03
21	1.82E-03	9.42E-03
22	2.64E-03	9.56E-03
23	1.46E-03	5.75E-03
24	1.36E-03	5.98E-04
25	8.69E-03	2.35E-03
26	5.80E-03	3.53E-03
27	5.50E-03	8.21E-03
28	1.45E-03	1.54E-04
29	8.53E-03	4.30E-04
30	6.22E-03	1.69E-03
31	3.51E-03	6.49E-03
32	5.13E-03	7.32E-03
33	4.02E-03	6.48E-03
34	7.60E-04	4.51E-03

Table 2 represents the network security and economical indices calculated for each contingency in the 30-bus system. The efficiencies of the contingencies are calculated using the data envelopment analysis which is shown in column 7 of the Table 2. The lowest and highest efficiencies correspond to the contingencies 34 (outage of the line between buses 6 and 28) and 9 (outage of the line between buses 6 and 7), respectively. The last column of Table 2 represents the ranking of all contingencies. The minimum and maximum efficiencies are 0.6619 and 1, respectively. The average efficiency for the 30-bus network is calculated to be 0.7726. In the 30-bus network, contingency 9 is the worst one.

Table 2

Summarizing the results of the proposed algorithm on IEEE 30-bus system

Contingency No.	LO^a from-to	PI_v	PI_MVA	PI_LMP	PI_CON	Efficiency	Rank
1	1–2	9.599	188.58	0.2835	64.312	0.7621	19
2	1–3	9.131	206.05	0.3197	73.909	0.7933	10
3	2–4	09.4075	200.75	0.3264	73.562	0.7973	08
4	2–4	8.980	204.32	0.3302	74.585	0.8070	07
5	2–5	10.2710	191.80	0.2795	64.464	0.7469	21
6	2–6	8.947	193.48	0.2854	65.002	0.7719	16
7	4–6	10.5780	161.52	0.2468	56.687	0.7327	28
8	5–7	10.2480	190.47	0.2800	064.021	0.7484	20
9	6–7	15.8420	237.03	0.7862	183.300	1.0000	01
10	6–9	12.1230	202.42	0.2503	057.802	0.6972	32
11	06–10	13.5170	194.07	0.2685	060.494	0.7043	31
12	09–10	12.7060	202.42	0.2528	057.803	0.6944	33
13	04–12	14.3330	241.59	0.5324	121.070	0.8532	03
14	12–14	10.5440	200.28	0.3185	071.257	0.7720	15
15	12–15	12.6300	213.66	0.3817	085.347	0.7869	12
16	12–16	19.7130	208.58	0.5354	123.460	0.8303	04
17	14–15	09.3161	189.38	0.2893	063.968	0.7716	18
18	16–17	13.0220	195.67	0.4055	091.332	0.8131	05
19	15–18	16.4950	194.30	0.4155	095.966	0.7881	11
20	18–19	12.6360	186.20	0.366	082.658	0.7941	09
21	19–20	10.5900	257.66	0.2972	066.245	0.7283	29
22	10–17	11.1170	202.02	0.2855	064.301	0.7359	24
23	10–21	09.2113	179.55	0.2826	062.672	0.7732	14
24	10–22	17.2340	228.73	0.7217	167.130	0.9488	02
25	21–22	10.0930	198.30	0.2645	062.296	0.7349	25
26	15–23	10.7160	190.12	0.2558	062.945	0.7339	27
27	22–24	08.7469	169.30	0.2048	047.973	0.7101	30
28	23–24	19.8380	203.75	0.4485	100.690	0.7798	13
29	24–25	08.4129	0055.619	0.1208	023.122	0.7360	23
30	25–27	14.4120	055.89	0.1601	016.149	0.7719	17
31	27–29	10.9410	231.85	0.3115	069.935	0.7447	22
32	27–30	11.1320	254.49	0.3126	071.068	0.7340	26
33	29–30	08.4816	208.59	0.3249	072.224	0.8104	06
34	06–28	15.4360	101.66	0.1516	016.554	0.6619	34

^aLO: Line outage from bus number-to bus number.

The efficiency of this contingency is 1, and it is ranked 1st, meaning that it is the strongest and worst contingency in the network based on network security and economical indices. Contingency 34 (outage of the line between buses 6 and 28) is ranked last, where the voltage violation index 15.436, line flow violation index is 101.66, the LMP index is 0.1516 and the congestion cost index is 16.554, which are much less than the ones of contingency 9. For better comparison, Fig. 2 represents the voltage violation indices of the contingencies versus their ranking. Although contingency ranked 1st (contingency 9) has the highest efficiency, Fig. 2 shows that its voltage violation index (as one of the network security indices) is less than many contingencies. The voltage violation index of contingency 28 (outage of the line between buses 23 and 24), ranked 13th, is 19.838, which is the highest voltage violation index.

Fig.2

Voltage violation indices of the contingencies versus their ranking for IEEE 30-bus system.

5 Conclusion

In this paper, the contingency ranking has been done by evaluating network security and economical indices. An integrated algorithm was proposed to rank the contingency using neural networks and data envelopment analysis. Initially, neural network algorithm was used to estimate magnitudes and angles of all system buses. Hence, three neural networks are represented to estimate LMPs, bus voltage magnitudes and angles in normal conditions and different contingencies in the power system. The proposed algorithm for contingency ranking is applied to IEEE 30-bus power test system. Results on IEEE test system show that predicted quantities comparable in accuracy to actual values and maximum absolute error is 10^–3. The efficiency of each of these contingencies was calculated using data envelopment analysis. In this research, the voltage violation index and line flow violation index are considered as inputs, and the LMP index and congestion cost index are considered as outputs. The average efficiency for the 30-bus network is calculated to be 0.7726. The efficiency of each contingency shows its severity and indicates that it affects network security and economic indices concurrently. The proposed method is capable of producing fast and accurate network security indices and economic indices, so it can be used for online ranking.

Footnotes

Acknowledgments

This work was extracted from research entitled by ‘Contingency ranking of power system considering destroyer effect of power system with Nero-fuzzy network’, which is granted and supported by the Fars Science and Research Branch, Islamic Azad University, Fars, Iran.

References

Charnes

, Cooper

W.W.

and Rhodes

, Measuring the efficiency of decision making units, European Journal of Operational Research 2(6) (1978), 429–444.

Flueck

A.J.

, Gonella

and Dondeti

J.R.

, A new power sensitivity method of ranking branch outage contingencies for voltage collapse, Power Systems, IEEE Transactions on 17(2) (2002), 265–270.

Yang

and Lu

W.M.

, Assessing the performance and finding the benchmarks of the electricity distribution districts of Taiwan Power Company, IEEE Transactions On Power Systems PWRS 21(2) (2006), 853.

Baghaee

H.R.

and Abedi

, Calculation of weighting factors of static security indices used in contingency ranking of power systems based on fuzzy logic and analyticalhierarchy process, International Journal of Electrical Power & Energy Systems 33(4) (2011), 855–860.

dos Santos

J.V.C.

, Costa

I.F

and Nogueira

, New genetic algorithms for contingencies selection in the static security analysis of electric power systems, Expert Systems with Applications 42(6) (2015), 2849–2856.

Chaturvedi

K.T.

, Pandit

, Srivastava

, Sharma

and Bhatele

R.P.

, Hybrid fuzzy-neural network-based composite contingency ranking employing fuzzy curves for feature selection, Neurocomputing 73(1) (2009), 506–516.

Verma

and Niazi

K.R.

, Supervised learning approach to online contingency screening and ranking in power systems, International Journal of Electrical Power & Energy Systems 38(1) (2012), 97–104.

Pandit

, Srivastava

and Sharma

, Voltage contingency ranking using fuzzified multilayer perceptron, Electric Power Systems Research 59(1) (2001), 65–73.

Salehizadeh

M.R.

, Rahimi-Kian

and Oloomi-Buygi

, A multi-attribute congestion-driven approach for evaluation of power generation plans, International Transactions on Electrical Energy Systems 25(3) (2015), 482–497.

10.

Salehizadeh

M.R.

, Rahimi-Kian

and Oloomi-Buygi

, Security-based multi-objective congestion management for emission reduction in power system, International Journal of Electrical Power & Energy Systems 65 (2015), 124–135.

11.

Simab

and Haghifam

M.R.

, Using integrated model to assess the efficiency of electric distribution companies, Power Systems, IEEE Transactions on 25(4) (2010), 1806–1814.

12.

Amjady

and Esmaili

, Application of a new sensitivity analysis framework for voltage contingency ranking, Power Systems, IEEE Transactions on 20(2) (2005), 973–983.

13.

De Moura

R.D.

and Prada

R.B

, Contingency screening and ranking method for voltage stability assessment, IEE Proceedings-Generation, Transmission and Distribution 152(6) (2005), 891–898.

14.

Misra

R.K.

and Singh

S.P.

, Efficient ANN method for post-contingency status evaluation, International Journal of Electrical Power & Energy Systems 32(1) (2010), 54–62.

15.

Matarucco

R.R.

, Neto

A.B.

and Alves

D.A.

, Assessment of branch outage contingencies using the continuation method, International Journal of Electrical Power & Energy Systems 55 (2014), 74–81.

16.

Singh

and Srivastava

, Line flow contingency selection and ranking using cascade neural network, Neurocomputing 70(16) (2007), 2645–2650.

17.

Krishnan

and McCalley

J.D.

, Contingency assessment under uncertainty for voltage collapse and its application in risk based contingency ranking, International Journal of Electrical Power & Energy Systems 43(1) (2012), 1025–1033.

18.

Hussain

, Chen

and Thogersen

, Fast and precise method of contingency ranking in modern power system, pp, In Applied Electrical Engineering and Computing Technologies (AEECT), 2011 IEEE Jordan Conference on (2011), 1–7.