Neural Comput & Applic (2010) 19:139–150 DOI 10.1007/s00521-009-0267-x
ORIGINAL ARTICLE
Contingency evaluation and monitorization using artificial neural networks G. Joya Æ Francisco Garcı´a-Lagos Æ F. Sandoval
Received: 6 November 2006 / Accepted: 6 April 2009 / Published online: 28 April 2009 Springer-Verlag London Limited 2009
Abstract In this paper, different neural network-based solutions to the contingency analysis problem are presented. Contingency analysis is examined from two perspectives: as a functional approximation problem obtaining a numerical evaluation and ranking contingencies; and as a graphical monitoring problem, obtaining an easy visualization system of the relative severity of the contingencies. For the functional evaluation problem, we analyze the use of different supervised feed-forward artificial neural networks (multilayer perceptron and radial basis function networks). The proposed systems produce a very accurate evaluation and ranking, and so present a high applicability. For the graphical monitoring problem, unsupervised artificial neural networks such as self-organizing maps by Kohonen have been used. This solution allows both a rapid, easy and simultaneous visualization of the severity level of the complete contingency set. The proposed solutions avoid the main drawbacks of previous neural network approaches to this problem, which are explicitly analyzed here. Keywords Artificial neural network Contingency analysis Contingency ranking Kohonen’s self-organizing maps Multilayer perceptron Performance index Power system Power network security Radial basis function
G. Joya F. Garcı´a-Lagos (&) F. Sandoval Departamento de Tecnologı´a Electro´nica, ETSI Telecomunicacio´n, Universidad de Ma´laga, Campus Teatinos s/n, 29071 Malaga, Spain e-mail:
[email protected];
[email protected] G. Joya e-mail:
[email protected] F. Sandoval e-mail:
[email protected]
1 Introduction The current energy market is involved in an important process of evolution due to the new conditions imposed by circumstances such as deregulation, increasing energy consumption, and economical, social and ecological constraints in the building of new grids. These conditions force the system to work near its security limits, producing less and less conservative operation points, and hindering the action of human operators. Consequently, as a part of a power system security assessment, a continuous system monitoring becomes necessary to detect dangerous situations as soon as possible. In this context, the contingency analysis operation must inform whether the current state is secure, critical or insecure with respect to a possible fault in a particular component of the system. In this paper, we are considering line outages as the only contingencies. To decide about the security level of the system, every bus voltage and line power flow must be compared to its corresponding maximum and minimum tolerable values. Thus, each time, the system security level will be a function of both the number of grid elements (buses and lines) the limit values of which are surpassed, and of the percentage of this surpass. In any case, the target of contingency analysis is not to analyze the security of the present state, for which we suppose all voltage and flow variables are known, but to analyze the security of a future state, where the system would evolve after the hypothetic occurrence of a particular contingency. The complexity of the problem is due to both the difficulty of calculating this future state and the high number of contingencies that can be considered. Contingency analysis can be approached using both functional and graphical methods. The first one characterizes the severity of each contingency with a numerical
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value, and the second one allows the contingency evaluation in a visual way. In both cases, the majority of methods are based on the evaluation by means of some performance index (PI). These PIs, generally inform on the relative surpassing of the security limit of the grid components with respect to a particular magnitude (voltage or power flow). Two PIs have been considered in our experiments: the active power performance index, which evaluates the severity of a contingency derived from the current overload of lines, and the reactive power performance index, which evaluates the severity of a contingency derived from the voltage violation in buses. Currently, most of the extended functional contingency analysis techniques are based on the solution of the load flow problem for the system, which may be described as follows [1, 2]: For the current state of the system: (1) solve a load flow problem for each contingency to be evaluated, therefore establishing the next state of the system arrived after the contingency would have occurred. (2) Calculate the numerical value of a selected PI indicating the seriousness of each contingency. (3) Rank the contingencies from their PI values. This process, especially the first step, requires a very high computational cost, making real-time response impossible. To reduce this computational time, several techniques have been proposed, such as more efficient algorithms to solve the load flow problem, selection of a reduced set of contingencies to be analyzed, and parallelization or distribution of processes. Thus, in references [3–6], efficient load flow algorithms are presented, which are based on fast decoupled load flow. However, since these methods always use fast converging load flow algorithms, the convergence is not guaranteed in heavily loaded systems. In reference [7] an algorithm for identification of the most problematic bus is proposed, which allows the analysis to be circumscribed to a reduced subnetwork. The drawback of this method is that the specific subnetwork is different for each contingency and its definition is not direct. In reference [8] the security and operation cost analysis is parallelized by means of a set of Monte Carlo simulations, but very high hardware and design costs are necessary because the parallelization process is specific for each network. In reference [9], two main modifications are proposed to the flow problem solution: on the one hand, the number of arithmetic operations is reduced with respect to the classical method; on the other hand, a heterogeneous client/server structure based on a net of computers communicating by sockets is proposed. This system has three main advantages: flexibility, because the clients may be different both in hardware and software configuration, and their number may change dynamically; efficiency, because a strategy of task distribution allows the computational load to be equilibrated in a continuous way; and reliability, because when a
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client goes out of service, its task will be automatically assigned to another. Among their drawbacks we can relate the following: the possibility of operating in real time becomes limited by the communication delays; high hardware cost; system reliability is dependent on external factor such as the current state of the data net or the current set of connected clients; the connection time is not predetermined but dependent of the connected clients. On the other hand, a high variety of non-classical, analytical or numerical, methods for security assessment have been developed. Thus, in reference [10] a rule-based expert system is proposed for making a contingency ranking from the current state of the system. Its drawbacks are the difficulty of looking for a sufficiently expert operator for describing each system; difficulty in generating a sufficiently precise rule set; and non-portability to other systems. Fuzzy logic based methods are proposed in [11, 12] in order to avoid the need for solving the flow problem. They are essentially based on a fuzzy classifier of contingencies, which has been trained offline from a set of present states for which the next state produced by each contingency has been previously obtained. These classifiers may have a binary output (secure–insecure) or a ranking propose output. Their advantage of not having to solve the flow problem is obscured by the difficulty in obtaining the membership functions for each particular grid, and consequently, their low portability. The main objective of this paper is to show the applicability of another computational intelligent paradigm, artificial neural network (ANN) models, to the contingency analysis operation, defined both as a functional and visual evaluation process. We try to point out the ability of these models for obtaining a precise contingency ranking, respecting real-time and easy visualization constraints. Consequently, this paper is organized as follows: Sect. 2 is centered in the revision of previous works applying artificial neural networks to contingency analysis, and remarks on their main drawbacks. Section 3 describes the PIs used in this work as well as the generation process of the different pattern sets. Section 4 describes our proposal for solving the evaluation and ranking of contingencies by means of supervised feed-forward artificial neural networks. Section 5 presents our proposal for visual monitoring of contingencies by means of self-organizing maps. Finally, Sect. 6 summarizes the main results and conclusions.
2 Artificial neural network for contingency analysis Artificial neural network techniques present interesting capabilities such as extracting classification criteria from unsupervised or supervised analysis of a set of complex
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patterns, approximation of complex non-linear function, or fault tolerance. These features seem to convert ANN in appropriated methods to face the contingency analysis task as well as a high number of other energy management system problems (see references [13, 14]). Besides, one can find different neural models to carry out both functional and visual Contingency Analysis. Thus, on the one hand, feed-forward supervised artificial neural networks such as, multilayer perceptron (MLP) with Backpropagation learning algorithm or radial basis function (RBF) networks, may be used to approximate the numerical function that evaluates the security level of each contingency, permitting the construction of a contingency ranking. On the other hand, unsupervised artificial neural networks, such as Kohonen’s self-organizing maps (SOM), take advantage of their ability to extract unknown criteria from a set of patterns to obtain a visual classification of contingencies according to their severity level. Regarding temporal constraints, these neural models are characterized by having an off-line, and probably slow, learning process phase, but a real-time response during the operation phase. Finally, these models approach the contingency analysis problem without solving the problematic flow problem, at least, during the operation phase. A considerable number of works applying ANNs to contingency analysis may be found in the specialized bibliography. Among these, we consider the following to be the most significant: in reference [15], RBF networks are used for contingency analysis in planning studies. The output of the RBFs is an index that determines the capacity of a power system to support a peak demand under a contingency. The method is not feasible for real-time operations, but this requirement is not necessary in the planning context. In reference [16], a mixed method based on both expert systems and ANNs is described. At the ANN stage, three feed-forward ANNs, each one specialized in a particular contingency, determine, in a binary way, the level of danger (normal, alert and emergency) of a selected set of contingencies. In reference [17], a MLP is used to calculate two different PIs. Each input vector is composed of the system power injections (quantitative variables) plus a number indicating the outage element (qualitative variable). This input vector is preprocessed using the fast Fourier transform (FFT). Thus, only one ANN is used, which must evaluate all contingencies. In reference [18], a first experimental approach to the analysis of some of the main artificial neural paradigms (MLP, RBF, SOM) applied to contingency analysis is performed. Both MLP and RBF paradigms are used for functional approximation of several PIs, while SOMs are analyzed as visual contingency classifiers. The main drawbacks of previously referred works are the following:
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Drawback 1 In most of the reported neural solutions, the output is a bivaluated function classifying each input pattern into two possible states: secure and insecure [16]. In general, this classification does not provide enough information either to elaborate a contingency ranking or to follow the system evolution with respect to its security level. Drawback 2 Some approaches result in excessively complex artificial neural systems [15]. Thus, only a reduced number of possible contingencies can be generally considered. Drawback 3 Some applications propose a single neural network to evaluate the complete set of contingencies [17]. This solution imposes the use of heterogeneous input patterns containing different types of components: measurements of electrical quantities (analogue variables) indicating the actual electrical state of the system, and qualitative variables (discrete variables) indicating the number of the contingencies to be currently evaluated. However, feed-forward paradigms such as MLP have serious difficulties dealing with variables of different nature and the approximation of multiple functions by only one network implies a very high number of hidden neurons and connection weights. The capacity of generalization of these approaches is therefore questioned. Drawback 4 The number of components of the input patterns generated for a real power system (active and/or reactive power flows at every line) may be excessively high. A feature selection process is needed, but the currently-proposed methods are very complex or not effective. In this paper, we present a second and deeper approach to the methods proposed in reference [18], specially oriented to avoid the previous four drawbacks. With respect to the application of MLPs and RBFs to functional approximation, we describe in detail the implementation of each paradigm remarking on the more important abilities and limitations of each one. With respect to the role of visual classifier of SOMs, we describe a one-dimensional model, which is a direct evolution of the bi-dimensional one presented in [18] making the operator’s visualization easier. The experiments have been carried out on big enough a grid, the standard IEEE-118 network.
3 Performance indexes, generation of patterns and preprocessing inputs In this section, first we describe the specific PIs used in our contingency evaluation operations. These indexes are the target functions when the contingency analysis is focused
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3.1 Performance indexes As mentioned above, for testing purposes we have used the standard IEEE-118 bus network. Two different PIs may be used taking advantage of the decoupling principle between active and reactive power. The active power performance index, PIp, evaluates the danger derived from the overload of lines caused by a contingency. The reactive power performance index PIv evaluates the danger derived from the voltage violation in buses caused by a contingency [2]. These indexes are, respectively, defined by (1) and (2), X 2 PIp ¼ wpL PL PL;lim ð1Þ a
PIv ¼
X
wvi Vi Vi;lim =Vi;lim
ð2Þ
b
where PL is the active power flow on line L, PL,lim is the active power flow limit on line L, wpL is the coefficient weighting the severity of an overload in the line L, Vi is the voltage magnitude at bus i, Vi,lim is the voltage magnitude limit at bus i, wvi is the coefficient weighting the severity of the voltage limit violation of bus i, and a and b are the sets including only elements (lines or buses, respectively) whose magnitudes (power flows and voltages, respectively) exceed their security limits. Hereafter, we indicate with PIp,j and PIv,j the respective active power performance index and reactive power performance index of the particular contingency j. 3.2 Generation of power flow patterns and final input vectors The previous PIs suggest using active (reactive) power flows of each line of the analyzed power system as input patterns to the neural system. However, the number of these power flows may be excessively high for a real power system (e.g., while the IEEE-14 has 20 lines, the IEEE-118
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has 179 lines). In order to avoid this difficulty, which is referred to as drawback 4 in the previous section, some feature selection (or extraction) technique is needed. Here, we use reduced power flow patterns obtained from a PCA [19, 20]. This technique orthogonalizes the components of a set of patterns to make them uncorrelated, and orders the resulting orthogonal components (principal components) as a function of its variance. This ordering can be used to select a reduced number of components, just the ones with the strongest contribution to the total variance in the data set. Consequently, starting from the base load pattern of a particular electrical grid, in our case the IEEE-118, the input pattern set to the ANN consists of 1,000 vectors (41 whole days and 16 h), which have been generated by means of the following procedure: (1) the global load of each hour h is obtained by multiplying the base load pattern by a coefficient in the interval [0.5, 1.8], just this corresponding to abscissa h in Fig. 1. This figure simulates a nearly real curve of energy demand variation in the Spanish electrical environment. (2) Each obtained hourly global load is distributed among all buses according to their load capacity. (3) A noise with a normal distribution function with 0 mean and 2% standard deviation is applied to each bus load. (4) For each load distribution, a complete load flow solution is carried out obtaining the active (reactive) power flows. (5) Finally, a PCA is applied to all active (reactive) flow patterns of the subset which we will use as the training set. That is, the test set patterns are not used to obtain the PCA matrix. Thus, in the IEEE-118 case, the original 179 components were reduced to 18 components. We must remark that our final input set avoids drawback 4 mentioned in the previous section. Although the PCA is not only important for reducing the input vector dimension, it may also display some characteristics or relations between components that one cannot extract from the original data.
3.5 3
Base Load Coefficient
as a functional approximation problem and solved by MLP and RBF networks. Second, we describe the process of generation of the experimental pattern sets. This process is referred to both the generation of 1,000 active (and reactive) power flow vectors (each one having 179 components for the IEEE-118 network) and the reduction of the vector dimension through principal component analysis (PCA), obtaining a final set of 18-component patterns. One should remark that the application of PCA involves finding a new system of reference with a lower dimension. This process implies a reduced loss of information (minor than 1% in our case). This loss of information is compensated by the simplification of the input space, so getting a better training process.
2.5 2 1.5 1 0.5 0
100
200
300
400
500
600
700
800
900
1000
Hour
Fig. 1 Variation rate curve of the base load for the IEEE-118 network
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The output corresponding to each of these input patterns, which will be used during the ANN training process, is obtained as follows: for each load distribution representing a current state of the system (this is obtained in step 4 of above paragraph) each one of the considered contingencies is simulated, and a load flow problem is solved to obtain the next system state generated by each contingency. The obtained load flow solutions allow to calculate the PIs corresponding to each new state. Thus, an input pattern is constituted by a current system state. For each input pattern and for each contingency j, the associated outputs will be PIp,j and PIv,j, i.e., the PIs corresponding to the next state generated by the contingency j. During the operation phase (once the training phase is finished), for each current system state, the ANN associated to contingency j must calculate the performance index PIp,j (PIv,j) before the contingency occurrence, that is, without knowing the system state after the contingency occurrence. Thus, the problematic solution of a flow problem for each state and contingency must only be made during the training phase, which may be as long as necessary, but not during the operation phase, which requires real-time response. Moreover, the trained networks will remain valid as long as no new power lines are added to the power system network, which is very unusual in a real system.
4 Functional evaluation and contingency ranking with supervised feed-forward artificial neural networks A functional approaching to the contingency analysis consists of obtaining, for the current state of the grid, a numerical evaluation of the danger of each particular contingency. Thus, one can make a contingency ranking that can be used by the system operator to be prepared to face the more dangerous contingencies. Feed-forward supervised artificial neural networks, such as MLP with backpropagation and RBF networks, are characterized by their ability for approaching complex nonlinear functions, with a possible relatively long training time, but a real-time response in operation mode. Consequently, we have selected both paradigms for estimating the previously defined PIpi and PIvi performance indexes for every contingency of the IEEE-118 standard network. For both paradigms, MLP and RBF, one neural network is implemented for each contingency and each PI, which addresses drawbacks 2 and 3 previously mentioned. The input patterns (input layer of neural networks) are the reduced active power flows obtained after the PCA (drawback 4 is thus avoided). Only one hidden layer is implemented for each neural network, although the number of hidden neurons as well as their activation functions is, obviously, dependent on the contingency and the paradigm
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used. Each neural network has only one output neuron. This neuron has a linear activation function approximating the PI of the corresponding contingency as a continuous function, in contrast with drawback 1. The generated set of patterns has been divided into a training set (900 patterns) and a test set (100 patterns). Although the learning phase of each neural network is an off-line and lengthy process, the operation phase is carried out in a negligible time. Thus, the evaluation of the complete set of contingencies and their ranking fulfills the most important requirement, realtime processing. Particular results obtained for each paradigm are described below. 4.1 Functional evaluation and ranking with multilayer perceptrons Multilayer perceptron is the most used artificial neural model in practically all application areas, especially when the problem can be formulated as an operation of functional approximation. That is due to its easy implementation and the existence of a high number of well-known and efficient training algorithms. As it has been mentioned, the contingency analysis operation can be focused as the problem of assigning a severity function to each contingency. Thus, the application of MLP seems to be a good election. In this work, we have implemented one specific MLP network for each contingency and PI, i.e., for each contingency i two MLPs have been implemented: one for implementing the function PIp,i, and the other one for implementing the function PIv,i. The structure of each ANN is the following: one input layer containing 18 neurons (corresponding to the 18 principal components obtained in the preprocessing stage); one hidden layer containing between 20 and 35 hyperbolic tangent neurons (the number of hidden neurons is dependent on each particular contingency); and one linear output neuron. These numbers of hidden neurons were found by try-and-error method. Thus, the networks finally used are those with a better behavior for both the training and test phases. The training process was carried out using a Levenberg–Marquadt learning algorithm [21] with the cross-validation strategy [22]. Thus, among the 900 patterns devoted to training, 800 were used for weight adaptation, and 100 for validation. Finally, the remaining 100 patterns were devoted to testing process. The best results have been obtained for a reduced number of epochs (about 30), in all cases. The experiments have been designed taking into account two aspects of the contingency analysis: on one hand, the possibility of obtaining an accurate numerical evaluation of the severity of each contingency; on the other hand, the possibility of using this evaluation for elaborating a useful contingency ranking. Thus, Table 1 shows the global mean absolute percentage error (MAPE) and its standard deviation
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Table 1 Mean absolute percentage error and standard deviation for training and test for the 20 most dangerous contingencies for the IEEE-118 network ANN model
MAPE
Table 2 MLP Contingency ranking of the 20 most dangerous contingencies of the IEEE 118 for the state #996 Contingency PI actual PI obtained Ranking
Standard deviation
Actual Obtained Difference
Training
Test
Training
Test 048
?4.240
?4.238
01
01
0
MLP
0.49
0.7
1.98
3.37
042
?2.838
?2.833
02
02
0
RBF
3.86
7.5
5.82
14.69
014 036
?2.572 ?1.629
?2.570 ?1.640
03 04
03 04
0 0
004
?1.445
?1.448
05
05
0
009
?1.251
?1.251
06
06
0
003
?0.984
?1.082
07
07
0
035
?0.895
?0.895
08
08
0
044
?0.767
?0.789
09
09
0
028
?0.663
?0.664
10
11
-1
040
?0.655
?0.690
11
10
?1
029
?0.581
?0.580
12
12
0
047
?0.465
?0.476
13
13
0
016
?0.386
?0.386
14
14
0
007
?0.347
?0.348
15
15
0
027
?0.325
?0.323
16
16
0
045
?0.317
?0.317
17
17
0
031 034
?0.259 ?0.237
?0.258
18
18
0
?0.236
19
19
0
006
?0.216
?0.215
20
20
0
(SD) for training and testing for the set of 20 most dangerous contingencies for the IEEE-118. One can note that obtained errors are significantly low, which means that MLP networks have produced a very accurate approximation of the PIs, so resulting in a very appropriate tool for numerical contingency evaluation subjects. Two contingencies exist (the so called 6 and 40) presenting abnormally high MAPE and SD values for a high number of grid states. However, these contingencies present PI values of almost zero in practically all cases. This fact, on one hand, explains the difficulty of the MLP to reduce the corresponding relative errors, and, on the other hand, reduces the importance of a possible misclassification of these contingencies, because their danger level is practically null. Tables 2 and 3 show the contingency ranking obtained for two different system states of the test set (patterns #996 and #1000). We dispose of the contingency ranking for the 1,000 system states, but we show only these two, especially representative, for evident space reasons. We can see that in both cases, the obtained ordering is correct for almost all the contingencies, especially for the most dangerous ones. For the pattern #966 (Table 2), the only disordered contingencies are the 40 and 28, both having a very similar PI. For the pattern #1000 (Table 3), for a three fractional digit precision, the obtained ranking is 100% correct. It must be remarked that for contingencies 06, 40 and 47, the MLP produces negative PI values; however, the order of these values is correct. Summarizing, MLP appears as a very valid technique both for numerical evaluation and ranking of contingencies on a power system. Its implementation requires an important preprocess of the input information, which consists of two main characteristics in our proposition: first, original input patterns must be preprocessed in order to reduce their dimension (this is done by means of PCA), and to allow the application of the method to bigger power systems; second, one neural network is specifically implemented for each PI and each contingency in order to handle homogeneous input patterns and avoid using qualitative variables such as ‘‘contingency number’’, since their numeric value has no significance for the neural network. Additionally, the training process is offline and has a high computational cost, but the operation process is real time and very efficient.
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4.2 Functional evaluation and ranking with RBF networks Radial basis function networks are widely referenced in function approximation applications, especially because of their local learning capability. In these networks, each particular piece of the objective function is approximated by a reduced set of neurons, therefore allowing a better adjustment of the function. Moreover, once the problem of distributing neurons into the space of definition of the function has been solved, the weight set is directly obtained from the resolution of a linear equation system. This feature avoids the need for a probabilistic learning process that does not guarantee an optimal solution. In our experiments, the implemented RBFs have the following characteristics: one input layer with 18 input neurons, one hidden layer with Gaussian function neurons, and one linear output neuron. The number of hidden neurons and their distribution in the input space were calculated from the orthogonal least squared algorithm for RBF networks [23]. In this algorithm, the radial function centers are estimated by solving an orthogonal least squared regression problem. At each step, a set of centers is selected that must maximize the expected output variance. This algorithm presents two main advantages over other methods based on a random selection: it calculates the radial function centers in a systematic way,
Neural Comput & Applic (2010) 19:139–150
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Table 3 MLP contingency ranking of the twenty most dangerous contingencies of the IEEE 118 for the state #1000
Table 4 RBF Contingency ranking of the twenty more dangerous contingencies of the IEEE 118 for the state #996
Contingency PI actual PI obtained Ranking
Contingency PI actual PI obtained Ranking
Actual Obtained Difference
Actual Obtained Difference
048
?2.734
?2.733
01
01
?0
048
?4.240
?4.240
01
01
?0
014
?1.266
?1.265
02
02
?0
042
?2.838
?2.832
02
02
?0
042 009
?0.778 ?0.770
?0.776 ?0.771
03 04
03 04
?0 ?0
014 036
?2.572 ?1.629
?2.568 ?1.623
03 04
03 04
?0 ?0
004
?0.764
?0.764
05
05
?0
004
?1.445
?1.436
05
05
?0
003
?0.760
?0.760
06
06
?0
009
?1.251
?1.241
06
06
?0
036
?0.509
?0.509
07
07
?0
003
?0.984
?0.996
07
07
?0
044
?0.438
?0.438
08
08
?0
035
?0.895
?0.887
08
08
?0
028
?0.361
?0.361
09
09
?0
044
?0.767
?0.721
09
09
?0
016
?0.299
?0.299
10
10
?0
028
?0.663
?0.658
10
10
?0
007
?0.269
?0.268
11
11
?0
040
?0.655
?0.483
11
12
-1
029
?0.251
?0.252
12
12
?0
029
?0.581
?0.576
12
11
?1
027
?0.251
?0.251
13
13
?0
047
?0.465
?0.463
13
13
?0
045
?0.245
?0.245
14
14
?0
016
?0.386
?0.338
14
14
?0
031
?0.204
?0.202
15
15
?0
007
?0.347
?0.200
15
15
?0
035
?0.000
?0.000
16
16
?0
027
?0.325
?0.188
16
16
?0
034
?0.000
?0.000
17
17
?0
045
?0.317
?0.183
17
17
?0
006 040
?0.000 ?0.000
-0.000 -0.001
18 19
18 19
?0 ?0
031 034
?0.259 ?0.237
?0.149 ?0.137
18 19
18 19
?0 ?0
047
?0.000
-0.001
20
20
?0
006
?0.216
?0.125
20
20
?0
and avoids the problem that stems from an excessive number of hidden neurons and a bad numerical conditioning. Another parameter to be considered in the network design is the spread, which determines, for each radial function unit the distance between the input vectors producing 1 and 0.5 output values. A higher or lower value of this parameter determines the higher or lower number of radial units that may contribute to the output value for a particular input. Several methods have been described to find an adequate value of this parameter, but, in general, they are very dependent on the function to be approached. Here, we have looked for an optimal value for this parameter by means of a statistical analysis of the results obtained for different values of the parameter. Concretely, the values used have been those fulfilling (3),
Table 5 RBF contingency ranking of the twenty more dangerous contingencies of the IEEE 118 for the state #1000
meanðdÞ 2stdðdÞ spread meanðdÞ þ 2stdðdÞ
ð3Þ
where mean(d) represents the mean value of distances between every two patterns of the input set, and std(d) represents the SD of these distances [24]. Although the spread value is dependent on each contingency, the obtained values are in the same order and very similar in all cases; thus, we can use only one value for all neural networks. The experiments have been designed in the same way as in the MLP case. Thus, Table 1 shows the MAPE and SD values for training and testing, and Tables 4 and 5 show the
Contingency PI actual PI obtained Ranking Actual Obtained Difference 048
?2.734
?2.739
01
01
?0
014
?1.266
?1.310
02
02
?0
042
?0.778
?0.880
03
03
?0
009
?0.770
?0.855
04
04
?0
004
?0.764
?0.764
05
05
?0
003
?0.760
?0.764
06
06
?0
036
?0.509
?0.542
07
07
?0
044
?0.438
?0.421
08
08
?0
028
?0.361
?0.358
09
09
?0
016
?0.299
?0.303
10
10
?0
007
?0.269
?0.272
11
11
?0
027
?0.251
?0.256
12
12
?0
029
?0.251
?0.246
13
13
?0
045
?0.245
?0.241
14
14
?0
031 034
?0.204 ?0.000
?0.180 ?0.007
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contingencies ranking obtained for the same patterns: #996 and #1000. We can see that the apparently high MAPEs of the RBF do not affect the ranking quality. Thus, for the state #966 only the actually consecutive contingencies 29 and 40 are interchanged. With respect to the state #1000, the same remarks as for the MLP results can be made. The high MAPEs and SDs can be explained by the difficulty of the RBF networks to approximate near zero values in our function, but these values correspond to not dangerous contingencies.
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w j = ( w j1 , w j 2 ,..., w jn )
xi=(x i1,xi2,...x in)
4.3 Comparison between MLP and RBF networks Both methods classify correctly almost all the contingencies with PI higher than zero. Alteration in the ranking generally occurs between contingencies with zero or near to zero PI. Simulation results show larger mean percentage relative errors, but also a larger percentage of correct ordering in the final contingency ranking, for RBF networks in contrast to MLP networks. This apparent contradiction may be explained because RBFs produce larger errors than MLPs for those patterns with a PI equal or near to zero. Errors in these patterns, which represent non-dangerous states, do not affect the final ranking.
Fig. 2 Schematic representation of the mapping of a n-dimensional input space to a bi-dimensional (alternatively linear) output space by means of a Kohonen’s Self-Organizing Map
- Choice initial values of the parameters lr (learning rate) and hj ( ) (neighborhood function of each neuron j) - Initialize weight vectors in a random way - Repeat until final condition - Present an input pattern k
x k = ( x k 1 , x k 2 ,..., x kn ) 5 Visual monitoring of contingencies with self-organizing maps Kohonen’s self-organizing maps (SOMs) are characterized by their capacity to classify a set of complex patterns in an unsupervised way, extracting both non-obvious and nonexplicit classification criteria. This classification is carried out by distributing a high dimension input space VI on a reduced (1 or 2) dimension output space VO, while preserving the topological relations between input patterns [25] (Fig. 2). The output space is constituted by a set of neurons ordered in a plane or a line, in which a physical spatial neighborhood function is defined. After an unsupervised learning process is carried out (Fig. 3), each input pattern x = (x1,…xn) will activate a particular output neuron i, whose weight vector wi = (wi1,…,win) is the nearest to x. Thus, the vector wi can be considered as a prototype of the input space region whose vectors activate the same neuron i. Also, two similar input patterns will activate either the same neuron or two close neurons in the output space. From these characteristics, the SOM appears as a promising method for contingency monitorization tasks, where sometimes it is more important to know the relative position of the current system state in a contingency
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- Obtain the winner neuron j fulfilling
max i ( w i ⋅ x k ) - Update the weight vector of each neuron i according the winner neuron neighbor function hj ( )
wi (t + 1) = wi (t) + lr h j (i)(xk − w j ) - Normalize updated weight vectors w - Update values of lr and hj ( ) Fig. 3 Kohonen’s Self-Organizing Map learning algorithm
ranking than to know the exact value of its PI for this contingency. Moreover, it has recently been remarked the possibilities of SOM for supplying information about the temporal evolution of the analyzed system [26]. In this work we analyze both capabilities: visual classification and evolution analysis. The SOMs described in our experiments use the update weight rule given by (4),
Neural Comput & Applic (2010) 19:139–150
ð4Þ
where x is an input vector, wi is the weight vector of neuron i, and hji, the current value of the neighborhood function, which determines the weight increment of each neuron as a function of the proximity between the neuron i and the winner neuron j. In our simulations, the neighborhood function is a step function with value 1 into a square (or segment) of the neural map, which is centered in the winner neuron, and value 0 out of this area. The edge of this square (or segment) will decrease during the training. The coefficient lr is a dynamic learning rate, which evolves along the training according to equation (5), lr ðtÞ ¼ lr0
. ct 1þ nn
ð5Þ
being lr0 the initial learning rate (0.3 in our experiments), c a constant (0.2), t the current iteration and nn the number of neurons [27]. In previous works [18, 24], we have described a first approach to the visual contingencies monitoring with SOMs, especially centered on the bi-dimensional SOM model, using as a benchmark the IEEE-14 standard network. With respect to this bi-dimensional model, the experiments show the high applicability of this neural paradigm for both a rapid visualization of the severity level of a particular contingency and an easy surveillance of the system evolution toward a possible dangerous operation point. Thus, we have found that, for each contingency, our trained SOM (a 10 9 10-neuron map), carries out a very clear pattern classification in correlation with its PI. Moreover, our final classes appear ordered in a continuous way along a main diagonal in increasing order of this PI. This result points out the possibility of using linear SOMs, which would facilitate the monitoring task for two reasons: on the one hand, the one-dimensional representation permits a more direct and fast visualization of a contingency severity and a greater number of contingencies can be simultaneously screened and visualized. On the other hand, the surveillance of the system evolution with respect to the severity of a contingency is easier if this evolution is projected on a line than on a plane. Consequently, in this paper we focus our attention in this last structure, using as case of study the previously mentioned IEEE-118 standard network. In any case, the methodology described here may be directly translated to the bi-dimensional case. One linear map with 30 neurons has been trained using 900 patterns from our complete set with 1,000 patterns, which has been described in Sect. 3. After training, the Kohonen network should be self-organized in such a way that each contingency will generate a different and welldefined clustering of the patterns (current electrical system states) as a function of their severity. For testing this
hypothesis, we have calculated the PI of every state vector with respect to every contingency. Then, for each contingency, we have divided the pattern set into groups of states with their PI value in a particular interval. It is obvious that the distribution of states in each group will be different for each contingency. The used intervals are: 0, (0, 1.70), [1.70, 3.40), [3.40, 5.20), [5.20, 6.80), [6.80, 8.60) and [8.60, ?). Each one of these groups has been assigned with a symbol and a color (this last characteristic is translated here into different tones of grey). The assigned symbols are: dot (.), hyphen (-), ampersand (&), plus (?), equis (x), asterisk (*) and sharp (#), respectively. It must be remarked that, because the SOM has an unsupervised training, the PI values are not used either during its training or during its operation phase; thus, these values and their associated symbols are only useful for visualizing the resultant distributions. Regarding this point, Fig. 4 shows several pattern distributions carried out by the linear SOM for 10 different contingencies (contingencies 3, 4, 7, 8, 9, 11, 14, 17, 26, and 33). Each line in Fig. 4 represents the distribution for a particular contingency (it must be remarked that only one SOM has been trained, but this one produces a different distribution of patterns for each contingency). One can observe that the linear SOM effectively clusters the patterns as a function of their severity for each contingency, and this clustering occurs in a continuous way and in an increasing severity order. Figure 4 shows the linear SOM’s capability for classifying the current states of an electrical network regarding their danger with respect to the occurrence of a particular contingency. Here, we propose two practical applications of this capability: (a) simultaneous monitoring of a high number of contingencies in the same screen as well as a very easy and fast discrimination of the more dangerous Global Contingency Visualization
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wi ðt þ 1Þ ¼ wi ðtÞ þ lr hji ðx wi ðtÞÞ
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Fig. 4 Linear patterns distribution for ten different contingencies. The number of each contingency is indicated on the Y axis
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contingencies for the current system state; and (b) following of the system state evolution regarding its severity for each contingency, and the assistance for forecasting the system evolution toward an either dangerous or nonproblematic future situation. The first application can be seen in Fig. 5, which shows the SOM activation produced by a particular system state regarding each one of the above selected contingencies. It must be remarked that, in all lines, the active neuron (that marked with character #), is obviously the same, because we have only one SOM, but the difference is in the subsegment of severity associated to this neuron in each case. Thus, one can see that contingencies 3 and 4 are the most dangerous ones for this operation point, while contingencies 8 and 26 are the least dangerous. The second application can be seen in Fig. 6, which shows the graphic representation of the PI of contingency 3 for 26 ordered in time test patterns (Fig. 6a), and the different images of the linear SOM when it is used to represent contingency 3 (Fig. 6b). It must be again remarked that in our original representation, the visualization of the activated neuron is based on a color system, which has been replaced here by asterisk symbols. Thus, each line shows the activation of the SOM for each one of the operation points corresponding to the patterns numbered from 1 to 26 in Fig. 6a. One can see that the activation profile for this contingency reproduces the PI value evolution curve. Additionally, if the inputs are presented to the SOM following a constant timing, then, the distance between two consecutively activated neurons will provide a notion about the velocity of the increase or decrease of the danger level of the system operation point. As a final advantage of this contingency monitoring method, it should be noted that only one SOM has to be trained for a considered power network. Thus, the
02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
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Fig. 6 a Graphical representation of the PI for a subset of test patterns. b Representation of the neuron activation for the selected test patterns regarding contingency 3 (Y axis represents the number of the pattern exiting the SOM at each moment)
complexity of the off-line training process does not significantly increase with the power system size. That is, the complexity of the method does not significantly increase with the number of contingencies to be considered.
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Fig. 5 Representation of the severity of each contingency for an input pattern
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6 Conclusions In this work we analyze the applicability of artificial neural networks (ANN) to the contingency analysis in power systems. Contingency analysis can be approached from two different perspectives: on one hand, as a functional approximation problem, in which, the severity of each contingency is numerically evaluated by means of a performance index, therefore allowing the elaboration of a contingency ordering or ranking; on the other hand, as a visual monitoring problem, in which, the severity of each contingency is presented to operators in a graphical (nonnumerical) way, thus obtaining a qualitative evaluation of the contingency severity.
Neural Comput & Applic (2010) 19:139–150
First, a revision of previous studies has been carried out. From this, a list of main drawbacks of the applicability of ANNs to this problem has been presented and analyzed. We consider that these drawbacks must be solved in order to make ANNs feasible tools in this task. Second, we present several ANN implementations oriented to approach this task in its two different aspects: functional and visual evaluation. These implementations try to overcome the above-referred drawbacks. For numerical evaluation, ANNs based on feed-forward architecture and supervised learning have been implemented. Thus, two different artificial neural systems are described based on multilayer perceptron and radial basis function (RBF) networks, respectively. Both neural paradigms need a long off-line training process, but they guarantee real-time response during their operation mode; therefore, avoiding the main drawback of classical methods based on the solution of a power flow problem. The main characteristics of these systems are: (1) each contingency is evaluated by a specific neural network, avoiding using a heterogeneous input set of quantitative and qualitative variables; and (2) the input vector is composed of a reduced number of components obtained from a principal component analysis on the complete set of active (or reactive) power flows. Simulation results show a high ability of both MLP and RBF paradigms for producing an accurate PI approximation, and consequently, a valid contingency ranking. In particular, the MLP based system produces lower mean absolute percentage errors than that based on RBF, but this produces better final ranking. This apparent contradiction is explained by the difficulty of RBF networks for approximating a function in the region near zero. However, this limitation is not significant in contingency analysis because near zero values of the PI mean a non-dangerous contingency. For visual contingency monitoring, we propose a system based on one-dimensional self-organized maps by Kohonen. In previous works we showed the viability of a two-dimensional SOM to contingency analysis, and we analyzed their characteristics making the one-dimensional SOM proposed here advisable. Simulations show a perfect pattern classification as a function of their severity level for the 179 possible contingencies in the standard IEEE-118 network. Moreover, the linear nature of the graphical representation provides the following interesting characteristics: (1) each contingency is represented by one segment and this is divided in successive sub-segments representing an increasing severity scale. Thus, a very high number of contingencies may be simultaneously presented in only one screen, and the analysis of the complete contingency set may be done in a greatly reduced time by a human operator. (2) The representation by a SOM of the successive
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power system operation points permits to follow the system tendency with respect to each particular contingency, and to show when it is approaching a dangerous zone well in advance. (3) Each contingency produces a different distribution of the intervals of severity in the original SOM, but there is only one training process and it is the same for all the contingencies. Thus, the complexity of the off-line training process does not significantly increase with the power system size. Finally, the process in the operation model is carried out in real time. Acknowledgments Authors acknowledge the interesting comments and suggestions of the reviewers. This work has been partially supported by the Spanish Ministerio de Educacio´n y Ciencia (MEC), project no. TIN2005-01359.
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