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Probabilistic-based Overload Estimation for real-time Smart Grid Vulnerability Assessment J. C. Cepeda, Graduate Student Member, IEEE, D. O. Ramírez, and D. G. Colomé
Abstract--In recent years, important efforts to improve monitoring, protection and control of power systems have been explored. In this connection, several novel approaches for assessing vulnerability in real time have been developed. However, most of the work is commonly focused on tackling stability phenomena, while the possible overloads are often treated as negligible in real-time power system security. But sometimes, high electric post-contingency currents might provoke overloads which could increase the system vulnerability problem. This paper presents a novel method for assessing the possibility of fast post-contingency overloads using Statistical Distribution Factors (SDFs) that allow computing an Overload Index (OVI) in real time. First, Monte Carlo-based contingency analysis is performed to iteratively calculate ac Distribution Factors (ac-DFs). After, SDFs are defined by the mean and standard deviation of ac-DFs. These SDFs are then used together with principal component analysis (PCA) and support vector machine classifier (SVM-C) in order to structure a table-based real-time post-contingency overload estimation algorithm, which allow computing OVIs depending on the actual operating state and the type of contingency. The proposal is tested on the IEEE New England 39-bus test system. Results show the feasibility of the methodology in alerting about fast possible overloads. Index Terms--Monte Carlo, overload, principal component analysis, security, smart grid, support vector machine, vulnerability assessment.
I. INTRODUCTION
E
LECTRIC Power Systems are commonly subjected to different types of perturbations. Depending on the operating state and the perturbation magnitude, cascading events, that may eventually lead the system to blackouts, might occur [1]. For this reason, the design of some Smart Grid applications that perform real time adaptive control actions, with the objective of improving the system security and reducing the risk of power system blackouts is required [2]. Thus, an intelligent scheme which provides critical information in real time, assesses vulnerability quickly, and performs timely self-healing and adaptive reconfiguration actions has to be structured (i.e. Self-Healing Grid) [3]. A fundamental task of this smart functionality is the vulnerability assessment (VA), since it has the function of detecting the necessity of performing global control actions. This work was supported in part by the German Academic Exchange Service (DAAD), and the National University of San Juan. J. Cepeda (e-mail:
[email protected]), D. Ramírez (e-mail:
[email protected]), and D. Colomé (e-mail:
[email protected]) are with the Institute of Electrical Energy, National University of San Juan, J5400ARL San Juan, Argentina.
A vulnerable system is a system that operates with a “reduced level of security that renders it vulnerable to the cumulative effects of a series of moderate disturbances” [4]. Vulnerability is a measure of the system weakness upon cascading events [4]. The concept of vulnerability involves the system security level (i.e. static and dynamic security) and the tendency of changing its conditions to a critical state [5] that is called the “Verge of Collapse State” [6]. Although there are a lot of vulnerability causes, which vary from natural disasters to human failures [7], the system vulnerability is characterized by four different symptoms of system stress, such as angle instability, voltage instability, frequency instability, and overloads [6]. Hence, vulnerability assessment might be performed through analyzing the system status as regards these symptoms of vulnerability. While much work has been directed towards the development of methods for assessing the three causes of vulnerability regarding stability issues [8]-[10], the possible overloads have often been treated as negligible in vulnerability assessment tasks. However, sometimes high electric postcontingency currents might provoke overloads which could increase the system vulnerability problem [11]. So, a system can be stable following a contingency, yet insecure due to post-fault system conditions resulting in equipment overloads [12]. Some approaches tackle the problem of overload via offline vulnerability assessment. For instance, an overload risk index (ORI) that describes the system vulnerability level based on overload of transmission lines is proposed in [13]. The overload is estimated using Distribution Factors (dc-DFs), which show the approximate line flow sensitivities (dc power flow assumptions) due to a system topology change. The ORI is a measure that reflects the likelihood and severity of line overloads given current system loading levels and component failure rates. The methodology incorporates both deterministic and stochastic calculations to determine the overload probability through Monte Carlo (MC) based simulations. The risk of instability and islanding is incorporated to the ORI in [14]. On the other hand, a method for assessing the possibility of overloads after a contingency occurs in real time, based also in dc-DFs, is introduced in [15]. The methodology considers that dc-DFs can be calculated via SCADA/EMS applications using pre-contingency steady-state data in order to update changes in grid topology. After a contingency occurs, re-distributed flows (post-contingency steady state power flows) can be estimated in each branch using the last computed dc-DFs and
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the pre-contingency power flows obtained from the state estimator. Then an overload index (OVI) is calculated for each branch. However, since dc-DFs are commonly based on dc power flow linear approximations, dc-DF-based overload estimation presents significant errors. To overcome this drawback, the present paper proposes a methodology to estimate the possible overloads in real time based on statistical ac-DFs (SDFs), which depend on the system response to different type of contingencies (i.e. power injection changes or branch outages) considering all the probably operating scenarios (i.e. MCbased simulation). Afterwards, the proposed approach is combined with a knowledge-based intelligent classifier to structure the function of a real-time overload estimator. The input features are a set of real-time monitored system signals describing the pre-contingency operating state. Principal component analysis (PCA) is then applied for input dimensionality reduction. In this structure, the off-line MCbased simulation results serve as a system knowledge base which is used for two key purposes, namely: i) to obtain the SDFs, and ii) to train an intelligent classifier (Support Vector Machine Classifier, SVM-C) that enables determining, in real time, the best SDFs depending on the pre-contingency operating state (from SCADA/EMS) and the contingency that actually occurred (from different types of intelligent electronic devices – IEDs). The paper is organized as follows. Section II analyses the power system overload timeframe and its relation to power system vulnerability. Section III depicts the dc-DF-based overload estimation procedure. Section IV presents the proposed SDF-based overload estimation methodology. Section V shows the simulation results on a test power system that demonstrates the performance of the proposal. Finally, the conclusions are presented in section VI. II. POWER SYSTEM OVERLOAD TIMEFRAME An important aspect to consider in VA formulation is the duration of the events involved in the timeframe of interest. Due to the complex tasks related to the power system operation, which comprise modeling, simulation, analysis, and control actions, the time-scale varies from microseconds to several hours [3]. Table I presents some power system actions and operations and their corresponding timeframes. TABLE I ACTIONS AND OPERATIONS WITHIN THE POWER SYSTEM Action or operation Electromagnetic transients Switching overvoltage Fault protection Electromagnetic effects in machine windings Electromechanical transients – stability Electromechanical oscillations Frequency control Overloads Economic load dispatch Thermodynamic effects Energy Management System applications
Timeframe µs – ms ms 100 ms ms – s ms – s ms – min Real-time VA 1 s – 10 s timeframe 5s–h 10 s – 1 h s–h Steady state; ongoing
Based on the four symptoms of system vulnerability, realtime VA addresses the post-contingency short-term phenomena, which develop in a timeframe of 15 to 20 seconds after a contingency occurs. This time window includes the socalled short-term stability phenomena, which comprises transient stability, short-term voltage stability, oscillatory stability and short-term frequency stability; and also, the possible overloads provoked by variations in power injections or changes of system topology. This paper addresses the estimation of overloads for any critical power system equipment, but it is mainly oriented to allow the real-time assessment of possible overloads that should be controlled within few seconds. This type of overload commonly corresponds to special series equipment which presents fast overload control requirements (e.g. electronic devices). On the other hand, equipment that presents slow overload control requirements (i.e. minutes or hours) would permit an on-line assessment via contingency analysis as part of SCADA/EMS applications. In these cases, the methodology presented in this paper might not be required. Power system series equipment presents different overload timeframes due to the variety of particular features such as physical limits, thermal capability, stability constrains, among others. The physical limits present particular interest in some series electronic equipment because of the special care required for semiconductor devices of power converters of HVDCs, and photovoltaic or wind facilities. In the following subsections, an analysis of the permissible overload periods of some series devices permits identifying the need of assessing overload in real-time as part of smart-grid VA tasks. A. Overload of ac-transmission lines and transformers In short-time emergency conditions, mineral oil immersed transformers can be loaded over nominal values with an aging permissible limit provided by its designer [16]. In contrast, actransmission line overload is usually specified by three types of limits: thermal, angle stability and voltage, from which the temperature commonly does not rise so fast to reach the maximum designed limit [17]. However, angle stability and voltage limits might be quickly reached after a contingency. On the other hand, long-time overload of transforms and lines is defined by their amount of loading within a time period greater than 30 minutes and 1 hour [17], [18]. B. Overload of high voltage direct current (HVDC) links The overload capability for HVDC is mainly subjected to thermal rating of main equipment, such as converter transformer, thyristor valves and smoothing reactor; also, it depends on the ambient temperature and status of redundant cooling system [19]. The overload capability is classified according to the time duration of overload. The overload capability of HVDC link includes three levels, i.e. continuous overload, short-time overload (transient overload) [19] and long-time overload. Overload ratings of some HVDC schemes [20] are described in Table II.
3 TABLE II PROJECT RATING MW SHORT-TIME AND LONG-TIME OVERLOAD Rating MW
Project Rihand - Dadri Vindhyachal Intermountain Gesha Itaipu Tian Guang 3 Gorges Changzhou 3 Gorges - Guangdong Thailand – Malaysia Cobara Bassa
1500 500 1600 1200 3150 1800 3000 3000 300 1920
Short time Overload Per Duration unit 5s 1.33 5s 1.2 1s 2 10 s 1.25 5s 1.25 3s 1.5 5s 1.5 5s 1.5 10 min. 1.5 None
Long time Overload Per Duration unit 2h 1.1 2h 1.1 Cont. 1.5 2h 1.1 20 s 1.15 Cont. 1.1 2h 1.13 2h 1.13 None None
C. Overload of series capacitors Series capacitor banks are designed to withstand higher currents, such as those experienced during emergency loading (typically 30 minutes rating), system swings, and faults. Series capacitor loading design is specified by the purchaser [21]. D. Overload of thyristor controlled series capacitors (TCSC) TCSCs are defined by a continuous overload characteristic (i.e. temporary overload and dynamic overload), which operates in capacitive boost mode. In continuous overload the line current and nominal TCSC reactance must be under their maximum levels, temporary overload has a long-time overload of 30 min as typically timeframe, and dynamic overload are define in a short-time overload of 10 s [22]. For instance, a short-time overload of 10 s is feasible to cover stress conditions during power oscillations as they may occur immediately after clearing a fault [23]. However, each system may require its specific overload cycles depending on the system characteristic during contingency situations [23]. The TCSC operating range is shown in Fig. 1. It is designed to operate in capacitive and inductive impedance mode for continuous and temporary overload line currents. cap.
XTCSC, pu
3.0
Continuous 30 min 10 s
2.0 1.0 0.0
1.0
2.0
3.0
IL, pu
ind. Fig. 1. TCSC overload characteristic [23]
III. DC-DF-BASED OVERLOAD ESTIMATION Distribution factors are linear approximations of the sensitivities of branch active power flows with respect to changes in nodal injections and withdrawals [24]. These factors can be used to predict the possible overload of grid elements after the occurrence of a disturbance [13]-[15], [24]. Dc-DFs are commonly calculated using the method presented in [24]. Consider a power network as G = (V, E), where all nodes make up the set V, and all edges make up set
E, with the bus 0 being the slack bus. Each branch ek ∈ E has associated the pair of nodes (ik, jk). As a convention, the direction of the real power flow fek on the edge ek is from ik to jk. The Injection Shift Factor (ISF) ψiek of a branch ek is the sensitivity of the change in edge ek real power flow fek with respect to a change in the power injection Pi at some node i ∈ V and the withdrawal of an equal injection at the slack bus. The Line Outage Distribution Factor (LODF) ς(eq)ek specifies the fraction of the pre-outage real power flow on the cut branch eq redistributed to the edge ek [24]. A method to assess the possibility of overloads after a contingency occurs using dc-DFs is presented by the authors in [15]. This method is based on the option of calculating dcDFs via SCADA/EMS applications. After a contingency occurs, re-distributed flows (post-contingency steady state power flows fek) can be estimated in each branch using the last computed dc-DFs, the available perturbation data, and the precontingency steady state power flow (fek pre-c), as follows. ⎧ fek pre−c + ΔPi ×ψ eik if contingencyis Power Injection Changein bus i (1) ⎪ fek = ⎨ (e ) f ek pre−c + feq pre−c × ς ek q if contingencyis Branch eq outage ⎪⎩
Since there are, in general, three categories of power transfer limits: thermal, angle stability and voltage, branch overload threshold values have to be pre-defined for each system. Thus, it is necessary to pre-establish the power transfer limit (restricted by one or more of the three limits) corresponding to each element of the system. Then, an overload risk band can be structured in order to calculate an overload index. Fig. 2 shows an example of a risk band for a specific element. Thermal Limit
Supper
Voltage Limit Angle Stability Limit
Slower Fig. 2. Overload Risk Band
Once the re-distributed flows are estimated, an Overload Index (OVI) can be calculated for each branch using the function defined by (2). ⎧ 0 ⎪ 1 ⎪ OVI ek = ⎨ f ek − Slower ⎪ Supper − Slower ⎪ 1 ⎩
(
f ek ≤ Slower
)
(2)
Slower < f ek < Supper f ek ≥ Supper
where Supper and Slower are the upper and lower limits of the overload risk band for each grid element. IV. STATISTICAL AC-DF-BASED OVERLOAD ESTIMATION Since the huge volume of uncertainties greatly influence the power system response, it is necessary to apply mathematical tools which allow considering all the probable scenarios. One of the main classes of probabilistic techniques is the Monte Carlo-based (MC) simulation, which provides the possibility of obtaining more realistic results, mainly for complex system analysis [25], since it avoids using surrogate
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models. Monte Carlo method is a repetitive procedure that consists of evaluating, at each repetition, the system response, using a set of input variables which are generated randomly from their probability distribution functions (PDFs). Hence, numerical random output values are obtained [26]. These numerical outputs are used to determine the SDFs for the bulk power system under analysis, which then allow computing an Overload Index (OVI) in real time. First, using the probabilistic models of input parameters based on a short-term operating scenario and via optimal power flow (OPF) computations, MC-based ac contingency analysis is performed to iteratively calculate ac-DFs. After, SDFs are defined by the mean and standard deviation of acDFs, considering two types of filters that permit improving the accuracy of SDFs. For real-time implementation, these SDFs are then joined with an intelligent classifier (based on PCA and SVM-C) in order to structure a table-based postcontingency overload estimation algorithm, which allows computing OVIs depending on the pre-contingency operating state and the actual contingency. Fig. 3 depicts the overall structure of the proposed approach.
Fig. 3. Methodological framework
A. SDF Computation This paper proposes using MC-based simulation as a method to obtain pre and post-contingency power flow results (i.e. contingency analysis), considering several possible operating conditions, and N-1 contingencies (i.e. branch outages and variation in power injections). These considerations allow including the most severe events that could lead the system to potential overload conditions, and further N-2 contingencies, which are considered as the beginning of a cascading event. ISFs and LODFs are calculated using the results of each MC simulation (i.e. results from ac power flows: ac-DFs), as shown in (3) and (4).
ac -ψ eik = (e ) ac -ς ek q =
f ek − f ek pre −c ΔSi f e − f e pre − c k
k
(3) (4)
f eq pre − c
where k, q are branches, fek is the post-contingency apparent power flow, fekpre-c is the pre-contingency apparent power flow, and ΔSi is the change of apparent power injection in bus i. Then, the SDFs are calculated based on the probabilistic attributes represented by the mean and standard deviation of all MC-based DFs (i.e. ISFs and LODFs, respectively).
⎧ mean {ac-DFk j } ⎪ k =1…n (5) SDFk j =( eq ) = ⎨ j ac -DF = ac -ψ ej=i or ac -ς e k k {ac-DFk } ⎪⎩ kstd =1…n where ac-DF is the distribution factor (ISF or LODF) of branch k provoked by contingency j (i.e. variation of power injection or branch outage), and n is the number of scenarios. j
B. Improvement of SDF accuracy Since ac-DFs directly depend on each operating scenario, the complete short-term possible scenarios have to be previously filtered in order to improve the accuracy of SDFs (i.e. more representative mean and lower standard deviation), with the aim of obtaining more robust results. This paper proposes two types of scenario filters: i) to establish representative clusters of scenarios depending on operating similitudes, and ii) to reduce the sample of scenarios to those with the highest probability of overload. B.1. Clusters of scenarios The MC-based operating scenarios are clustered to effectively represent all the possible states embraced by the probabilistic simulation. Thus, a combined technique of Principal Component Analysis (PCA) and Fuzzy C-Means (FCM) is applied to properly select the reduced subset of relevant scenarios to be considered in the formulation of the real-time overload estimation. The n operating scenarios generated by MC simulation (i.e. observations) are represented by multivariate vectors of p features, which include active and reactive power nodal injections, and voltage phasors of system buses. The feature vector set constitutes a (n × p) multivariate data matrix (X). First, normalization is carried out to the data matrix for avoiding the influence of different measurement units of original variables. Then, PCA is applied to the normalized data matrix in order to reduce the dimensionality of the data, maintaining as much as possible of the variation presented in them. This is achieved by transforming the data to a new set of variables, the principal components (PCs), which are uncorrelated and ordered so that the first few components keep most of the variation of the original variables [27]. The sum of the PC eigenvalues (λi) is equivalent to the total variance of the data matrix, and each PC eigenvalue offers a measure of the corresponding explained variability (EVi). So, the number of the chosen PCs depends on the desired explained variability, which is specified as more than 98% in this paper.
5
EVi =
λi p
∑λ
× 100
(6)
i
i =1
Afterwards, Fuzzy C-means algorithm (FCM) is applied to the selected PC scores in order to determine the N clusters of operating states. B.2. Sample of scenarios Since the objective is to determine the best SDFs in order to predict possible overloads, using all the MC-based scenarios might introduce unnecessary errors, such as high standard deviations or means far from medians. Hence, a filter to enhance SDF values that mainly represent critical overload scenarios for each cluster is proposed. This filter is defined as follows:
functions (based on the so-called “kernel functions”) that classify an input into one of the given classes through training using input–output (features-label) pair data. The optimal decision function is called the Optimal Hyperplane (OH), and it is determined by a small subset of the training set which are called the Support Vectors (SV), using the concept of VC (Vapnik-Chervonenskis) dimension as the theoretical basis [29]. Several kernel functions can be used, such as linear, polynomial, radial basis function (RBF), among others. Fig. 4 shows an illustration of an SVM-C solution for a two-class data classification problem using a linear kernel function, where the SV and the OH have been determined. The classified vectors are shown in a two-dimensional plane whose axes represent the first and second PCs obtained from PCA of operating states.
⎧mean {ac-DFk j } ∀k | fe > fe pre−c k k MAX ⎪ k =1…nN (7) =⎨ SDFk j j =( eq ) = j i = … 1 i N ac -DF = ac -ψ or ac -ς {ac-DFk } ∀k | fek > fek pre−cMAX ⎪ k =std ek ek ⎩ 1…nN where fek pre-c MAX is the maximum value of the MC-based precontingency power flows of branch k (since OPF ensures this value is the maximum admissible steady-state branch loading that satisfies all the restrictions), j is the contingency, and nN is the number of scenarios belonging to cluster i. j
C. Table of SDFs for real-time assessment Once SDFs have been calculated, a table for real-time implementation has to be structured. This table includes the computed SDFs, structuring a three-dimensional matrix whose dimensions represent: i) the branch that shows the power flow sensitivity, ii) the type and location of the contingency, and iii) the clusters of scenarios. For real-time applications, a classifier has to be capable of orienting the selection of the best SDFs considering the precontingency operating state (that can be obtained from SCADA/EMS), and the actual contingency (that can be quickly alerted by local IEDs). D. SVM-based Operating State Classification For real-time implementation, an N-class classifier that sorts the power system operating state into the N clusters is used. This classifier is trained using the vectors obtained from PCA as inputs and Support Vector Machines (SVM) as the classification method. Then, in real-time application the classifier will use the selected PC scores of the actual precontingency operating state as inputs in order to predict its corresponding cluster (i.e. the third dimension of the table). SVM is a machine learning technique for solving problems in classification (C), regression (R), and novelty detection [28]. SVM belongs to a set of algorithms, namely kernel methods, and it employs structural risk minimization (SRM) as the optimization principle. So, it is usually more robust to avoid over-fitting problems [25]. Due to this feature, SVM has been considered of great potential for power engineering applications [25]. For these reason, SMV has been chosen as the classifying tool for the developed N-class classifier. SVM-C is a nonparametric classifier that acquires decision
Fig. 4. Support vectors and optimal separating hyperplane
In this paper, RBF kernel is used because this function is capable of handling possible nonlinear relations between labels and features [30]. The classification needs a priori an off-line learning stage, in which the classifier has to be trained using the results of PCA applied to all the MC-results (i.e. training set). Each element in the training set contains one “target value” (class labels) and several “attributes” (features). The objective of SVM-C is to yield a training data based model, which predicts the target values of the test data given only the test data features [30]. One of the main challenges using SMV is to determine its optimal parameters which permit obtaining the best classification accuracy [30]. Traditionally, k-fold crossvalidation procedure and grid-search methodology are used for this purpose. This algorithm iteratively generates a grid of parameters and obtains the accuracy for each parameter setting [31]. Then, the parameters with the highest accuracy are selected. Nevertheless, k-fold cross-validation presents the problem of being a high-time-consuming task. This paper applies a method to optimally determine the SVM parameters based on the fact that such parameter setting task can be tackled as an optimization problem. For this purpose, Particle Swarm Optimization (PSO) [32] is used to optimize an accuracy-based objective function.
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V. SIMULATION RESULTS The methodology is tested on the IEEE New England 39 bus test system [33] lightly modified in order to satisfy the N1 security criterion. Simulations consist in several contingencies generated by applying MC method. Two types of events are simulated: branch outages and variation in power injections. Several operating states have been considered by varying the load of PQ buses, depending on three different daily load curves. Then, optimal power flow (OPF) is performed in order to establish each operating state, using the MATPOWER package [34]. After that, contingency analysis has been performed using the DIgSILENT Power Factory software, so that the pre and post-contingency power flows can be solved. Fig. 5 shows the single-line diagram of the test system.
Fig. 6. Mean and CV plot for scree homogeneity criteria: (a) mean of postcontingency power flow estimation errors, (b) CV of post-contingency power flow estimation errors
Fig. 7. Main PC scores and clustering formation TABLE III SDFS RESULTING FROM BRANCH OUTAGES
Fig. 5. IEEE New England 39 Bus test system single-line diagram [33]
A total number of 10,000 different operating states have been considered for the MC-based contingency analysis. The output constitutes the pre and post-contingency ac power flow results, which are used to calculate the corresponding ac-DFs. Based on MC results, and the pre-defined overload risk bands, the most critical branches are established. In this case, twelve edges present risk of overload and these are the critical branches to which SDFs have to be calculated. In order to consider both filters defined in section IV.B, the adequate number of clusters has to be firstly specified. Thus, PCA is applied to the normalized operating state data matrix, from which the first six PCs offer 98.08% of EV. Then, FCM is used to define the clusters of scenarios, where the number of groups is determined depending on statistical indicators of homogeneity which are based on the mean and coefficient of variation (CV) of the post-contingency power flow estimation errors. Fig. 6 shows a scree-plot that illustrates the decrease of mean and CV errors according to the number of clusters. From the figures it is determined that six groups offer good homogeneity. Fig. 7 shows the spatial distribution of the three main PC scores and the clustering formation. Then, SDFs are calculated for both types of contingencies, getting a maximum std{ac-DF} of 0.1. In the cases where SDFs are not computed, the variation of power flow does not surpass the filter limit, so it neither provokes overloads, then the corresponding SDFs are not included in the SDF-table for real time assessment. In these cases, OVIs are automatically set to zero. Table III shows some of the computed SDFs (i.e. mean{ac-DF}) that correspond to different branch outages.
Outage Branch (eq)
Branch (ek)
1
L15-16 L01-02 L22-23 T06-31 T06-31 L01-02 L04-14 L08-09 T10-32 L16-24 L16-21 L21-22
L03-04 L03-04 L23-24 L21-22 L15-16 L15-16 L03-04 L15-16 L15-16 L21-22 L23-24 L23-24
0.793 0.664 0.317 0.014 0.102 0.331 0.314 0.352 0.052 1.029 1.016 -
Cluster / SDF (mean{ac-DF}) 2 3 4 5 0.809 0.706 0.399 0.008 1.036 1.017 -
0.792 0.709 0.185 0.359 0.289 0.325 1.036 1.018 1.009
0.812 1.009 1.003 0.998
0.798 0.645 0.265 0.349 0.285 0.308 0.324 0.337 0.044 0.975 1.013 1.006
6 0.988 0.979
Afterwards, a six-class SVM-C is trained using the PC scores as input features and the corresponding cluster indices as targets. First, optimal values of SVM parameters are identified using PSO for optimizing an adequate accuracy objective function. The parameter optimization permits obtaining a 10-fold cross validation accuracy of 99.88%. Then, several performance tests are carried out in order to verify the accuracy of the classifier and SDFs for predicting possible overloads. Fig. 8 and 9 present histograms of postcontingency power flow estimation errors for two different transmission lines and both types of contingencies, where (a) corresponds to estimation based on SDFs, and (b) represents the estimation using dc-DFs. Both cases highlight the huge improvement of the accuracy estimation when SDFs are used.
Fig. 8. Estimation error histograms for branch L16-17 due to branch outages: (a) using SDFs, (b) using dc-DFs
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Fig. 9. Estimation error histograms for branch L04-05 due to variation in power injections: (a) using SDFs, (b) using dc-DFs
Table IV presents a comparison of the accuracy of the estimation using SDFs and dc-DFs for the twelve critical branches. The values represent the mean of all simulated cases. It is also seen the better performance when SDFs are used. TABLE IV DC-DFS VS. SDFS - COMPARISON OF ACCURACY
Branch (ek) L03-04 L05-06 L05-08 L06-07 L07-08 L14-15 L15-16 L16-17 L16-21 L17-18 L21-22 L23-24
SDF estimation error (%) Outage Injection 1.09 0.20 3.22 3.20 0.94 0.88 0.88 1.16 1.51 8.61 0.45 0.19 0.60 0.08 0.27 0.10 0.94 0.73 0.47 0.07 1.07 0.05 4.76 0.09
dc-DF estimation error (%) Outage Injection 47.58 7.62 37.60 55.78 8.91 2.59 14.06 4.29 18.23 53.60 49.56 22.01 23.13 12.16 44.56 0.81 47.21 18.19 41.84 3.10 15.69 5.62 63.16 0.70
The previous analysis shows the advantage of the proposed methodology as regards a steady state reference frame. Nevertheless, since post-contingency quasi-steady power flows result from the stabilization of a dynamic transient state, it is necessary to verify the performance of the proposal considering also dynamic simulation. Fig. 10 highlights the good accuracy of the proposed static estimation for predicting post-contingency quasi-steady power flows.
Once estimation has been completed, the final step is to calculate the corresponding OVIs, which permit ranking the level of system vulnerability as regards overloads. These indices might orient the selection of corrective control actions (e.g. focalized load shedding) if needed. Table V shows a summary of the number of cases that correspond to three OVI ranges. Most of the cases show excellent accuracy in estimation, mainly in the range (0.5, 1] where possible corrective control actions might be required. TABLE V OVIS FOR BRANCH OUTAGES - SUMMARY OF NUMBER OF CASES Branch (ek) L03-04 L05-06 L05-08 L06-07 L07-08 L14-15 L15-16 L16-17 L16-21 L17-18 L21-22 L23-24
OVI = 0 Sim. SDF est. 234,994 234,993 234,993 234,993 234,986 234,983 234,986 234,986 235,000 235,000 234,943 234,945 233,952 233,959 233,224 233,226 234,458 234,479 235,000 235,000 231,681 231,682 234,718 234,717
0 < OVI ≤ 0.5 Sim. SDF est. 6 7 7 7 14 17 14 14 0 0 57 55 972 965 1,293 1,291 542 521 0 0 2,876 2,875 282 283
0.5 < OVI ≤ 1 Sim. SDF est. 0 0 0 0 0 0 0 0 0 0 0 0 76 76 483 483 0 0 0 0 443 443 0 0
In order to verify the robustness of SDFs, a case that does not satisfy the MC premises is considered. For this purpose, the original data of the test system are used [33]. Fig. 11 contrasts a branch outage estimation case with its simulation, showing a good accuracy even in this not-contemplated case.
Fig. 11. Not-contemplated case estimation: fek L21-22 due to L23-24 outage
VI. CONCLUSIONS
Fig. 10. Comparison of static estimation with dynamic simulation: (a) fek of L05-08 due to L06-07 outage, (b) fek of L16-17 due to L14-15 outage, (c) fek of L21-22 due to L23-24 outage
Real time vulnerability assessment is a fundamental task within a Self-Healing Grid structure. In this context, some approaches for mainly addressing stability phenomena have been developed. However, the possible overloads have often been treated as negligible. Accordingly, this paper proposes a novel probabilistic post-contingency overload estimation method for assessing the possibility of overloads that require fast control actions. The methodology uses statistical distribution factors (SDFs) that allow computing an overload index (OVI) in real time. SDFs are defined by the mean and standard deviation of Monte Carlo based ac-DFs. These SDFs are then used to structure a table-based real-time postcontingency overload estimation algorithm. For this purpose, principal component analysis and support vector machines are used in order to organize an intelligent classifier capable of orienting the selection of the best SDFs depending on the actual system state. This tool has been proved in a test power system, and it has shown excellent performance due to its high
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estimation accuracy (overcoming the drawbacks of dc-DFbased estimation). Finally, the proposal might provide an indicator (e.g. OVI) for triggering global corrective control actions (such as focalized load shedding) depending on the real time event evolution to avoid possible cascading events due to actuation of short-time local overload protections. VII. REFERENCES [1]
[2] [3] [4] [5] [6]
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VIII. BIOGRAPHIES Jaime C. Cepeda (GSM’10) was born in 1981. He got his Electrical Engineer degree in 2005 from National Polytechnic School, Quito, Ecuador. He is currently a Ph.D. candidate at Institute of Electrical Energy, National University of San Juan, San Juan, Argentina, favored with a scholarship from the German Academic Exchange Service (DAAD). His research experience includes an internship at Institute of Electrical Power Systems, University Duisburg-Essen, Duisburg, Germany, as part of the DAAD scholarship. His special fields of interest comprise power system stability analysis, security assessment, and vulnerability assessment. Diego O. Ramírez was born in 1983. He acquired his Electrical Engineer degree in 2008 from National Polytechnic School, Quito, Ecuador. At the moment he is a Ph.D candidate at Institute of Electrical Energy, National University of san Juan, Argentina, favored with a scholarship from the German Academic Exchange Service (DAAD). His main fields consist on monitoring and control of electric power systems.
Delia G. Colomé was born in 1959. She obtained her Electronic Engineer degree in 1985 and her Ph.D. degree in Electrical Engineering in 2009, both from National University of San Juan, San Juan, Argentina. Since 1983, she has been a researcher and a Professor at Institute of Electrical Energy at National University of San Juan. During this time, she has worked as project manager and as senior engineer in numerous technical support projects in Argentina and different Latin–American countries. Her main fields are control and supervision of power systems, modeling and simulation of power systems, and the development of computational tools for engineering teaching.
