Computational framework combining static and transient power system security evaluation using uncertainties

Electrical Power and Energy Systems 71 (2015) 151–159

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Computational framework combining static and transient power system security evaluation using uncertainties Amélia Yukie Takahata a,b,1, Marcelos Groetaers dos Santos a,c,1, Glauco Nery Taranto b,2, Marcus Theodor Schilling c,⇑ a b c

National Operator of the Electrical System (ONS), Rua Julio do Carmo 251, Cidade Nova, Rio de Janeiro, RJ, Brazil Federal University of Rio de Janeiro (COPPE/UFRJ), Dept of Electrical Engineering, Caixa Postal 68504, Ilha do Fundão, Rio de Janeiro, RJ, Brazil Fluminense Federal University (UFF), TCE/TEE/IC, Caixa Postal 33024 Leblon, CEP: 22440-970, Rio de Janeiro, RJ, Brazil

a r t i c l e

i n f o

Article history: Received 29 October 2014 Received in revised form 26 December 2014 Accepted 20 February 2015

Keywords: Computational tool Probabilistic method Reliability Security Uncertainty

a b s t r a c t This paper presents a simplified but effective procedure to represent power system uncertainties that allow the development of a computational tool to tackle the power system probabilistic security problem from both the small signal stability (SSS) perspective, and the transient stability (TRS) analysis perspective. A set of examples using the New England test-system is presented and discussed. Among the advantages of the suggested method, the following points are evident: (i) the ability to discriminate between the three types of uncertainties (scenarios, fault events, and noise types) that permeate power systems and that are relevant to the system security; (ii) the capacity to use existing traditional tools from both small signal dynamic analysis and transient stability analysis to adapt them easily to the well-established concept of probabilistic adequacy assessment, without resorting to abstruse and hard-to-implement theoretical techniques; (iii) the enormous advantage of usual availability for the required statistical data (no hard-to-collect data are required); and (iv) proposal of a conceptual procedure that renders a highly combinatorial problem amenable to current state-of-the-art hardware resources, within acceptable limits of computational burden. Important practical results that one may wish to highlight are related to the effective representation of noise uncertainties through a straightforward combination of weighted histograms, and the successful performance of the new Apparent Stability Index – ASI. Ó 2015 Elsevier Ltd. All rights reserved.

Introduction According to [1], (sic) ‘‘the concept of uncertainty is just one category or type of the more general and broader concept of ignorance. It encompasses the ideas of ambiguity (the possibility of having multi-outcomes for processes or systems), approximations (a process that involves the use of vague semantics in language, approximate reasoning, and dealing with complexity by emphasizing relevance), and likelihood (defined by its components of randomness, statistical and modelling). The last subcategory (i.e. likelihood) can be understood in the context of chance, odds, and gambling, having aspects of randomness and sampling.’’ Therefore, this is the subtype of uncertainty that is most commonly

⇑ Corresponding author. Tel.: +55 21 2274 1391. E-mail addresses: [email protected] (A.Y. Takahata), [email protected] (M.G. dos Santos), [email protected] (G.N. Taranto), theodor1@netflash.com.br (M.T. Schilling). 1 Tel.: +55 21 34449640, 21 34449814. 2 Tel.: +55 21 25628615. http://dx.doi.org/10.1016/j.ijepes.2015.02.042 0142-0615/Ó 2015 Elsevier Ltd. All rights reserved.

used in power systems reliability assessment, and therefore is the only one that will be discussed in this paper. A broad overview of the literature covering the subject of both power systems adequacy and security, under the influence of probabilistic uncertainties, may be observed in [2,3], indicating that considerable maturity in this field has already been attained. Adequacy analysis considering the treatment of power system uncertainties under the realm of fuzzy variables has also been introduced [4–6], but the probabilistic approach still remains clearly dominant, even in more recent publications [7,8]. Probabilistic security analysis, despite recent advances [9], still lags behind adequacy analysis in useful applications for actual large scale power systems [10]. Several reasons could be mentioned to explain this situation, ranging from unavailability of data and computational tools to theoretical hurdles involving concepts, models, procedures and criteria for diagnosis. One remarkable difficulty related to probabilistic security analysis stems from the intrinsically combinatorial nature of the problem. It is well known that, if these problems are treated naïvely, combinatorial problems may easily render a so called trans-computational problem [11]. In

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this situation even a fictitious ideal computer, working with a hypothetical lightning speed processing capability and unlimited memory and energy resources, would not provide a feasible solution. This happens due to the Bremermann‘s limit, which was derived on the basis of quantum theory, and is expressed succinctly as, sic [1]: ‘‘no data processing systems, whether artificial or living, can process more than 1.36 E47 bits per second per gram of its mass’’. This statement is valid even when one considers a virtual Turing machine [12]. However, the practical treatment of a large-scale power system probabilistic security problem may be feasible if a set of simple artifices is recalled. Therefore, the objective of this paper is to present a simplified, but effective procedure to represent power system probabilistic uncertainties that allow the development of computational tools to tackle the power system probabilistic security problem, from both the small signal perspective, and the transient behaviour perspective. A conceptual framework is proposed and a set of examples is given and discussed.

Probabilistic small signal and transient stability analyses: previous approaches Small signal stability (SSS) analysis has been performed in a deterministic and classical way, considering only unitary probabilities or certainties [13], or considering probabilistic state spaces. The probabilistic approach may be addressed with both analytical techniques [14–17], and numerical techniques [18–21] based on standard enumeration, Monte Carlo simulation, or even ad-hoc strategies such as the structured singular value theory (l-analysis) [22,23]. Regarding probabilistic transient stability (TRS) analysis, earlier references appeared in the seventies [24,25]. Between 1978 and 1981, a series of seminal papers based on a new proposal were published [26–28], and were only interleaved by an alternative approach [29] in 1979. A new research avenue [30–32] exploring protection uncertainties and Monte Carlo simulation appeared later. Several authors also proposed joint formulations tackling, both the adequacy, and the security problem [33–41], simultaneously. A large set of publications dealing with several other aspects of probabilistic reliability, but also useful to transient stability, were also delivered, covering a plethora of new and diversified strategies, algorithms and concepts. Significant works have been developed and utilized based on arcane or original techniques such as the conditional probabilities [42], health analysis [43], bisection method [44], collocation method [45], Bayes classifier [46], cascading failure analysis [47], copula theory [48], cross-entropy methods [49], and bootstrap techniques [50]. Detailed comparisons between several methods have already been published [51] and research efforts in this area continue thriving [52–54]. Notwithstanding, despite the significant amount of literature, it seems that probabilistic security analysis has not yet gained large acceptance in the industry environment. The applications in actual large systems are seldom found [55]. However, it has been observed that stratifying power system uncertainties according to their type may help develop a quite simplified statistical approach amenable to the probabilistic security assessment, and may still render useful results.

Decoupled view of power systems uncertainties A useful classification of power system uncertainties associated with power system states may encompass categories related to discrete and continuous/hybrid values, as shown in Table 1.

Table 1 Types of power systems uncertainties. Discrete state uncertainties

Continuous/hybrid state uncertainties

Scenarios Fault events

Noise (small signals) Operational settings

Uncertainties related to scenarios (S) may be seen as those associated with long term stationary probabilities, calculated through standard statistical techniques. Scenarios are defined as necessarily countable, meaning that they are related to discrete random variables, and desirably with a ‘‘slow dynamics’’ (i.e., its occurrence should be sufficiently slow as to be possible, for practical purposes, to ascribe a probability value to it). For instance, in this paper, a time frame over one minute is selected as the limit for a given state to be ascribed the scenario type uncertainty. In this category, one may find, for instance, the existing grid possible discrete topology states (nodes and branches represented by stations, transmission lines, and transformers). Therefore, scenarios provide information regarding the probabilities of a given element to be in a given operational state, such as in or out, and are extensively used in traditional adequacy analysis. In this paper, the system topology uncertainties will only be circumscribed to single and double outages of transmission lines and transformers, but higher contingencies may be taken into account if the corresponding statistical data are available. For the sake of simplicity, detailed nodal topologies will not be modelled and each study will be conditioned to a given set of generators in operation. Load level probabilities will also be classified in the scenario category, because the approximate representation of the system load curve through three or four discrete load levels is usually accepted as reasonable. In the developed computational prototypes, up to five load levels can be considered. Hydrological profiles, and other countable uncertainties characterized by slow physical dynamics, may also fall into the scenario category. In this paper, up to five modes of inter-area tie flows representing uncertainties of different hydrological profiles will be considered. Therefore, modelling the uncertainty of a given specific scenario is relatively easy, because it is given by the product of probabilities for the existing elements in each scenario. For instance, let a hypothetical scenario be composed of a given grid topology (i.e., lines and transformers), a specific load level and a given generation dispatch representing one of the system accepted operational profiles. The product of the probability associated with the selected load level times the probability of the network topology times the probability of the chosen operational mode will render, in this case, the final scenario probability. It is recognized that the automatic combinatorial generation of scenarios may be easy, but it has the drawback of eventually creating unacceptable or unrealistic probabilistic states from the standpoint of the actual stationary system. Disregarding those states would cause a loss of convexity (in this paper, the convexity concept is used in the sense that if unrealistic states are unaccounted for, summing up all of the remaining scenarios probabilities would result in a value less than one). An unviable state may be detectable if, for instance, no convergence is attained when a standard power flow solution is searched, or if a clearly unrealistic configuration is generated. In this paper, the unrealistic states are carefully avoided in the examples given, but when they occur in other situations, they may be counted separately, generating specific probabilistic indexes. Although the recovery of full convexity could be attempted, using skilfully selected repartition heuristics, this approach was not further investigated here. Uncertainties related to fault events (F) are those associated with sudden severe disturbances, such as short-circuits, transient line outages, or any other sudden events, and with possible major

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Fig. 1. Decoupled view of power systems uncertainties types.

and severe consequences and a fast physical dynamics characteristic. Uncertainties related to fault events may be addressed by two strategies: unconditioned and conditioned. The former approach is very general and encompasses the rigorous evaluation of all transition rates depicted in Fig. 1. A major hurdle in this alternative is the lack of reliable data and the subsequent numerical evaluation. On the other hand, the conditioned approach considers that a given situation will be analysed only if a given disturbance has occurred (this means that a given disturbance has occurred with probability one). The advantage of this approach relies on the usual availability of statistics describing countable fault events, such as the types, durations and locations of disturbances. Noise (N) uncertainties usually have a very high frequency nature, are normally small in magnitude, are related to fast physical dynamics, and are normally associated with minor impacts. They are essentially associated with the concept of small signals, are not countable and are mathematically hard to represent. Frequently, they are represented by continuous random variables or an extremely large number of discrete random variables. Thus, the typical load random ripple is a good example of this type of uncertainty. Although several types of noise could be modelled, such as those influencing the values of parameters, only those types affecting the load will be considered. In this paper, N uncertainties will only be addressed by adding or subtracting values to/ from the load averages. Furthermore, they will only be considered for a given specific scenario (this means that for a specified scenario, noises uncertainties may or may not be considered). Operational (O) setting uncertainties are those related to the changing nature of power system operational states. Both countable and uncountable values may be recognized such as fixed transformers load tap positions, voltage values, control and protection settings or instantaneous dispatched power. Because of the usual lack of statistics and the complex nature, this type of uncertainty is also quite difficult to treat and, in this paper, it will not be further addressed. However, if necessary, they can be taken into account by a conditional approach (i.e., if probabilities can be ascribed to a given set of operational settings).

Combining uncertainties Considering the four types (S, F, N, O) of uncertainties previously defined, fifteen combinations are possible. However, from this set, only three combinations (S, F, SF) may be easily treated as countable sets and, therefore, are amenable to modelling through traditional statistics. Other combinations essentially yield uncountable sets, whose computational treatment will be more demanding. In this paper, only the following combinations will be investigated: (i) (S, N, SN) in the realm of probabilistic small signal analysis

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and; (ii) (S, F, SF) in the realm of transient stability analysis. SN means the joint combination of a given scenario with noises (for instance, load fluctuation); SF means the joint combination of a given scenario submitted to faults. The other ten combinations are recognizably too complex or do not generate interest, and will not be further investigated here. From the above selection, it is observed that the probabilistic transient analysis will only deal with countable sets, while the small signal analysis will consider two sets influenced by N uncertainties. In both cases, because the probabilistic state space is automatically generated, infeasible states may possibly be created, even though they should be discarded, and must be accounted for anyway. In [56] a proposal to circumvent this hurdle is advanced, but it will not be further discussed in this paper. The main hypotheses of the computational tools that were developed in [56,57] are briefly described in the following sections: Probabilistic state space for small signal stability For the SSS prototype, a given scenario (S) will be composed of the following components: up to eight profiles associated with power flows through the system main inter-area ties. This amount was selected because it is sufficient to associate most of the usual hydrological conditions that occur in Brazil with a selected set of discrete random variables, whose probabilities of occurrence can be derived from statistics. Thus, each profile has a known probability of occurrence on a yearly horizon. The system topology is represented by all transmission lines and transformers. The set of transmission configurations that may be taken into account in the SSS prototype comprises all single and double contingencies, plus the grid without any outage. This simple approach avoids an unmanageable growth in the search space. Therefore, for a system with n branches (n P 2), a total of (n2 + n + 2)/2 probabilities are to be calculated on a yearly horizon. The yearly load is modelled by up to five levels, with their respective probability of occurrence. Stations and generators are treated deterministically. This means that each evaluation will be conditioned to a given generation configuration, including the number of machines and, power dispatches. Thus all scenario combinations are easily countable and their respective probabilities are available. Regarding N type uncertainties, the following aspects may be modelled in the proposed computational procedure: spatial location (i.e., defining specific electrical areas or specific load buses for introducing noise uncertainties); noise magnitudes as determined by samples from a set of given probabilistic densities functions (including Gaussian, rectangular, triangular, Weibull, lognormal, Laplace and exponential distributions) with specified standard deviations, which are composed with load levels averages; coupling (meaning that the noise can be taken into account only at the active and/or reactive part of the loads, in a coupled or uncoupled way); conformity (specifying whether a sampled noise magnitude will be or not be composed with each specific load or the whole set of all system loads), load nature (referring to the noise influence at specific load components such as the percentages of constant impedances, constant currents, constant power, and their combinations); and noise emulation (this relates to the definition of the number of samples in a specified processing batch and the number of batches). Regarding the sampling requirements and accuracies of the probabilistic estimates, they are established through a convenient selection of the number of batches and samples combined with chosen values of coefficients of variation b. As previously mentioned, the SN combination will be always conditioned to a given scenario (because S probabilities of occurrence are easily calculable). N uncertainties will be always accounted for using histograms, which will be conventionally weighted by corresponding scenario probabilities later.

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Probabilistic state space for transient stability Regarding scenarios, both TRS and SSS prototypes are based on the same hypotheses, as previously described. Regarding F uncertainties, the following aspects were considered: (i) Location of transmission lines where faults are to be applied: probabilities may be provided considering that a given line will suffer a fault based on the following choices- probability equal to the ratio of the line length and the total sum of line lengths of a specific type (this model explores the spatial exposition of lines); probabilities given by 1/tl, where tl is the number of transmission lines (if used, this model allows an approach equivalent to a deterministic rationale); (ii) Location of fault in a given line: three probabilities may be informed for fault occurrences- at the line middle and at both extremities; (iii) Types of faults: probabilities of five choices of faults are permitted, including one, two, and three phase ground short-circuits and, phasephase short-circuits and open line trips (1U, 2U, 3U, UU, LT); (iv) Fault durations: modelled by Gaussian distributions N(l, r);and (v) Fault elimination: probabilities representing successful fault elimination through fast tripolar after-fault line reclosing may be used. Security reliability evaluation At this point it should already be clear that the problem being tackled here encompasses a coordinated generalization of the power system probabilistic security assessment, using decoupled uncertainties (using a new taxonomy defined in Table 1) based on available statistical data and a skilful adaptation of the existing software tools for both small signal stability analysis, and transient stability analysis. Thus, the procedures for both SSS and TRS analyses follow the traditional steps for reliability evaluation: (i) State selection: in this step each state is identified as feasible or infeasible from a stationary perspective (in regard to acceptable ranges of voltage violations and transmission overloads) through a standard power flow assessment and their probabilities and/or histograms are obtained; (ii) State evaluation: for the SSS analysis the eigenvalue evaluation was based on the PSAT (versions:1.3.4; 2.0) algorithms [56,58], while for the TRS analysis, a SIME procedure was recalled [57,59–68]; (iii) Indices updating before remedial measures: a set of indices, indicated by (1)–(3), were developed for the SSS analysis, while others that were conceptually described by (1) and (4) were proposed for the TRS analysis; (iv) Activation of remedial measures and indices updating: while no remedial measure was provided for the SSS analysis, a fast tripolar after-fault line reclosing option is available for the TRS analysis; and (v) Stop criteria: both prototypes go back to step (i), if the chosen stopping criteria are not met. In both SSS and TRS analyses the programs end when an enumerable probabilistic state space is exhausted. When noise uncertainties are modeled, the procedure may end when the specified coefficients of variations are met or when the set of proposed batches are exhausted. Application and results The methodology was applied to the New England test system shown in Fig. 2, with 39 busses, 10 generators and 46 branches (34 transmission lines + 12 transformers) [69,70]. All additional data that was used in both SSS and TRS experiments are available in [56,57], in addition to many examples using other test-systems and several other experiments. While a comprehensive set of additional simulations for both SSS and TRS analyses may be provided in a future paper, this work will focus

only on the New England related results, because of space restrictions. Probabilistic small signal stability analysis For the SSS analysis all generators were represented by PSAT 5.2 models [58], and loads by using constant power models represented by three levels. A type II PSAT voltage regulator (rotating exciter) was also used [58]. No power system stabilizers and governors were included. All 21 combinations of 3 load levels and 7 topologies were analysed. Neither transmission overloads nor voltages violations outside the range (0.950–1.050 pu) were observed. Generator G1 was used as swing machine and the power dispatch (MW) was defined as (G2 = 700; G3 = 750; G4 = 650; G5 = 750; G6 = 650; G7 = 750; G8 = 650; G9 = 950; G10 = 350). Regarding scenario probabilistic data to high (7380 + j1691) load, medium (6150 + j1410) load, and low (5305 + j986) load (MW + jMVar) levels, probabilities (0.2, 0.4, 0.4) were ascribed respectively. Uncertainties were attributed to only three lines (see busses 35–11, 21–22, 38–1 in Fig. 2). Because only single and double outages were considered, the respective probabilities for the full grid, n  1 and n  2 configurations (total of 7 states) are given as: full grid: 9.04E-1; n  1 topologies: 1.24E2, 2.89E2, 5.16E2, n  2 topologies: 4E4, 6.81E4, 1.53E3. All (3  7 = 21) scenario probabilities are obtained by combining load levels and topology probabilities. In the given example, no uncertainty is attributed to specific profiles associated with power flows through the system’s main inter-area ties. Regarding noise probabilistic data, uncertainties were only applied to the heavy load level using a Monte Carlo technique to generate histograms of all resulting critical damping values, and real values of critical eigenvalues. Occurrences of right-side eigenvalues and/or damping less than 10% were used as failure modes. Table 2 presents three types of experiments to illustrate the probabilistic SSS analysis. Experiment 3S3 considers only scenario uncertainties in both load levels and topologies using the previously given data. It is then observed that the countable combination of 3 load levels with 7 topologies renders Ns = 21 probabilistic states. Experiment 5N2 illustrates the sole influence of noise. In this example, the uncertainty is applied in N‘ = all load buses and is characterized by magnitudes Nt = mg of Ns = 1000 samples extracted from a Gaussian, Dt = G, distribution with an average value equal to each bus’ heavy load level and Nsd = 10% standard deviation. The uncertainty is simultaneously applied to both the active and reactive NC = pq components of all loads in a fully conformed St = cf way. Experiment 7SN is a hybrid type, combining each of the 21 scenarios and sets of 1000 samples representing noise uncertainties as previously described. As previously noted, countable probabilistic spaces are amenable to an easy assessment through analytical indices, while spaces suffering from the influence of noise uncertainties may be preferably analysed with the help of a statistically weighted combination of histograms. In the first case, for the SSS analysis, two types of indices are suggested based on probabilities and central-class measurements. Regarding the former, they are: probabilities of systemic instability (POI), systemic stability (POS), systemic damping range (PDR), and oscillation frequency range (PFR). They reflect whether the system is unstable or stable and whether the system dwells in a given range of damping or oscillation frequency, respectively. All four indices have the same structure as given by (1), where pstate is the probability of the i-th countable state whose characteristic is under measurement, such as the existence of eigenvalues in the right or left real semi-plane (POI, POS), damping range (PDR), or frequency range (PFR), and W is the set of all probabilistic states with the characteristic under consideration

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Fig. 2. The New England 39 busses test system [69,70].

Table 2 Probabilistic small signal stability experiments. Ex

Lo

To

N‘

Nt

Nc

St

Dt

Nsd

Ns

3S3 5N2 7SN

Y no Y

Y no Y

no all all

– mg mg

– pq pq

– cf cf

– G G

– 10% 10%

21 103 21  103

Ex = experiment; Lo = modelling of load levels uncertainties; To = modelling topology uncertainties; N‘ = noise location (all busses); Nt = noise type (mg-adding magnitudes); Nc = noise coupling (pq-influencing active and reactive load); St = sampling type (cf-conform, i.e. all loads change in the same direction); Dt = distribution type (G-Gaussian); Nsd = noise standard deviation; Ns = number of states and/or samples; Y = yes.

(instability, stability, damping and frequency ranges). It is worth noting, that although the indices defined by (1) may be seemingly equal, each one is dependent on a specific set W. It may also be argued that if the probabilistic state space is amenable to full enumeration in the sense that all states are feasible (i.e., power flows are calculable), then it follows that POI = 1POS, and one of these indices would be useless. However, if some states do not exist in practice, the evaluation of both indices POS and POI is useful to measure the extent of the probabilistic space that has been discarded due to physical infeasibility.

POI; POS; PDR; PFR ¼

X pstate ðiÞ

ð1Þ

The critical margin (CMA) and critical damping (CDA) averages are given by (3), where n is the number of counted states. The median of the critical margins (MCM) and the median of the critical damping (MCD) are obtained similarly. Although these last four l indices do not discriminate the states probabilities, they also help render a preliminary assessment of the system’s global robustness.

CMA; CDA ¼ kcritical ; fcritical ¼

n X ð:Þcritical =n

ð3Þ

i¼n

The results obtained for the countable experiment 3S3 of Table 2 were as follows:

i2W

The other six proposed indices are central-class measurements types. Their magnitudes indicate system robustness. The critical margin and critical damping expectations (CME, CDE) are given by the expectations of the real part of the critical eigenvalues and the critical damping of all probabilistic states as expressed by (2), respectively.

CME; CDE ¼ Eð:Þ ¼

X i2W

pstate ðiÞxð:Þ

ð2Þ

POI = 1.72%; POS = 98.26%; PDR < 0% = 1.72%; PDR >0%, < 5% = 98.26%; PDR >5%, <20% = 0%; PFR >0.1Hz, <3.0Hz = 100%; CME = 0.077 s1; CDE = 1.12%; CMA = 0.052 s1; CDA = 2.85%; MCM = 0.070 s1; MCD = 1.24% It was verified that all 21 scenarios related to experiment 3S3 were physically viable, which means that all 21 combinations of load levels and topologies rendered convergent power flows.

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Risk Region Security Region

Fig. 3. SSS analysis for scenarios (S), experiment 3S3.

However, six cases showed the presence of eigenvalues situated in the right semi-plane, or a damping rate outside the accepted limits, as illustrated in Fig. 3, and indicated by POI = PDR = 1.72%. The sum of POI and POS reaches near 100% as expected (99.98%, to be exact, because all states were physically viable). All positive damping rates were situated in the range between 0% and +5% and all detected electromechanical oscillations modes were in the range between 0.1 and 3.0 Hz. The indices CME and CDE appeared to be unable to detect the presence of any failure mode, hinting at an optimistic assessment. Because both indices are probabilistically weighted, they suggest a low risk situation. The same indication was obtained through the results of MCM and MCD. On the other hand, the two average-based indices CMA and CDA clearly indicate the existence of failure modes due to the location of eigenvalues in the right semi-plane and the occurrence of negative damping. It is worth noting that Fig. 3 depicts only a small set of all eigenvalues for experiment 3S3, which are distributed in the complex plane. This figure shows, at a glance, that the system is clearly subject to a given degree of risk, indicated by the existence of eigenvalues with damping rates less than 10%, and eigenvalues with positive real values. The set comprising these eigenvalues is highlighted by a closed ellipsoid situated outside the security zone. The risk zone is situated where the damping rate is below 10%, and/or the eigenvalues’ real part is positive. While the figure indicates the system visual qualitative profile, the set of probabilistic indices defined by equations (1) to (3) provides a corresponding quantitative evaluation that can be easily compared with other circumstances. For instance, it is quite interesting to note that when only transmission line uncertainties are considered combined with the heavy load level, the resulting 7 probabilistic states give rise to a POI equal to 8.31%, which is apparently worse than the situation comprising 21 probabilistic states, with POI = 1.72%. This happens because the probabilistic spaces are different, and if compared, should take into account this difference. The SSS analysis under the influence of only noise uncertainties or the combination of scenarios and noise uncertainties is based on histograms, such as those results from experiment 5N2 in Fig. 4, and experiment 7SN in Fig. 5. According to Table 2, experiment 5N2 illustrates the effect of a typical load noise applied on a given load level of the system with a fixed topology. In this case the uncertainties represented by a 10% standard deviation extracted from 1000 samples of a Gaussian distribution, applied in all load busses, and influencing simultaneously and coherently the magnitude of both active and reactive

Fig. 4. SSS analysis for noises (N), experiment 5N2.

Fig. 5. SSS analysis, hybrid (scenarios + noises), experiment 7SN.

load components. The histogram showing the distribution of the real part of the critical eigenvalues is depicted in Fig. 4, where the average value is 0.0987 and the standard deviation is 0.00013. Therefore, it seems that there is no critical eigenvalue with a positive real part, meaning that the sole presence of noise uncertainties in experiment 5N2 does not turn the system unstable. A similar histogram, not shown here, giving the distribution of critical damping rates indicates an average value of 1.537% and a standard deviation of 0.00198. From both results, it can be concluded that the system is quite robust against this type of uncertainty, because the probability of occurrence of eigenvalues with positive real parts and/or with damping rates less than 10% is practically nil. Experiment 7SN simultaneously combines both the scenario and noise uncertainties used in experiments 3S3 and 5N2. The resulting histogram representing the distribution of the real part of critical eigenvalues is shown in Fig. 5, where there is a non-null probability of occurrence of unstable situations equal to 28.58%. It is interesting to note that experiment 3S3 is related to scenario uncertainties and indicates a low risk situation, while experiment 5N2 is related to noise uncertainties and indicates a safe situation. When both types of uncertainties are simultaneously combined, a significant increase in the risk level arises. This effect, seemingly

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Only scenario uncertainties

Only noise uncertainties

Scenario + noise uncertainties

<0 0
1.72 98.26

0 100

28.58 71.42

exaggerated, was however purposely designed in this experiment to highlight a situation that could otherwise remain unnoticed. This result, as summarized in Table 3, emphasizes the relevance of considering hybrid mode uncertainties, as indicated by experiment 7SN.

Probabilistic transient stability analysis For the TRS analysis, dynamic modelling was restricted to rotor transitory effects and the field voltage of the synchronous machine (three dynamic states per generator, associated with electro-mechanical phenomena). Voltage regulators were also modelled, but no power system stabilizers and governors were considered. Preliminary stationary conditions observed no voltage violations outside the range (0.950–1.050 pu) and no overloads. Two hydrological dispatching profiles (in MW) were used, and were referred to as A and B. In both cases, generator G2 was always used as the swing machine in the load flow computation. Dispatch A was defined as (G1 = 1000; G2 = 520.3; G3 = 650; G4 = 632; G5 = 508; G6 = 650; G7 = 560; G8 = 540; G9 = 830; G10 = 250), while dispatch B was defined as (G1 = 1000; G2 = 1011.3; G3 = 650; G4 = 532; G5 = 408; G6 = 550; G7 = 460; G8 = 540; G9 = 730; G10 = 250). Probabilities 0.2 and 0.8 were attributed to dispatching profiles A and B, respectively. Because in one of the experiments a 3U, 80 ms deterministic short-circuit fault was applied in the middle of the line between busses {14–15}, the corresponding probabilities for the full grid topology and for all remaining 33 ‘‘n  1’’ topologies were calculated. Those probabilities are given in [57] and are related to those long-term events indicated in Fig. 1. When the dispatching profiles are combined with the possible n and n  1 topologies, a total of (2  34 = 68) probabilistic states representing scenarios are created. Only one load level equal to (6097.3 + j1409.1) MVA was analysed without uncertainties. Regarding the evaluation of probabilities for the location of faults, one simple but valid strategy is based on the argument that the longer the transmission line, the greater its exposition to faults. Therefore, for each one of the possible 34 ‘‘n  1’’ transmission line topologies, the probability of a fault location in a given line equals to the rate of the given line length to the total system line length (see last right column of Table VI-34 of [57]). For the sake of simple illustration, the other uncertainties related to a fault description were disregarded in the following examples. The only failure mode considered in the TRS analysis was the loss of synchronism. Table 4 presents four experiments to illustrate the probabilistic TRS analysis.

For instance, experiments 13a and 13b have no scenario uncertainties related to hydrological profiles (Dp) or diverse topologies (To). However, both consider geographical location uncertainties (Loc1), meaning that there are probabilities indicating in which transmission line the fault will occur. In both experiments, there will be faults in all 34 transmission lines. Specifically, in case 13a, all probabilities will be equal to 1/34, aiming to emulate a traditional deterministic rationale. On the other hand, the probabilities of experiments 13b and 15 are proportional to each line exposure. Experiment 14b indicates that there will be no type of fault uncertainty. Loc2 indicates that there will be no uncertainty related to the position where faults will occur in a given line (all faults will be applied in the middle m of all lines). Ft and Dur also indicate that there will be no uncertainties regarding the type and duration of faults (only 3U, 80 ms faults are simulated). The probabilistic TRS analysis was based on the evaluation of three indices. Two of them (POS and POI) are conceptually represented by (1). A third and new index, (Apparent Stability Index – ASI) is a non-dimensional number given by (4):

ASI ¼ f½EðM 1 Þ2 þ ½EðM 2 Þ2 g

1=2

ð4Þ

where E(M.) is the modulus of the expectancies of type 1 (M1) and type 2 (M2) negative margins [57,65]. ASI is greater or equal to zero and its magnitude is inversely proportional to the system’s dynamical robustness. A type 1 negative margin is defined by equation (2.19) of [65] and is given in (MW.rd). It is known that the concept of stability margin expresses how far a given system is from an unstable zone. However, during extremely severe circumstances, the stability margin based on the generalized equal areas criterion [65] is non-existent because one of the following conditions: (i) curves representing mechanical and electrical power do no intersect each other; and (ii) the rotor angle associated with the fault clearance is beyond the angle that defines instability. When one of those conditions appear, a type 2 negative margin [57] is given in (MW), and is defined as the minimum value of the accelerating power Pa that occurs during the system’s transient behaviour, expressed by (5), where M is the generalized one machine infinite bus (OMIB) inertia coefficient and d is the rotor angle [57,59–68]. Because it is not possible to know in advance which type of margin (type 1 or 2) will be found, the formulation given by (4) is justified because it captures both possibilities at once and combines the two types of negative margins into a single index that inversely conveys the overall system stability level. 2

2

Pa ¼ Mðd d=dt Þ

ð5Þ

The results obtained for the experiments proposed in Table 4 are shown in Table 5. Experiments 13a and 13b were devised to show the influence of distinct fault uncertainties in the presence of deterministic scenarios. As the fault locations in experiment 13a are considered to be all equiprobable and there are 34 transmission lines, the corresponding fault location probabilities are equal to (1/34 = 0.029). Therefore, this evaluation is similar to a conventional deterministic assessment in disguise. Because there are 7 unstable cases, the POI index reaches 7/34, as shown in

Table 4 Probabilistic transient stability experiments. Ex

Dp

To

Loc1

Loc2

Ft

Dur

RM

Probability Loc1

13a 13b 14b 15

no no no Y

no no Y no

all all {14–15} all

m m m m

no no no no

80 ms 80 ms 80 ms 80 ms

no no no no

Equiprobable Proportional 1.0 Proportional

Ex = experiment; Dp = dispatching uncertainties; To = topology uncertainties; Loc1 = geographical location uncertainties; Loc2 = element location uncertainties; Ft = type of fault uncertainties; Dur = fault duration ; RM = using remedial measures Y = yes.

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Table 5 Probabilistic transient stability results. Experiment

POI (%)

POS (%)

ASI

13a 13b 14b 15-A (20%) 15-B (80%) 15 (S + F)

20.59 29.14 0.027 32.99 16.34 19.67

79.41 70.86 99.91 67.02 83.66 80.33

53.80 57.83 0.14 58.46 6.47 16.66

Table 5. On the other hand, experiment 13b is clearly able to highlight the concept of risk associated with the combination of distinct degrees of severity for each fault location with their respective diverse probabilities. Comparing cases 13a and 13b it is suggested that in this particular example, the conventional deterministic analysis (in disguise) would be optimistic (ASI = 53.80), making a strong case in favour of the probabilistic analysis, which gives a precise assessment of the actual risk (ASI = 57.83). Experiment 14b takes the opposite stance: the applied fault is now deterministic between busses 14 and 15 (fault probability equal to one) and the system is dynamically stable under a 3U, 80 ms fault condition, when the grid has its 34 lines in operation. However, all 33 remaining ‘‘n  1’’ possible topologies are also analysed with their respective probabilities of occurrence. It is then observed that some n  1 topologies are unable to withstand the fault, thus rendering an ASI = 0.14. Finally, experiment 15 exemplifies a case where both scenario and fault uncertainties are combined together. Table 5 yields the separate results for each of the chosen dispatching profiles, clearly indicating the severe risk that the system undergoes when the condition with a probability equal to 20% prevails. These results indicate why the proposed technique may be a useful decision tool for those in charge of the system operations. Conclusions This paper has presented a practical, simple, and effective way to introduce the influence of probabilistic uncertainties in power system security studies. Among the advantages of the suggested method, the following points are evident: (i) the ability to discriminate several types of uncertainties that permeate power systems and that are relevant to system security; (ii) the capacity to use existing traditional tools from both small signal dynamic analysis and transient stability analysis in such a way that easily adapts them to the well-established concept of probabilistic adequacy assessment without resorting to abstruse and hard-to-implement theoretical techniques but still maintains a state-of-the-art and accurate dynamics modelling; (iii) the enormous advantage of usual availability for the required statistical data; and (iv) proposal of a conceptual procedure that renders a highly combinatorial problem amenable to current state-of-the-art hardware resources within acceptable limits of computational burden. It was also indicated that the method has the potential for the introduction of several interesting future developments such as [71]: the evaluation of a varied set of new probabilistic indices; ample gamut of remedial measures, including the effect of protection reliability; using parallel and distributed computation, and representing ageing phenomena [72]. Among the obtained results that one may wish to highlight, two results may deserve attention, related to the SSS and TRS analyses respectively: the effective representation of noise uncertainties through a straightforward combination of weighted histograms and the successful performance of the new ASI index. The first claim is supported by the result shown by experiment 7SN, where an easy combination of histograms associated with noise

uncertainties can be achieved using simple weights related to scenario probabilities. The second assertion is based on the coherent behaviour of the proposed ASI index, which is capable of capturing security nuances influenced by uncertainties and translating them to comparable numerical values that are useful to management. Using the computational approach suggested in this paper, it is believed that the generalized power system probabilistic security problem would greatly benefit if further investigation tackles the establishment of modelling standards and procedures, similar to those indicated in [73] for the adequacy problem and advances through the security diagnosing and security managing phases, as discussed in [74]. The research in this direction will facilitate the effective diffusion of the probabilistic security analysis in the realm of power company’ management. Acknowledgments The authors thank Prof. Federico Milano, College, Dublin, Ireland for his essential help, providing us with PSAT(1.3.4, 2.0), used in the SSS analysis, and Prof. Zhao Yang Dong, U. of Queensland, Australia, for his bibliographical help. This work was supported in part by the National Council of Scientific and Technological Development – CNPq. References [1] Ayyub BM, Klir GJ. Uncertainty Modeling and Analysis in Engineering and the Sciences. USA: Boca Raton: Chapman & Hall/CRC; 2006. p. 52–6. [2] Billinton R, Fotuhi-Firuzabad M, Bertling L. Bibliography on the application of probability methods in power systems reliability evaluation 1996–1999. IEEE Trans PWRS 2001;16(4):595–602. [3] Schilling MTh, Billinton R, Santos MG. Bibliography on power systems probabilistic security analysis 1968-2008. Int J Emerging Electric Power Syst 2009;10(3):1–48. article 1(online). [4] Mielczarski W, editor. Fuzzy logic techniques in power systems. PhysicaVerlag Heilderberg; 1998. [5] El_Hawary ME, editor. Electric power applications of fuzzy systems. Piscataway: IEEE Press; 1998. [6] Li W. Probabilistic transmission system planning. New Jersey: Wiley, IEEE Press; 2011. [7] Anders G, Vaccaro A, editors. Innovations in power systems reliability. London: Springer-Verlag; 2011. [8] Haarla L, Koskinen M, Hirvonen R, Labeau PE. Transmission grid security, a probabilistic security analysis approach. London: Springer-Verlag; 2011. [9] Li W. Risk assessment of power systems. Piscataway: Wiley, IEEE Press; 2005. [10] CIGRE TF 38.03.12 Report. Power System Security Assessment: A Position Paper, ELECTRA, vol. 175, 1997. [11] Bremermann HJ. Complexity and transcomputability. In: Duncan R, WestonSmith M, editors. The encyclopaedia of ignorance. Oxford: Pergamon Press; 1977. p. 167–74. [12] M. Turing. On computable numbers, with an application to the Entscheidungsproblem. In: Proceedings of the London Mathematical Society, sr.2, vol. 42, 1937. p. 230–65, errata, vol. 43, p. 544–46. [13] Pai MA, Sengupta DP, Padiyar KR. Small signal analysis of power systems. Harrow, UK: Alpha Science Int Ltd.; 2004. [14] Burchett RC, Heydt GT. Probabilistic methods for power system dynamic stability studies. IEEE Trans PAS 1978;PAS-97:695–702. [15] Anders GJ. Probability concepts in electric power systems. New York: John Wiley & Sons; 1990. [16] Mori H, Tuzuki S. New advances of power system probabilistic dynamic stability using the bilinear transform of the system matrix. In: Proc. of IEEE Montech, October 1–3, 1986. [17] Pang CK, Dong ZY, Zhang P, Yin X. Probabilistic analysis of power system small signal stability region. In: Proc 5 th IEEE Int Conf on Control and Automation (ICCA 2005), Budapest, June 27–29, 2005. p. 503–09. [18] Nwankpa CO, Shahidehpour SM, Schuss Z. A stochastic approach to small disturbance stability analysis. IEEE Trans PWRS 1992;7(4):1519–28. [19] Chung CY, Wang KW, Tse CT, Bian XY, David AK. Probabilistic eigenvalue sensitivity analysis and PSS design in multimachine systems. IEEE Trans PWRS 2003;18(4):1439–44. [20] Xu Z, Dong ZY, Zhang P. Probabilistic small signal analysis using Monte Carlo simulation. In: Proc IEEE PES General Meeting. San Francisco; 2005. [21] Xu Z, Ali M, Dong ZY, Li X. A novel grid computing approach for probabilistic small signal analysis. In: Proc of IEEE PES General Meeting. Montreal; 2006. [22] Castellanos RB, Messina AR. Robust stability and performance analysis of large scale power systems with parametric uncertainty. New York: Nova Science Publishers Inc; 2008.

A.Y. Takahata et al. / Electrical Power and Energy Systems 71 (2015) 151–159 [23] Castellanos RB, Messina AR, Sarmiento HU. A l-analysis approach to power system stability robustness evaluation. Electric Power Syst Research 2008; 78(2):192–201. [24] Patton AD. Short-term reliability calculation. IEEE Trans PAS 1970;89(4): 509–14. [25] Light R. Transient stability aspects of power system reliability (CEGB system). In: Int Conf on Reliability of Power Supply Systems, IEE Conf Publication no. 148, London, UK, February 1977. [26] Billinton R, Kuruganty PRS. A probabilistic index for transient stability. In: Proc of IEEE PES Winter Meeting, A 78 231–3, New York, NY, January 29-February 3, 1978. [27] Kuruganty PRS, Billinton R. Protection system modeling in a probabilistic assessment of transient stability. IEEE Trans PAS 1981;PAS-100(5): 2163–70. [28] Billinton R, Kuruganty PRS. Probabilistic assessment of transient stability in a practical multimachine system. IEEE Trans PAS 1981;PAS-100(7): 3634–41. [29] Loparo K, Blankenship G. A probabilistic mechanism for dynamic instabilities in electric power systems. In: Proc of IEEE PES Winter Meeting, A 79 053–0, New York, NY, February 4–9, 1979. [30] Anderson PM, Bose A. A probabilistic approach to power system stability analysis. IEEE Trans PAS 1983;102(8):2430–9. [31] EPRI, A Probabilistic Approach To Stability Analysis-Volume 1: Mathematical Models, Computing Methods, And Results, EPRI Report EL-2797, RP 1764–4, Electric Power Research Institute, Palo Alto, California, 1983. [32] Anderson PM. Reliability modeling of protective systems. IEEE Trans PAS 1984;103(8):2430–9. [33] Wu FF, Tsai Y, Yu Y. Probabilistic steady-state and dynamic security assessment. IEEE Trans PWRS 1988;3(1):1–9. [34] Porretta B, Kiguel DL, Hamoud GA, Neudorf EG. A Comprehensive approach for adequacy and security evaluation of bulk power systems. IEEE Trans PWRS 1991;6(2):433–41. [35] Billinton R, Khan E. A security based approach to composite power system reliability evaluation. IEEE Trans PWRS 1992;7(1):65–72. [36] Leite da Silva AM, Endrényi J, Wang L. Integrated treatment of adequacy and security in bulk power system reliability evaluations. IEEE Trans PWRS 1993;8(1):275–85. [37] Billinton R, Aboreshaid S. Security evaluation of composite power systems. IEE Proc– Gen Trans Distribution 1995;142(5). [38] McCalley JD, Fouad AA, Agrawal BL, Farmer RG. A risk-based security index for determining operating limits in stability-limited electric power systems. IEEE Trans PWRS 1997;12(3):1210–9. [39] Aboreshaid S, Billinton R. A framework for incorporating voltage and transient stability considerations in well-being evaluation of composite power systems. In: Proc of IEEE Power Eng Society Summer Meeting, vol. 1, 18–22 July 1999. p. 219–24. [40] Leite da Silva M, Jardim JL, Rei AM, Mello JCO. Dynamic security risk assessment. In: Proc of IEEE Power Eng Society Summer Meeting, vol. 1, 18–22 July 1999, p. 198–205. [41] Rei AM, Leite da Silva AM, Jardim JL, Mello JCO. Static and dynamic aspects in bulk power system reliability evaluations. IEEE Trans PWRS 2000;15(1): 189–95. [42] Hsu Y, Chang C. Probabilistic transient stability studies using the conditional probability. IEEE Trans PWRS 1988;3(4):1565–72. [43] Billinton R, Fotuhi-Firuzabad M. A basic framework for generating system operating health analysis. IEEE Trans PWRS 1994;9(3):1617–9. [44] Aboreshaid S, Billinton R, Fotuhi-Firuzabad M. Probabilistic transient stability studies using the method of bisection. IEEE Trans PWRS 1996;11(4):1990–5. [45] Hockenberry JR, Lesieutre BC. Evaluation of uncertainty in dynamic simulations of power system models: the probabilistic collocation method. IEEE Trans PWRS 2004;19(3):1483–91. [46] Kim H, Singh C. Power system probabilistic security assessment using Bayes classifier. Electric Power Syst Research 2005;74:157–65. [47] Dobson I, Zhang P. IEEE PES CAMS Task Force on Understanding, Prediction, Mitigation and Restoration of Cascading Failures. Initial review of methods for cascading failure analysis in electric power transmission systems. In: Proc of IEEE PES General. Meeting, Pittsburgh, PA, USA, July 2008. [48] Papaefthymiou G, Kurowicka D. Using copulas for modeling stochastic dependence in power system uncertainty analysis. IEEE Trans PWRS 2009; 24(1):40–9. [49] da Silva AML, Fernandez RAG, Singh C. Generating capacity reliability evaluation based on Monte Carlo simulation and cross-entropy methods. IEEE Trans PWRS 2010;25(1):124–37.

159

[50] Kasim SR, Othman MM, Ghani NFA, Musirin I. A power system risk assessment using bootstrap technique. In: Proc of 2010 Int Conf on Intelligent and Advanced Systems (ICIAS), Kuala Lumpur, Malaysia, June 2010, p. 1–6. [51] Berizzi A, Bovo C, Fumagalli E, Grimaldi EA. Security assessment in operation: a comparative study of probabilistic approaches. In: Proc of IEEE Bologna Power Tech Conf, Bologna Italy, June 23–26, 2003. [52] Dissanayaka A, Annakkage UD, Jayasekara B, Bagen B. Risk-based dynamic security assessment. IEEE Trans PWRS 2011;26(3):1302–8. [53] Odun-Ayo T, Crow ML. Structure-preserved power system transient stability using stochastic energy functions. IEEE Trans PWRS 2012;27(3):1450–8. [54] Dong ZY, Zhao JH, Hill DJ. Numerical simulation for stochastic transient stability assessment. IEEE Trans PWRS 2012;27(4):1741–9. [55] Vaahedi, Li W, Chia T, Dommel H. Large scale probabilistic transient stability assessment using B.C. Hydro’s on-line tool. IEEE Trans PWRS 2000;15(2): 661–7. [56] Takahata AY, Power System Small Signal Security Analysis Considering Uncertainties, D.Sc. Thesis, COPPE/UFRJ, Electrical Engineering Dept., Rio de Janeiro, Brazil, May, 2008 (in Portuguese). . [57] dos Santos MG. Power Systems Transient Security Considering Uncertainties, D.Sc. Thesis, Fluminense Federal University (IC/UFF), Niterói, Brazil, December, 2009 (in Portuguese). . [58] Milano F. An open source power system analysis toolbox. IEEE Trans on PWRS, vol. 20, no. 3, p.1199–206, August 2005, (PSAT down load: . [59] Xue Y, Van Cutsem Th, Ribbens-Pavella M. A simple direct method for fast transient stability assessment of large power systems. IEEE Trans PWRS 1988;3(2):400–12. [60] Xue Y, Van Cutsem Th, Ribbens-Pavella M. Extended equal area criterion justifications, generalizations, applications. IEEE Trans PWRS 1989; 4(1):44–52. [61] Rahimi FA. Generalized equal-area criterion: a method for on-line transient stability analysis. In: Proc of IEEE Int Conf on Systems, Man, and Cybernetics, Los Angeles, California, November 1990, p. 684–88. [62] Xue Y, Pavella M. Critical cluster identification in transient stability studies. IEE Proceedings-C 1993;140(6):481–6. [63] Xue Y, Rousseaux P, Wehenkel L, Gao Z, Pavella M, Trotignon M, Duchamp A, Heilbronn B. Dynamic extended equal-area criterion. part ii: recent extentions. In: Proc of Athens Power Tech, NTUA-IEEE/PES Joint Int Power Conf, September 1993. [64] Chiodo, Lauria D. Transient stability evaluation of multimachine power systems: a probabilistic approach based upon the extended equal area criterion. Trans Distrib 1994;141(6):545–53. [65] Pavella, Ernst D, Ruiz-Veja D. Transient stability of power systems – a unified approach to assessment and control. Massachusetts: Kluwer Academic Publishers; 2000. [66] Machado Ferreira M, Dias Pinto JA, Maciel Barbosa FP. Dynamic security analysis of an electric power system using a combined Monte Carlo-hybrid transient stability approach. In: Proc of IEEE Porto Power Tech Conf, Porto, Portugal, 10–13 September 2001. [67] Pizano-Martinez, Fuerte-Esquivel CR, Ruiz-Veja D. Global transient stabilityconstrained optimal power flow using the SIME method. In: Proc of IEEE Power & Energy Soc Gen Meeting. Canada: Calgary; 2009. p. 26–30. [68] Tan HM, Vowles DJ, Zivanovic R. Implementation of the enhanced binary-SIME method for finding transient stability limits with PSS/ETM. In: Proc of IEEE Bucharest Power Tech Conf, Bucharest, Romania, June 28th–July 2nd 2009. [69] Athay T, Podmore R, Virmani S. A practical method for direct analysis of transient stability. IEEE Trans PAS 1979;98(2):573–84. [70] Pai MA. Energy Function Analysis for Power System Stability. Boston/ Dordrecht/London: Kluwer Academic Publ., Appendix A, 1989. p. 223–27. [71] Schilling MT, Coutto MB, Leite da Silva AM, Billinton R, Allan RN. An integrated approach to power system reliability assessment. Int J Electrical Power Energy Syst 1995;17(6):381–90. [72] Schilling MT, Praça JCG, Queiroz JF, Singh C, Ascher H. Detection of ageing in the reliability analysis of thermal generators. IEEE Trans Power Systems 1988;PWRS 3(2):490–9. [73] Schilling MT, Souza JCS, Coutto MB. Power system probabilistic reliability assessment: current procedures in Brazil. IEEE Trans PWRS 2008;23(3): 868–76. [74] Schilling MT, Leite da Silva AM. Conceptual investigation on probabilistic adequacy protocols: Brazilian experience. IEEE Trans PWRS 2014;29(3): 1270–8.