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With increasing amounts of asynchronous generation being deployed to meet system energy demands, many transmission corridors may become constrained by angular stability criteria rather than steady-state thermal limitations. In such a system, it is paramount to have the capability to rapidly evaluate the stability margins of the system, particularly in a threatened post-fault state. The use of single machine equivalents (SIME) has been shown to be a powerful and flexible hybrid stability analysis method which can be computed directly from measured PMU data. However, due to the nature of the system reduction employed by SIME, as well as the method of extrapolation to estimate stability margins, there are many cases where the swing of the system is not accurately modelled by the traditional methods until a significant amount of data has been collected, at which point it may be too late to respond to the threat. In this paper, we address some of the limitations imposed by the traditional methods of reduction and prediction. We propose a method where rather than identifying a single critical SIME model for stability prediction, a spectrum of SIME representations of the post-fault system is developed, yielding a more timely and accurate estimate of post- fault stability conditions, through observation and feature extraction. . . . Keywords Transient stability analysis Hybrid stability analysis method Setofcriticalmachines Single machine equivalent models Kinetic energy trajectory Introduction system is sourcing more power from renewables. Since fewer conventional power plants will remain in operation under this Climate change and greenhouse gas emissions have become scenario, methods of establishing system security will need to recognized as problems of international significance in recent be updated to cope with the loss of system inertia. years. Though renewable energy sources are considered a po- On the other hand, the deployment of Phasor Measurement tential solution to these problems, their integration into Units (PMUs) on large scale in power systems creates the existing power grids brings a new set of challenges. The un- basis for near-real-time and accurate system state observa- predictable and uncontrollable nature of these sources of gen- tions. With high sampling rate and GPS time synchronisation, eration will require the development of advanced analytics PMUs enable a new stream of applications for stability anal- and distributed control algorithms in order to manage such a ysis. The data points generated by PMUs can be used to esti- highly distributed generation base. As integration of renew- mate the stability of the network and even the margin. able resources continues, though it assists in decarbonisation Stability of power systems has constituted a concern for of the power industry, further challenges arise when the system operators and several analysis techniques have been developed, such as time-domain simulation methods [1, 2]. These offer good accuracy, but they are computationally in- * I. Pisica tensive as they require solutions of ordinary differential equa- email@example.com tions. They do not provide any information on the margin. Direct methods are faster but can lead to narrow margins . School of Engineering and Design, Energy Futures Institute, Brunel Several strands of hybrid stability methods have been uti- University London, London, UK lized to ensure accuracy and low computational burden. The National Grid UK, London, UK one used in this paper is the single machine equivalent (SIME) [4–8]. Hybrid methods resemble direct methods by using en- School of Electrical Engineering and Information, Sichuan University, Chengdu, China ergy functions, but the input data consists of measurements 15 Page 2 of 10 Technol Econ Smart Grids Sustain Energy (2019) 4:15 −1 −1 δ ðÞ t ¼ M ∑ M δ ðÞ t taken during and immediately after the fault. Determining the C i i C δ ðÞ t ¼ M ∑ M δ ðÞ t ; ð4Þ NC j j NC i∈C j∈NC generators that are likely to go out of step during a fault is an important task in SIME methods [5, 9, 10, 11], with ap- and the frequencies of the critical and non-critical machines as proaches such as Preventive SIME and Emergency SIME be- determined respectively as ing applied . Several papers have focused on energy func- −1 tions for power systems property description [12–15]. The −1 ω ðÞ t ¼ M ∑ M ω ðÞ t C i i C ω ðÞ t ¼ M ∑ M ω ðÞ t : ð5Þ NC j j NC i∈C methodology described in this paper can help evolve power j∈NC systems towards smart grids through advancement of real- The rotor angle of the SIME model is then found as time monitoring and control algorithms [16–23]. This paper gives a brief overview of single machine δ ðÞ t ¼ δ ðÞ t −δ ðÞ t ; ð6Þ S C NC equivalents and describes the time-domain evolution of aSIMEmodel’s kinetic energy trajectory, and then anal- and the frequency as yses the features of post-fault swings. The results presented ω ðÞ t ¼ ω ðÞ t −ω ðÞ t : ð7Þ S C NC here prove the usefulness of the method application for large power systems. The mechanical input power to the SIME model is calcu- lated from the generator data as Single Machine Equivalents −1 −1 P ðÞ t ¼ M M ∑ P ðÞ t −M ∑ P ðÞ t ; ð8Þ m;S S m;i m; j C NC i∈C i∈NC The aim of SIME is to generate a simplified model of a power and the electrical output power is system in the post-fault state, employing a hybrid stability analysis approach where measured or simulated data of the −1 −1 swing of the generators serves as a set of inputs for generating P ðÞ t ¼ M M ∑ P ðÞ t −M ∑ P ðÞ t : ð9Þ e;S S e;i e; j C NC i∈C j∈NC the model. This technique creates a simplified power system model that is useful for extrapolation and prediction of system Subsequently, the accelerating power of the SIME model is behaviour, and that can be updated with real-time data. The found to be importance of SIME models in the proposed method warrants a brief treatment of the technique. The SIME model formula- P ðÞ t ¼ P ðÞ t −P ðÞ t : ð10Þ a;S m;S e;S tion begins by separating the generators into two distinct groups in the post-fault state, For any given SIME model of a system in the post-fault state, the system can be approximated as C ¼fg G ; NC G C ¼ ∩NC ¼ ∅; C∪NC ¼fg G ; ∀k ¼ 1…N ð1Þ i j k G S ¼ C; NC; M ; δ ðÞ t ; ω ðÞ t ; P ðÞ t ; ð11Þ S S S a;S where C is the set of critical generators, NC is the set of non- critical generators, G indicates the kth generator in the sys- where any S contains the set of data which describes a tem, and N is the total number of generators in the system. SIME system representation. In prior usage, the critical While a number of methods could be used to determine which SMIB refers to the SIME model which first goes unstable, generators belong to the critical set for a given fault, the most thus having the minimal stability margin of all possible often applied approach is through observation of the post-fault SIME representations. angular separation of the generators . Once the generators are split into these two sets, the inertia of the equivalent critical and non-critical machines can be calculated respectively as, Kinetic Energy Measures M ¼ ∑ M ; M ¼ ∑ M ; ð2Þ C i NC j i∈C j∈NC In prior research, SIME methods of estimating stability mar- gins in the post-fault state have focused on fitting and extrap- and the SIME inertia is then calculated as olation of the P – δ curve of the SIME model . Because the SIME representation is analogous to an SMIB structure, M M C NC M ¼ : ð3Þ analysis of the P – δ characteristics is an application of the M þ M C NC equal area criterion, which is also equivalent to the Lyupanov This renders the SIME representation equivalent to a single criteria for a single machine system [22, 23]. machine infinite-bus (SMIB) system. While forward-prediction of system stability using the Similarly, the rotor centre-of-angle for the equivalent criti- SIME P – δ curve is effective for cases where the curve is cal and non-critical machines is nearly sinusoidal, these cases only occur when a given fault Technol Econ Smart Grids Sustain Energy (2019) 4:15 Page 3 of 10 15 drives the generators to separate into two nearly-coherent sets. which are meant to provide additional insight into the post- For other faults where there is weaker coherency within the fault behaviour of the system. The set of SIME representations critical and non-critical sets, the SIME approximation’sP – δ chosen in the proposed method are deliberately selected curve is unlikely to have a sinusoidal shape, thus requiring a to observe the system behaviour for several post-fault more sophisticated extrapolation technique. conditions. Figure 1 demonstrates the concept of the In the proposed method, rather than observing the P – δ multiple SIME method as proposed in this paper. The curve to calculate a stability margin estimate, the time-domain spectrum of SIME representations selected for this study evolution of a SIME model’s kinetic energy trajectory forms are formed by first observing the angular frequencies ω the basis for the analysis and prediction of system stability. of the generators at the time of the fault clearing t . fc Initially, the kinetic energy of a chosen SIME representa- The generators are then sorted into an ordered set ac- tion S is known to be cording to their angular frequency as L ¼ …; G ; G ; … ω > ω ∀i; j∈1…N ; ð17Þ i j i j G E ¼ M ⋅ω ðÞ t : ð12Þ k;S S S sorted from greatest to least. Calculating the derivative of the SIME kinetic energy with Utilizing this ordered set L, the set of N – 1SIME repre- respect to time yields sentations are formed, where for k = 1… N – 1, the critical and non-critical generator sets are defined as dE dω ðÞ t k;S S ¼ M ⋅ω ðÞ t : ð13Þ S S dt dt C ¼fg L ; …; L ; NC ¼fg L ; …; L : ð18Þ 1 k kþ1 N From the swing equation, the relationship between change That is, for k = 1, the first generator in L is treated as in frequency and accelerating power is belonging to the critical set, and the remaining N – 1gener- ators form the non-critical set. For k = 2, the first two gener- dω ðÞ t M ¼ P ðÞ t ; ð14Þ S a;S ators in L are treated as belonging to the critical set, and the dt remaining N – 2 generators form the non-critical set. This is and substitution of this relationship into the time-derivative of repeated for all k = 1 … N – 1. kinetic energy yields With the critical and non-critical sets defined, the SIME equations can be applied to create an SMIB rep- dE k;S ¼ P ðÞ t ⋅ω ðÞ t : ð15Þ a;S S resentationofthesystem basedonthe kth critical and dt non-critical generator sets, which are then used to form system Finally, the time-derivative of the kinetic energy is inverse properties of this representation as S . Application of the weighted by the SIME model’s inertia, yielding SIME equations thus yields the spectrum of SIME represen- tations S ,k =1 … N – 1. k G 1 dE P ðÞ t ⋅ω ðÞ t k;S a;S S Each of the SIME system representations S contains Y ðÞ t ¼ ⋅ ¼ ð16Þ M dt M S S the set of data necessary for calculating the characteris- tic feature Y (t), and thus N – 1separate Y trajecto- Sk G For any SIME model, Y (t) forms the characteristic feature ries are calculated from the various generator set groupings. which is utilized in the proposed method for model extraction An interesting pattern emerges in the behaviour of these post- and stability prediction. fault characteristics Y for the spectrum of SIME models S .As Features of Post-Fault Swings When assessing stability estimates of a system in the post-fault state, additional insight into the behaviour of the system may be obtained by observing not only a single SIME representa- tion of the system, but also through the observation of the behaviour of several SIME representations. Some prior work exists that observes, in addition to the critical SIME model, the P – δ characteristics of other candidate SIME models that are similar but not identical to the critical SIME model . The approach taken in the proposed method involves the intentional simultaneous generation of a set of SIME models, Fig. 1 The concept of Multiple SIME representations 15 Page 4 of 10 Technol Econ Smart Grids Sustain Energy (2019) 4:15 seen in Fig. 2, while each of the N – 1 trajectories are unique, The procedure begins with the computation of a set of finite they tend to follow a post-fault group trend. differences, with the first-order finite difference defined as, The mean value of the group trend is calculated as the 0 Y ðÞ t −Y ðÞ t−Δt S S k k inertially-weighted average, Y ðÞ t ¼ ; ð20Þ Δt N −1 ∑ M Y ðÞ t S S k k k¼1 and the second-order finite difference defined as hi Y ðÞ t ¼ ; ð19Þ N −1 ∑ M k¼1 0 0 Y ðÞ t −Y ðÞ t−Δt S S k k Y ðÞ t ¼ : ð21Þ and if this mean trend is removed from the Y trajectories, the Δt remaining mean-adjusted trajectories behave as shown in Computing the relevant finite differences for the most re- Fig. 3. As seen by the post-fault behaviour of the mean-adjusted cently available data at time t over the spectrum of computed trajectories Y (t) – 〈 Y (t) 〉, these tend to follow a damped SIME models S yields a coefficient matrix, Sk S k oscillatory pattern prior to the system reaching instability and 2 3 Y ðÞ t −Δt Y ðÞ t −Δt S c c going out-of-step. The method proposed in this paper seeks to 1 S 6 0 7 6 7 Y ðÞ t −Δt Y ðÞ t −Δt extract the features of the post-fault characteristic trajectories S c c AðÞ t ¼ 2 ; ð22Þ 6 7 4 5 ⋮⋮ Y (t), by calculating a second-order state space model which Sk Y ðÞ t −Δt Y ðÞ t −Δt S c c encapsulates the group trend as well as the damped oscillatory k nature of these signals. and the corresponding second-order differences are used to form a column vector as 2 3 Feature Extraction Technique Y ðÞ t 6 7 Y ðÞ t 6 S 7 In the event of a system fault, if PMUs are deployed within a bðÞ t ¼ : ð23Þ 4 5 system, then post-fault estimation of system stability may be Y ðÞ t achieved through analysis of the discrete-time data set obtain- ed from measurement. Within this study, it is assumed that the The least-squares solution to the linear problem measured data obtained from the PMUs is received in discrete AðÞ t fðÞ t ¼ bðÞ t ð24Þ time intervals, spaced by a time step of Δt. For each time step, c c c a new set of measurements is obtained from the PMUs de- yields a feature extraction vector f(t ) which is used to form ployed at each generator, and this data is subsequently used to the second-order state space approximation. form the spectrum of SIME models presented in Section 4. This approach can be extended to not only include Utilizing the newly obtained generator data for the current samples from the most recently available data at time t , time step tc, the spectrum of SIME models S ,k=1 … NG but can be extended to incorporate further past data, – 1 is formed, and the corresponding Y (t) are calculated, up Sk increasing the dimension of the least-squares problem. to the most recently available time interval t . The sequence of This is achieved through vertical concatenation of pre- data from the Y (t) are used to compute a least-squares fitting Sk viously calculated A and b, forming the expanded least- of a second-order state space model, which can then be used squares problem for extrapolation and post-fault stability estimation. Fig. 2 The calculated characteristic feature Y (t) for fault number F14 Technol Econ Smart Grids Sustain Energy (2019) 4:15 Page 5 of 10 15 Fig. 3 The mean adjusted trajectories of Y (t) for fault number F14 indicates the state variable corresponding to the current slope 2 3 2 3 AðÞ t bðÞ t c c of Y (t). Thus, for example, δ would be the change in slope Sk sv 6 7 6 7 AðÞ t −Δt bðÞ t −Δt c c per unit time dependent on the current value of Y (t). 6 7 6 7 Sk 6 7 6 7 AðÞ t −2Δt bðÞ t −2Δt c c The state space behavioural model F(t ) is constructed as 6 7 6 7 fðÞ tc ¼ ð25Þ 6 7 6 7 ⋮ ⋮ 6 7 6 7 4 5 4 5 A t −m Δt b t −m Δt c p c p FðÞ t ¼ jfðÞ t ; ð30Þ c c which can be generalized as where the state space coefficients determining the change in value and slope of Y (t) with respect to the current value are Sk A⋅fðÞ t ¼ b: ð26Þ set to zero and unity respectively. The coefficients which are dependent on the slope, which form the second column of the Note that, in order to ensure that the problem is not state space model, are formed from the previously calculated underdetermined, there is a constraint on the first time instance values in the column vector f(t ). This forms the state space after fault clearing where this model can be derived, where behavioural model for the current time step, which can then be t ⩾t þ m Δt; ð27Þ used for extrapolation and prediction, and which can also be c fc T recalculated for each new time step where new data is given that m is equal to the maximum order derivative used received. in the formation of b, plus the number of previous time sam- ples m used in the formation of A and b. Calculating the feature extraction vector proceeds by com- Extrapolation and Prediction puting Having extracted a suitable behavioural model from the char- fðÞ t ¼ A b; ð28Þ acteristic features Y (t) of the multiple SIME representations Sk S , the method proceeds by utilizing the behavioural model where superscript indicates the pseudo-inverse, though any F(t ) for extrapolation to predict the system trajectory, and suitable method of least-squares solution, such as QR decom- c subsequently generate an estimate of the stability margin η position or SVD decomposition, is acceptable. under the relevant fault conditions. A framework that permits The desired system behavioural model for approximating trajectory sensitivity analysis was proposed in , but the the time-domain evolution of the characteristic signals Y (t) Sk method proposed in this paper uses parallel simulations of is a second-order state space model, thus being composed of SIME models and feature extraction to speed up the transient four elements in a two-by-two matrix. This behavioural mod- analysis. el, F(t ), contains elements which describe the change in one The first stage in the extrapolation procedure is identical to state variable per unit time with respect to a given state vari- a stage that is also present in the ESIME method , where the able. In this instance, the four elements of the model are goal is to determine the critical SIME model S*. At each time δ δ vv sv step of the analysis, an estimate of the composition of the FðÞ t ¼ ð29Þ δ δ vs ss critical machine is determined through observation of the rotor angle trajectories of the system generators. Specifically, a where the subscript υ indicates the state variable correspond- Taylor Series extrapolation of the generator rotor angles is ing to the current value of the signal Y (t), and the subscript s Sk 15 Page 6 of 10 Technol Econ Smart Grids Sustain Energy (2019) 4:15 Fig. 4 Extrapolated curve Y ðÞ t for fault number F15 calculated, and the system generators are classified into two system stability. While a set of SIME representations S were sets: critical and non-critical, according to the maximum sep- utilized to extract the behavioural model F(t ), the critical aration between the extrapolated rotor angles. SIME model S* determined from the generator angular sepa- In this stage, the critical SIME model S* is a separate sys- ration is not necessarily equivalent to one of those in the set S . tem representation which will be extrapolated to estimate With the critical SIME model S* chosen for the current time step t , the behavioural model F(t ) can then be used c c for extrapolation of the Y (t) trajectories. The vector of cur- S* rent feature values is hi vðÞ t ¼ Y ðÞ t Y ðÞ t ; ð31Þ c S* c c S* and for each discrete time step Δt, a forward prediction of the values is calculated as vðÞ t þ Δt −vðÞ t ⋅ðÞ 1 þ FΔt ; ð32Þ c c ^ ^ yielding extrapolated estimates of Y and Y , S* S* ^ ^ vðÞ t þ Δt¼ YðÞ t þ Δt YðÞ t þ Δt: ð33Þ c S* c c S* This forward extrapolation process is continued through repeated incrementation by Δt, ^ ^ vðÞ t þ½ m þ 1 Δt¼ vðÞ t þ mΔt ⋅ðÞ 1 þ Ft ; ð34Þ c c until the estimated value of Y crosses the zero threshold S* going from negative to positive. The time which this crossing is predicted to occur is defined as time t , which also repre- sents the stopping criteria of the extrapolation. From the ex- trapolation, a discrete time-series estimate of the critical tra- jectory Y is formed, starting from the most current received S* measurement time t until the estimated zero-crossing time t , c s ^ ^ ^ ^ YðÞ t ⩽t⩽t ≈ Y ðÞ t ; YðÞ t þ Δt ; YðÞ t þ 2Δt ; …; Y ðÞ t : ð35Þ S* c s S* c S* c S* c S* s This yields a time series of values over which a numerical integration can be performed to estimate the stability margin of the system during the currently analysed fault. An example of the calculated Y (t) for fault number F15 is shown in S* Fig. 5 Flowchart of the proposed stability margin estimation method Technol Econ Smart Grids Sustain Energy (2019) 4:15 Page 7 of 10 15 Fig. 6 Comparison of the calculated estimated stability margins Fig. 4. Due to Y (t) being defined as the inertia-corrected estimate η is thus a time-varying estimate of system stability, S* derivative of the system kinetic energy, then the numerical which is updated at each discrete time step when a new set of integration yields PMU data is collected. Figure 5 is the flowchart of the afore- mentioned steps for calculation of the stability margin esti- t t P ðÞ t ⋅ω ðÞ t t dE s s a;S* S* s k;S* E ≈M ∫ Y ðÞ t dt≈M ∫ dt≈∫ dt; ð36Þ mate η. c;S* S* S* S* tc tc tc M dt S* The proposed approach for obtaining these stability esti- which can thus be seen as the estimation of the amount of mates is thus an enhancement to the established ESIME meth- od , and will be shown in the subsequent section to yield accumulated kinetic energy which will be converted to poten- excellent results, matching the performance of the ESIME tial energy during the system swing. From the equal area cri- terion, it is a well-known constraint for post-fault system sta- technique and enhancing the accuracy of the stability estimate in many cases. bility that the accumulated kinetic energy during a fault con- dition must be sufficiently low, such that the excess kinetic energy can be fully absorbed as potential energy during the post-fault swing. Results Because the SIME representation yields an SMIB struc- ture, application of the equal area criterion is used to estimate, The results of testing the proposed method for post-fault sys- tem stability estimation are presented in this section, verifying at each time step, the margin of stability, the accuracy and validity of the method. The proposed ap- 1 t η ¼ E þ E ¼ M ⋅ω ðÞ t þ M ∫ Y ðÞ t dt; ð37Þ proach is applied to a number of post-fault test cases for both k;S* c;S* S* S* c S* S* the IEEE 9-bus and IEEE 39-bus test networks, and a set of whichisrecomputedateachtimestepby observing thedif- performance metrics for the approach are collected. For com- ference between the remaining excess kinetic energy and the parison, an identical set of performance metrics are collected estimated stable deceleration area. The stability margin Table 1 Fault list for 9-Bus system Fault Number Fault Bus and/or Generator Line Cleared F1 Bus 5 Line 5–7 F2 Bus 7 Line 5–7 F3 Bus 7 Line 7–8 F4 Bus 8 Line 7–8 F5 Bus 8 Line 8–9 F6 Bus 9 Line 8–9 F7 Bus 9 Line 6–9 F8 Bus 6 Line 6–9 F9 Bus 6 Line 4–6 F10 Bus 4 Line 4–6 F11 Bus 4 Line 4–5 F12 Bus 5 Line 4–5 Fig. 7 IEEE 9-bus system 15 Page 8 of 10 Technol Econ Smart Grids Sustain Energy (2019) 4:15 Table 2 Stability estimation for 9-Bus system For both methods, the periodic update of information from the PMUs located at each of the generator buses is assumed to Fault Number Proposed Method ESIME Method happen at a rate of once per cycle. In this case, the newly μ N N μ N N received data permits a discrete update of the post-fault stabil- E R + E R + ity estimate every 16.6 ms. Every time a periodic update of F1 0.85 8 0 0.33 11 0 PMU data is received, a stability estimation algorithm itera- F2 0.69 5 0 1.60 9 11 tion is performed, and the time-varying behaviour of the sta- F3 1.84 3 1 0.55 6 0 bility estimate η is recorded. F4 0.66 6 1 0.10 6 1 Comparison of the performance of the proposed method F5 0.60 3 0 0.25 3 0 against the ESIME method is observed through the computa- F6 0.68 1 0 0.36 1 0 tion of a set of metrics, derived directly from the time varying F7 2.07 2 1 0.63 7 0 stability estimates. The first metric μ is the mean error be- F8 1.28 3 0 0.07 6 0 tween all time-varying samples of η and the final converged F9 1.02 3 0 0.03 4 0 value of the stability margin estimate η . As seen in Fig. 6,both F10 1.84 4 0 0.09 7 0 methods converge towards an identical η as time advances F11 2.43 4 0 0.39 5 0 and more information is received, but the progression for each F12 1.19 4 0 0.36 5 0 method’s η may vary. The second metric N is thenumberofsamples,pri- or to the system going unstable, for which the stability for the same fault set, alternatively applying the ESIME meth- margin estimate η is within a ± 10% range of η.The od for stability estimation. This section compares the results of third metric indicates the number of time samples N the proposed method to the predecessor ESIME method de- for which the method yields a positive estimate for η, scribed in the literature, displaying that the proposed tech- despite that all tested faults are unstable, and should nique yields comparable results, while enhancing prediction yield a negative stability margin estimate. accuracy. Fig. 8 IEEE 39-bus system Technol Econ Smart Grids Sustain Energy (2019) 4:15 Page 9 of 10 15 9-Bus System Fig. 8 was simulated and a list of twenty different faults were created on generators and randomly selected busbars. The proposed method was tested by simulating a number of For a busbar fault, both self-clearing fault and the outage of fault conditions on the IEEE 9-Bus system as shown in Fig. 7, the connecting lines were considered. A detailed list of created then collecting the power, frequency, and angular characteris- fault scenarios has been provided in Table 3. A comparison of tics of the fault event from each generator bus once per cycle, results using proposed performance matrices for all the above yielding a data set identical to that which would be available if mentioned fault scenarios are shown in Table 4. a PMU were deployed at each generator bus. Utilizing this The strengths of the proposed method are clearly observ- discrete-time data set, the proposed method was applied to able for the 39-bus system test case, where for nearly every estimate the stability margin once per cycle with each new one of the twenty fault scenarios studied, the proposed method data update, as well as applying the ESIME method for sta- yields a more accurate estimate of the stability margin in the bility analysis on the same data. post-fault period. Furthermore, it provides an accurate esti- A detailed list of created fault scenarios is provided in mate more rapidly, as evidenced by N , giving a supervisory Table 1.Asseenin Table 2, the results of the proposed method control system more time to react in case some corrective yield comparable results to those obtained via the use of action is required to prevent an out-of-step condition. It can ESIME. For this particular system configuration, the ESIME be up to 100% faster in some cases. On the other hand, the approach tends to have a slightly reduced average error for the falsely estimated points N happen for only two samples in the stability margin estimate, enabled by the nearly sinusoidal F5 scenario whereas this number is as high as 40 samples for post-fault power angle swings in this simplistic network. four different fault scenarios. Because ESIME applies a quadratic approximation, it is par- ticularly well suited for extrapolative predictions in this simple network. The proposed method obtains stability margin esti- Conclusions mates which are nearly as accurate, but also avoids the signif- icant false-positive stability margin estimate during Fault 2. The increasing share of renewable energy sources within the As such, the proposed method need not be seen as a replace- generation portfolio, along with continual increase in demand, ment of the ESIME method, but certainly provides stability has created conditions which may drive transmission corridors estimation information which can enhance system insight. Table 4 Stability estimation for 39-bus system 39-Bus System Fault Number Proposed Method ESIME Method To verify the performance of the proposed method on a more μ N N μ N N E R + E R + realistic and larger network, IEEE 39 bus system as shown in F1 6.20 5 0 12.53 3 1 F2 10.90 15 0 24.81 10 0 Table 3 Fault list for 39-bus system F3 16.63 1 0 17.99 1 0 F4 4.16 14 0 9.74 7 0 Fault Number Faulted Bus and/or Generator Line Cleared F5 0.58 3 2 2.23 3 4 F1 Bus 1 / Gen. 1 – F6 13.21 14 0 10.13 16 0 F2 Bus 2 / Gen. 2 – F7 28.37 1 0 31.37 1 0 F3 Bus 3 / Gen. 3 – F8 14.59 1 0 22.00 1 0 F4 Bus 4 / Gen. 4 – F9 11.22 3 0 28.47 2 0 F5 Bus 5 / Gen. 5 – F10 17.73 13 0 28.12 13 3 F6 Bus 6 / Gen. 6 – F11 2.72 14 0 14.79 11 0 F7 Bus 7 / Gen. 7 – F12 2.36 10 0 10.82 10 0 F8 Bus 8 / Gen. 8 – F13 4.09 19 0 10.71 13 0 F9 Bus 9 / Gen. 9 – F14 16.75 12 0 13.36 19 0 F10 Bus 10 / Gen. 10 – F15 13.71 1 0 14.39 1 0 F11 / F12 Bus 15 Line 15–18 / – F16 12.88 3 0 22.97 3 0 F13 / F14 Bus 21 Line 21–36 / – F17 13.30 16 0 24.09 12 0 F15 / F16 Bus 28 Line 26–28 / – F18 14.25 16 0 22.10 12 0 F17 / F18 Bus 34 Line 34–35 / – F19 35.35 1 0 28.62 1 32 F19 / F20 Bus 37 Line 27–37 / – F20 19.92 19 0 20.82 15 0 15 Page 10 of 10 Technol Econ Smart Grids Sustain Energy (2019) 4:15 6. 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