power and energy system oscillation damping using multi
TRANSCRIPT
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Research Article
Power and energy system oscillation damping using multi‑verse optimization
Ramesh Devarapalli1 · Biplab Bhattacharyya1
Received: 7 March 2020 / Accepted: 9 February 2021 / Published online: 27 February 2021 © The Author(s) 2021 OPEN
AbstractPower system oscillations are the primary threat to the stability of a modern power system which is interconnected and operates near to their transient and steady-state stability limits. Power system stabilizer (PSS) is the traditional controller to damp such oscillations, and flexible AC transmission system (FACTS) devices are advised for the improved damping performance. This paper suggests a technique for controller parameters tuning of PSS and a shunt connected FACTS device to be operated in coordination. A static synchronous compensator (STATCOM) connected in a two-machine system is considered as a test power system for the system studies. A recent meta-heuristic algorithm, Multi-Verse optimizer (MVO) has been suggested and compared with the other state-of-the-art algorithms. Improvement in system damping has been achieved by minimizing the oscillating nature of the system states by framing the objective function as a func-tion of damping ratio and location of poles of the system. The Phillips-Heffron model of the test system has been designed by considering the system dynamics. The coordinated system behavior under the perturbation in system parameters has been observed satisfactory with the tuned controller parameters obtained from the suggested algorithm.
Keywords Power system oscillations damping · Power system stabilizers · STATCOM · Whale optimization algorithm · Gray wolf optimization · Multi-verse optimizer
1 Introduction
In the present day by day growing open-access power sys-tem regime, damping of power network oscillations play-ing a vital role for trustworthy power transfer to the load ends. Power networks are experiencing electromechani-cal oscillations due to inconsistent conditions prevailing in the network. In damping the electromechanical oscil-lations, power system stabilizers (PSS) are conventionally used [1]. The increase in power demand necessities the interconnection of various power systems via transmission lines. Due to the expansion in the power network, pertur-bations in system parameters introduces the oscillations in the whole system [2]. In an extensive power system, weak-ening in stabilizer performance takes place due to system
latency [3]. To balance the load requirements and to run reliably, PSS are operating to their maximum limits and to maintain the whole system in a stable and safe operating mode is becoming a challenging task [4].
In promoting the utilization of renewable energy sources in power generation, rapid growth in solar and wind energy generation has been observed [5]. The new non-conven-tional energy sources which are integrated into the power grid introduce fluctuations into the existing system. The sys-tem uncertainties are modeled using inverse output addi-tive perturbation structure from the study of DFIG effect on low-frequency oscillations to show the performance characteristics of PSS in improving transient stability [6]. The concept of reducing multiple machine control into multiple single machine control has been introduced by
* Ramesh Devarapalli, [email protected]; Biplab Bhattacharyya, [email protected] | 1Department of Electrical Engineering, Indian Institute of Technology (Indian School of Mines), Dhanbad, Jharkhand, India.
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decentralized nonlinear model predictive control approach, and power system oscillation damping has been achieved [7]. An ant-colony based STATCOM has been suggested for the power system stability enhancement in a multi-machine power system [8]. The proposed optimization method has been tested for the different system disturbances and the successful oscillation damping has been achieved. A multi-objective grasshopper algorithm has been proposed for the stability enhanced by the authors [9]. A STACOM has been suggested with higher order sliding mode controller for the stability enhancement [10]. The low-frequency oscillations and forced oscillations mitigation have been done with FACTS devices with energy source [11], by shifting the reso-nating frequency of the system using UPFC [12]. The oscilla-tions injected into the system due to the installation of wind farm and PV plant have been controlled by facts devises and coordinated with PSS using adaptive velocity update relaxa-tion particle swarm optimization (AVURPSO) algorithm and compared with genetic algorithm (GA) and gravitational search algorithm (GSA) [13] and by using energy-storage unit based on the supercapacitor (SC) [14]. For the power system stability improvement, a fuzzy lead‐lag controller for a coordinated structure combining SSSC and PSS has been designed, and the parameters have been tuned by modi-fied whale optimization algorithm (WOA) [15]. The lightning search algorithm has been applied in coordinating IPFC and PSS in the combined two‐area restructured ALFC and AVR system [16].
The applications of different meta-heuristic algorithms in the tuning of PSS parameters have been increasing, namely firefly algorithm [17], backtracking search algorithm (BSA) [18], hybrid particle swarm optimization algorithm [19] to minimize the settling time and overshoot of the low-frequency oscillations to improve the system stability. Frequency deviation and tie-line power deviation of an interconnected power system have been minimized by the biogeography‐based optimization (BBO) in a system with fractional order fuzzy PID control [20]. The applications of GWO [21], WOA [22], PSO and modified PSO [23–25] are get-ting increased in the recent time in the field of power sys-tems. GWO algorithm shows better performance character-istics in small-signal stability of a power system [26], and in transmission line expansion problem [27]. The contribution of WOA also predominant and proven as efficient optimiza-tion technique in optimal reactive power dispatch [28], in performance improvement of photovoltaic power systems [29] and distribution systems [30]. PSO with time-varying acceleration coefficients (PSO-TVAC) has been extensively used in achieving optimal parameters and efficient perfor-mance characteristics in generation schedule [31] and in other recent applications in power systems [32–34].
From the literature presented above, it has been iden-tified that the necessity of different power oscillation
damping devices is more in the present scenario. Also, various FACTS devices have been proposed by various researchers to enhance the performance of a power net-work. Hence, the installation of the newly proposed con-trollers with the existing traditional PSS is much needed without creating any king of incoordination among the controllers. By keeping all these concerns, the need for an appropriate technique to obtain the controller parameters of various devices is essential. The present manuscript includes these objectives and highlighted as follows:
• This paper considers a STATCOM as power system oscil-lation damping device alone with the existing PSS for a power network.
• A two-machine system connected with STACOM has been mathematically modeled by considering all sys-tem dynamics for the purpose of analysis.
• The objective of this paper is to derive an optimal solution for the controller variables in damping power system oscillations and is achieved by installing and coordinating the PSS and STATCOM.
• Different object functions on the basis of system eigen-values have been proposed and examined.
• A recently developed multi-verse optimization (MVO) has been suggested for finding the optimal controller parameters by incorporating superior features over other methods in the literature.
• The system model has been built with lesser number of control parameters and proposed MVO have faster convergence characteristics to achieve the objective.
• The implementation of MVO is not complex and results in consistent optimal solution over several trails. The proposed method includes exploration, exploitation and local search for achieving the best solution.
The organization of the paper is as follows: Briefing the severity of power system oscillations and present approaches in damping those in Sect. 1. Section 2 deals with the detailed system modeling and objective function to be minimized with constraints have been explained. Section 3 explains the proposed optimization algorithm and assessment made by comparing with other popular methods in Sect. 4. Section 5 gives the detailed system performance under different loading conditions followed by the conclusions made in the study. Finally ends with appendixes and references considered in the paper.
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2 Research methodology
2.1 STATCOM mechanism in damping power system oscillations
A STATCOM is a shunt connected FACTS device primarily used to provide reactive power support in a power sys-tem. It is widely employed to improve the voltage stabil-ity of a system. During the abnormalities in the system, the reactive power magnitude and direction varies in the transmission line based on the location of the abnormal-ity. The installation of STATCOM in an appropriate location will provide the necessary reactive power in either direc-tion. The reactive power absorption or delivery at the util-ity bus depends on the direction of the current provided by the voltage source converter (VSC) of the STATCOM. This further depends on the voltage difference between the converter terminal and utility bus. So, it mainly used for dynamic compensation for providing voltage sup-port, transient stability enhancement and to increase damping [35]. The valve switching action of the VSC is controlled by the pulse width modulation technique with
the modulation index (me), and the phase angle (de). The magnitude of current depends on the DC voltage of the VSC, usually a capacitor or energy-storage device (Cdc). Hence, the selection of me and de will define the purpose of STATCOM installation, which can be further performed as the controlling variable of STATCOM.
2.2 Mathematical modeling of test system
The considered test system is depicted as Fig. 1. The main components in each generator side are represented, and dynamics have been considered for the complete system modeling [36].
The magnitude of current and voltage at the generator can be represented as (1), (2).
where δj = ∠(E′qj, Vtj), i: is the generator; j: is the area 1, 2.From Fig. 1, the d, q components of the STATCOM cur-
rent can be expressed as in (3), (4).
(1)IiL = IiLd + jIiLq
(2)Vtj = Vtjd + jVtjq;Vtj = Vtj(sin �j + j cos �j)
(3)IL0d = xe
[E
�
q1
xdee1+
E�
q2
xdee2
]−meVdc sin de
[x2L
xdee1+
x1L
xdee2
]+ xe
[Vt2 cos(�1 − �2)
xdee1+
Vt1 cos(�2 − �1)
xdee2
]
(4)IL0q = meVdc cos de
[x2L
xqee1+
x1L
xqee2
]+ xe
[Vt1 sin(�2 − �1)
xqee2+
Vt2 sin(�1 − �2)
xqee1
]
Fig. 1 Test power system with STATCOM and PSS controllers
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The mechanism of reactive power flow control with me, de at the STATCOM can be realized from Eqs. (3) and (4).
2.3 Power system dynamics
The analysis of the test system can be done by the location of system eigenvalues. For finding the system eigenvalues, the dynamic behavior of the generator–excitation system can be represented by (5)–(8) [37].
where Δ� = (� − �0)∕�0 and system parameters are explained in Nomenclature.
The system state equations with the STATCOM can be represented as (9).
where A is the state matrix, B is the STATCOM control input
matrix. With [Δ �
1
Δ�1 ΔE�
q1ΔEfd1 ΔVdc Δ �2 Δ�2 Δ E
�
q2Δ Efd2
]T
are the state variables and [ΔmeDeltade
]T are the control
variables.
2.4 PSS mechanism in damping power system oscillations
The conventional PSS consists of a compensation block which is comprised of a first-order lead-lag system with gain, and a reset block. It is represented, as shown in Fig. 2. The mechanism of PSS is to provide a lead or lag signal depending on the rotor speed deviations. The signal is pro-vided to the excitation control unit to balance the speed of the rotor based on the rotor speed variation. Hence, in
(5)Δ.
� = �b�
(6)Δ.
� =(−ΔPe − DΔ�)
M
(7)Δ
.
E�
q=
(−ΔEq + ΔEfd)
T�
d0
(8)Δ.
Efd = −1
TAΔEfd −
KA
TAΔVt
(9).
X = Ax + Bu
PSS, the time constants and the gain of the compensa-tion block acts as the controlling parameters to damp the system oscillations. However, in the normal operation, the washout block functions to avoid the compensation effect in the system, which can be achieved by choosing a large value of Tw.
By considering the effect of PSS, Eq. (9) can be modified as in (10).
where A is system state matrix, B is STATCOM control input matrix and BE is the supplementary control matrix.
where A12 = w0; A21 = -K11/M1; A22 = -D1/M1; A23 = -K21/M1; A25 = -Kpd1/M1; A31 = -K41/Td011; A33 = -K31/Td011; A34 = 1/Td011; A35 = -Kqd1/Td011; A41 = -Ka1*K51/Ta1; A43 = -Ka1*K61/Ta1; A44 = -1/Ta1; A45 = -Ka1*Kvd1/Ta1; A47 = Ka1/Ta1; A51 = K71; A53 = K81; A55 = -K9; A58 = K72; A510 = K82; A61 = -K11/M1; A62 = -D1/M1; A63 = -K21/M1; A65 = -Kpd1/M1; A66 = -1/Tw; A71 = -KA11*K11*T11/(M1*T21); A72 = -D1*KA11*T11/(M1*T21); A73 = -KA11*T11*K21/(M1*T21); A75 = -Kpd1*KA11*T11/(M1*T21); A76 = (KA11/T21)*(1-(T11/Tw)); A77 = -1/T21; A89 = w0; A95 = -Kpd2/M2; A98 = -K12/M2; A99 = -D2/M2; A910 = -K22/M2; A105 = -Kqd2/Td012; A108 = -K42/Td012; A1010 = -K32/Td012; A1011 = 1/Td012; A115 = -Ka2*Kvd2/Ta2; A118 = -Ka2*K52/Ta2; A1110 = -Ka2*K62/Ta2; A1111 = -1/Ta2; A1113 = Ka2/Ta2; A125 = -Kpd2/M2; A128 = -K12/M2; A129 = -D2/M2; A1210 = -K22/M2; A1212 = -1/Tw; A135 = -Kpd2*KA12*T12/(M2*T22); A138 = -KA12*K12*T12/(M2*T22); A139 = -D2*KA12*T12/(M2*T22); A1310 = -KA12*T12*K22/(M2*T22); A1312 = (KA12/T22)*(1-(T12/Tw)); A1313 = -1/T22; And zero for the remaining elements.
B21 = -Kpe1/M1; B22 = -Kpde1/M1; B31 = -Kqe1/Td011; B32 = -Kqde1/Td011; B41 = -Ka1*Kve1/Ta1; B42 = -Ka1*Kvde1/Ta1; B51 = KAe; B52 = KAde; B61 = -Kpe1/M1; B62 = -Kpde1/M1; B71 = -KA11*T11*Kpe1/(M1*T21); B72 = -KA11*T11*Kpde1/(M1*T21); B91 = -Kpe2/M2; B92 = -Kpde2/M2; B101 = -Kqe2/Td012; B102 = -Kqde2/Td012; B111 = -Ka2*Kve2/Ta2;
(10).
X = Ax + Bu + BEuE
Ac =
⎡⎢⎢⎣
A11 … A113
⋮ ⋱ ⋮
A131 ⋯ A1313
⎤⎥⎥⎦and
Bc=
[0 0;B21B22;B31B32;B41B42;B51B52;B61B61;
B71B72;0 0;B91B92;B101B102;B111B112;B121B122;B131B132
]
Fig. 2 PSS representation
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B112 = -Ka2*Kvde2/Ta2; B121 = -Kpe2/M2; B122 = -Kpde2/M2; B131 = -KA12*T12*Kpe2/(M2*T22); B132 = -KA12*T12*Kpde2/(M2*T22);
The system constants are further defined in Appendix B.
2.5 Objective function and constraints
The detailed system model with STATCOM has been explained, and the complete model has been presented in the form of state-space representation with state vari-ables and control variables mentioned above. The eigen-values obtained from the designed test system can be represented as in (11).
where Re [λi], Im [λi] are the real and imaginary parts of the ith eigenvalue.
The damping ratio for the given eigenvalue can be cal-culated using (12).
(11)�i = Re[�i]+ j Im
[�i]
where i = 1, 2….s; and s = Number of system states.Any system with eigenvalues located in the LH-side of
s-plane and higher magnitude of damping ratio represents the desired features of a stable system. These two features can be framed as the objective function for the considered research problem as given in (13), (14).
where s = No. of system state variables, limiting values of decrement ratio (σ0) and damping factor (ξ0) are consid-ered as -3 and 0.3, respectively [38]. The common features of (13), and (14) can be achieved with the objective func-tion framed as given in (15).
(12)Damping Ratio (�i) =−Re
[�i]
√(Re
[�i])2 + (Im
[�i])2
(13)J1 =
s∑i=1
(�0 − Re[�i])2
(14)J2 =
s∑i=1
(�0 − �i)2
Fig. 3 Desired location of system eigen values with objective function a J1, b J2, and c J3
Table 1 Limits of control parameters
Time constant Gain STATCOM
PSS1 PSS2 PSS1 PSS2 me de
T11, T21 T12, T22 Kc11 Kc12
Min. limit 0.01 0.1 0 0Max. limit 2 50 1 1
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where the selection of multiplication coefficient (α) is based on the magnitudes of individual portions J1 and J2, as the magnitude of the square of the low damping ratios is comparatively less than the square of the real parts, the α is selected as 1000 [39]. The considered objective func-tions and their operating regions are represented in Fig. 3.
The controller parameters of the PSS and STATCOM can be selected by employing the proposed objective func-tions and meta-heuristic technique within limits as men-tioned in (16) and Table 1.
where T11, T21, T12, T22 are the time constants and Kc11, and Kc12 are the gains of PSS of generator 1 and generator 2, and me and de are the modulation index and phase angle of VSC based STATCOM.
3 Multi‑verse optimization
Multi-verse optimization [40] is a national inspired meta-heuristic algorithm, from the big bang theory, which says that the universe is formed in a highly dense and hot con-dition and there exist multiple universes in the space. The features and methodology of MVO are further explained in the following subsections.
3.1 Features
The multi-verse theory says that there are multiple uni-verses exist and they interact and might collide with each other. The main components involved in the interaction of universes are Black holes, White holes and Wormholes. Black holes are formed when there is a run out of hydrogen or other nuclear fuel to burn and begin to collapse in a giant star. This giant start, which is of around 20 times big-ger in size that of sun, forms a region of space from which nothing can escape, including light. White holes behave in contrast to the black holes that nothing including light can enter inside it. A wormhole is a kind of passageway that
(15)J3 =
{s∑
i=1
(�0 − Re[�i])2 + �
s∑i=1
(�0 − �i)2
}
(16)
T11,min ≤ T11 ≤ T11,max
T21,min ≤ T21 ≤ T21,max
Kc11,min ≤ Kc11 ≤ Kc11,max
T12,min ≤ T12 ≤ T12,max
Kc12,min ≤ Kc12 ≤ Kc12,max
T22,min ≤ T22 ≤ T22,max
me,min ≤ me ≤ me,max
de,min ≤ de ≤ de,max
⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭
connects very distant points in space as if there is no or very less distance between them. Every universe can cause its growth through space by means of its inflation rate [41].
3.2 Methodology
The MVO explore the search space with the concept of white holes and black holes, whereas exploits the search space with the concept of a wormhole. The fitness function value is analogous to the inflation rate, and the following rules are applied for the solutions of MVO:
• The larger fitness function value/inflation rate, then the universe will have a greater probability of possessing white holes and lesser probability of possessing black holes.
• A universe with the greater inflation rate, rise to move objectives through white holes and universe with lower inflation rate rise to accept more objects through black holes.
• Irrespective of inflation rates, the objects approach the best universe through wormholes in all universes.
Where each solution and variables are analogous to the universe and objects, respectively.
3.3 Mathematical interpretation of MVO
The mathematical interpretation of object exchange between the universe via white holes and black holes has been implemented by adopting the Roulette Wheel Mechanism. Based on fitness function value/ inflation rate, the sorted white holes for the best universe have been obtained by roulette wheel theory as,
where a is the universe; m = Number of universes (or solu-tions); n = Number of Parameters (Variables) and
where aij represents the jth parameter of the ith universe, r1 denotes a random number in [0, 1], NI(Ui) denotes a nor-malized inflation rate of the ith universe. And akj is selected by the roulette wheel selection mechanism.
Considering two coefficients wormhole existence prob-ability (WEP) and traveling distance rate (TDR) as,
(17)a =
⎛⎜⎜⎝
a11 … a1n⋮ ⋱ ⋮
am1 ⋯ amn
⎞⎟⎟⎠
(18)aij =
{akj , r1 < NI(Ui)
aij , r1 ≥ NI(Ui)
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where ‘WEP_min’ is the minimum and ‘WEP_max’ is the maximum values of wormhole existence probability, ‘ite’ denotes the present iteration, and ‘max_ite’ denotes the total number of iterations, ‘acc’ defines the accuracy of exploitation during the iterations.
As explained in the methodology, the exchange of objects between two universes will take place through space by means of wormholes. This exchange will take place until the objects move to the best universe. In the MVO algorithm, the formation of the wormhole to exploit the searchability is defined by,
where Xj specifies the jth parameter of best universe formed so far, lbj indicates the lower limit of jth param-eter, ubj is the upper limit of jth parameter, and r2, r3, r4 are arbitrary values in [0, 1].
3.4 MVO in Damping power system oscillations
Step 1 model the power system with the controllers math-ematically, as explained in Sect. 2.
Step 2 define the constants for all the components of the power system as given in Appendix A.
Step 3 define the number of universes (Search Agents) and time (maximum iterations) as given in Appendix A.
Step 4 assign the minimum and maximum limits for the unknown parameters (PSS and STATCOM parameters) as mentioned in Table 1 and create a universe given in (17) and (18).
Step 5 initialize the parameters within the search space randomly and calculate the inflation rate (Objective Func-tions) by using Eq. (13) to (15).
Step 6 initialize minimum and maximum WEP, best uni-verse, best universe inflation rate, as mentioned in Appen-dix A.
Step 7 compute WEP and TDR using (19) and (20), respectively.
Step 8 check whether the generated solutions are within the search limit or not (16) and Table 1.
Step 9 update the position of the universe from best to worst based on the inflation rate (Objective function value) from (21).
(19)WEP = WEP_min+ite ×
(WEP_max−WEP_min
max_ite
)
(20)TDR = 1 −ite1∕acc
max_ite1∕acc
(21)
aij =
⎧⎪⎨⎪⎩
�Xj + TDR × ((ubj − lbj) × r4 + lbj);r3 < 0.5
Xj − TDR × ((ubj − lbj) × r4 + lbj);r3 ≥ 0.5;r2 < WEP
aij ;r2 ≥ WEP
Step 10 check for maximum iteration condition and go to step 5 with the updated universe.
Step 11 optimal values of variables and the best inflation rate gives the solution of MVO.
4 Performance analysis of MVO
For the modeled power system and objective functions considered in Sect. 2, the MVO algorithm has been imple-mented as explained in Sect. 3 within the limits of control parameters. To demonstrate the supremacy of the pro-posed optimization algorithm, the analysis on the con-sidered system under the same operating environment has been performed and compared with the widespread optimization algorithms in the present scenario namely, WOA, GWO and PSO-TVAC. The optimal controller settings for the considered optimization algorithms are tabulated in Fig. 4, and the corresponding convergence characteris-tics for the objective functions with respect to the number of iterations is shown in Table 3. All the optimization algo-rithms that are proposed to damp out the power system oscillations have been analyzed for three different loading conditions and three choices in the selection of objective functions. The minimization of the considered objective functions for the considered algorithms has been pre-sented for a total number of 500 iterations.
From Table 2, it has been observed that for the objec-tive function J1, as explained above, all the optimization algorithms have given satisfactory controller parameters in the prescribed boundary limits, whereas all the pro-posed algorithms gave comparable value for the objective function. And from the convergence characteristics shown in Fig. 4, it has been observed that MVO has given better convergence characteristics in objective function minimi-zation in less number of iteration whereas GWO has shown the next better convergence characteristics followed by WOA and PSO-TVAC.
For the objective function J2, the performances of dif-ferent algorithms can be observed from Table 2 and con-vergence characteristics in Fig. 4. For light load condition, GWO minimized the objective function to the minimum values followed by MVO, WOA and PSO-TVAC optimization algorithms. Whereas for the nominal load condition, all the four algorithms have minimized to the almost similar value of the objective function of which GWO has given the bet-ter result. In case of heavy loading condition, unlike the other loading conditions, a certain difference in the per-formance of proposed algorithms can be observed. The optimization algorithm PSO-TVAC resulted in a relatively higher value of the minimized objective function in com-parison with other optimization methods. However, the convergence characteristics for the given minimized value
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of objective function show that MVO and GWO algorithms have been converged in the lesser number iterations for the minimized value.
For the objective function J3, the performance charac-teristics have been analyzed similarly as above, and it has been inferred that the performance of different algorithms has been variant for the different operating conditions. And in concise, with reference to the minimized value of objective function and number of iterations taken, MVO and GWO have shown better performance characteris-tics over WOA and PSO-TVAC. In all the similar cases that have been analyzed, the minimized value of the objective function has upturned in stabilizing the power networks toward the damping of power oscillations for the consid-ered objective function.
However, the analysis based on the optimization and number of iterations is no longer be sufficient for the selection of the optimal algorithm in designing the con-troller parameters for the complex systems like power systems. So, the study is further carried to comment on the better algorithm in controller selections by consider-ing the behavior of system states under different system operating conditions.
5 Results and analysis
Based on the convergence characteristics and boundary limits, a primary analysis has been prepared for the con-sidered power system in the previous section. Further, the analysis of the system stability has been carried out in this section to comment on the efficiency of the considered optimization algorithms for the corresponding objec-tive function. The preliminary eigenvalue investigation is presented in Table 3 with all the combinations of the optimization algorithm, objective functions and system loading conditions. The study and analysis on the system have been further extended for behavior of system states under perturbations and oscillation damp out charac-teristics under different loading conditions. The concise analysis will lead to the selection of a suitable combina-tion of optimization algorithm and objective function. The basic idea of analyzing the system performance based on location of system eigenvalues is the magnitude of real and imaginary parts of eigenvalues define the oscillating behavior of the corresponding system state. The eigen-values of complete system consist the oscillating and non-oscillating modes. Where the imaginary part of non-oscillating eigenvalues is zero, which in turn has damping
Fig. 4 Convergence characteristics under various case studies
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Tabl
e 2
Con
trol
ler p
aram
eter
s un
der d
iffer
ent l
oadi
ng c
ondi
tions
Obj
. Fun
cCo
ntr.
Var
Ligh
t loa
dN
omin
al lo
adH
eavy
load
WO
AG
WO
PSO
-TVA
CM
VOW
OA
GW
OPS
O-T
VAC
MVO
WO
AG
WO
PSO
-TVA
CM
VO
J1T1
11.
6153
0.11
231.
793
1.07
481.
0649
0.07
7693
0.01
0007
0.01
0.01
0.01
0816
0.01
0.01
T21
0.33
951
0.33
244
0.34
318
0.33
709
0.33
325
0.33
507
0.33
484
0.33
519
0.33
604
0.33
514
0.33
490.
3344
7
K10.
1064
10.
1820
40.
10.
1672
50.
10.
10.
10.
10.
10.
10.
10.
1
T12
0.94
920.
0330
541.
7812
0.58
361
0.01
0.01
2862
0.01
0006
0.01
0.04
9966
0.01
6865
0.01
0.01
T22
0.33
655
0.33
438
0.33
922
0.33
213
0.33
157
0.33
462
0.33
498
0.33
568
0.33
348
0.33
442
0.33
481
0.33
509
K20.
1000
90.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
1
me
11
11
11
11
11
11
de0.
6300
80.
6283
30.
6297
60.
6288
50
0.08
1971
0.08
1595
0.08
143
0.05
2014
0.05
2348
0.05
4076
0.05
5173
Obj
Fun
17,1
01.4
433
17,1
01.6
039
17,1
01.4
059
17,1
01.4
719
17,1
88.0
954
17,1
87.7
266
17,1
87.7
181
17,1
87.7
181
17,2
08.9
166
17,2
08.9
046
17,2
08.9
018
17,2
08.9
018
J2T1
11.
4989
81.
9281
21.
7805
72
1.64
923
1.72
801
1.32
109
1.32
664
1.96
013
1.41
913
1.99
971.
5858
8
T21
0.14
1955
0.05
7571
50.
0803
256
0.01
3006
50.
0118
152
0.01
0015
70.
0128
684
0.01
0746
60.
0530
368
0.01
0052
30.
1761
30.
01
K127
.710
126
.879
828
.581
49.2
072
34.4
963
45.7
098
39.0
587
42.3
737
12.5
8649
.699
8.62
0250
T12
22
21.
9984
41.
9660
41.
9976
52
21.
9612
32
21.
9534
3
T22
0.01
4731
30.
0106
827
0.03
6352
80.
010.
0200
487
0.01
4349
50.
010.
010.
010.
0178
508
0.18
361
0.01
4248
2
K242
.617
450
28.3
858
49.9
884
22.2
277
23.0
611
37.2
191
39.9
469
35.2
862
19.9
973
8.54
6122
.634
8
me
0.69
7237
0.87
7931
0.78
2375
0.91
2667
0.78
5894
0.89
5951
11
0.98
10.
9112
440.
1037
70.
9100
39
de0
0.59
5171
0.32
0186
0.62
9248
0.88
5266
0.08
1493
30.
8739
060.
8739
020.
1352
240.
0021
1332
1.74
48e-
050
Obj
Fun
2451
.194
122
96.6
8924
55.3
241
2312
.331
123
99.7
728
2343
.759
823
73.8
402
2374
.808
624
34.4
392
2373
.108
631
81.2
189
2373
.337
4
J3T1
12
1.99
226
1.99
616
1.94
252
22
21.
9996
0.01
2306
20.
7093
421.
9999
T21
0.35
4838
0.63
1084
0.53
1169
0.61
5527
0.32
303
0.20
802
0.34
586
0.34
346
0.76
391
0.12
949
0.16
8539
0.12
879
K115
.131
220
.004
817
.106
121
.036
85.
8294
4.82
996.
5267
6.62
870.
3829
53.
7016
19.6
737
3.66
92
T12
1.86
242
1.99
706
1.98
926
22
21.
7036
1.94
982
0.69
435
1.98
16
T22
0.33
9922
0.40
8557
0.37
6092
0.38
4432
0.23
584
0.18
249
0.23
173
0.22
369
0.14
066
0.14
394
0.17
4195
0.14
262
K26.
9557
27.
6825
17.
6703
68.
2172
84.
923
4.34
44.
0409
4.39
165.
2735
5.05
4519
.182
85.
0458
me
0.76
4352
10.
9368
51
0.71
676
10.
7163
10.
7163
50.
9514
71
0.23
5767
1
de0.
0134
767
0.55
3048
0.47
1697
0.55
4909
10.
0008
0834
0.99
996
10.
0828
440.
0007
4374
00
Obj
Fun
19,7
76.3
485
19,7
45.0
703
19,7
49.1
738
19,7
45.5
804
19,9
02.0
885
20,0
64.5
007
19,8
97.1
407
19,9
02.3
097
21,1
58.5
541
20,3
87.5
714
20,8
80.6
189
20,3
87.8
824
Vol:.(1234567890)
Research Article SN Applied Sciences (2021) 3:383 | https://doi.org/10.1007/s42452-021-04349-2
Tabl
e 3
Eig
enva
lues
and
cor
resp
ondi
ng d
ampi
ng ra
tios
unde
r diff
eren
t cas
e st
udie
s
WO
AG
WO
PSO
− T
VAC
MVO
Eige
n va
lues
Dam
ping
ratio
Eige
n va
lues
Dam
ping
ratio
Eige
n va
lues
Dam
ping
ratio
Eige
n va
lues
Dam
ping
ratio
Ligh
t loa
dJ1
− 0
.050
0 +
0.00
00i
− 0
.050
0 −
0.00
00i
− 1
.048
6 +
0.00
00i
− 0
.123
9 +
2.24
29i
− 0
.123
9 −
2.24
29i
− 3
.064
8 +
0.00
00i
− 0
.549
6 +
3.02
34i
− 0
.549
6 −
3.02
34i
− 3
.402
4 +
0.00
00i
− 3
.403
5 +
0.00
00i
− 3
.686
7 +
0.00
00i
− 9
5.31
08 +
0.0
000i
− 9
5.35
58 +
0.0
000i
1.00
001.
0000
1.00
000.
0552
0.05
521.
0000
0.17
880.
1788
1.00
001.
0000
1.00
001.
0000
1.00
00
− 0
.050
0 +
0.00
00i
− 0
.050
0 −
0.00
00i
− 1
.093
8 +
0.00
00i
− 0
.101
0 +
2.24
61i
− 0
.101
0 −
2.24
61i
− 2
.938
8 +
0.00
00i
− 2
.951
9 +
0.00
00i
− 0
.532
3 +
3.01
63i
− 0
.532
3 −
3.01
63i
− 3
.836
2 +
0.00
00i
− 3
.952
5 +
0.00
00i
− 9
5.30
78 +
0.0
000i
− 9
5.35
38 +
0.0
000i
1.00
001.
0000
1.00
000.
0449
0.04
491.
0000
1.00
000.
1738
0.17
381.
0000
1.00
001.
0000
1.00
00
− 0
.050
0 +
0.00
00i
− 0
.050
0 −
0.00
00i
− 1
.055
5 +
0.00
00i
− 0
.122
6 +
2.24
00i
− 0
.122
6 −
2.24
00i
− 0
.562
9 +
3.01
85i
− 0
.562
9 −
3.01
85i
− 3
.302
6 +
0.00
25i
− 3
.302
6 −
0.00
25i
− 3
.432
3 +
0.02
48i
− 3
.432
3 −
0.02
48i
− 9
5.31
09 +
0.0
000i
− 9
5.35
76 +
0.0
000i
1.00
001.
0000
1.00
000.
0546
0.05
460.
1833
0.18
331.
0000
1.00
001.
0000
1.00
001.
0000
1.00
00
− 0
.050
0 +
0.00
00i
− 0
.050
0 −
0.00
00i
− 1
.076
0 +
0.00
00i
− 0
.117
8 +
2.23
66i
− 0
.117
8 −
2.23
66i
− 3
.048
9 +
0.00
00i
− 0
.542
1 +
3.01
78i
− 0
.542
1 −
3.01
78i
− 3
.388
5 +
0.00
00i
− 3
.428
0 +
0.00
00i
− 3
.753
1 +
0.00
00i
− 9
5.31
09 +
0.0
000i
− 9
5.35
51 +
0.0
000i
1.00
001.
0000
1.00
000.
0526
0.05
261.
0000
0.17
680.
1768
1.00
001.
0000
1.00
001.
0000
1.00
00
J21.
0e +
02
* −
0.0
005
+ 0.
0000
i −
0.0
005
+ 0.
0000
i −
0.0
088
+ 0.
0000
i −
0.0
033
+ 0.
0098
i −
0.0
033
− 0.
0098
i −
0.0
041
+ 0.
0127
i −
0.0
041
− 0.
0127
i −
0.0
435
+ 0.
1342
i −
0.0
435
− 0.
1342
i −
0.1
593
+ 0.
5076
i −
0.1
593
− 0.
5076
i −
0.9
723
+ 0.
0000
i −
1.3
542
+ 0.
0000
i
1.00
001.
0000
1.00
000.
3224
0.32
240.
3052
0.30
520.
3081
0.30
810.
2995
0.29
951.
0000
1.00
00
1.0e
+ 0
2 *
− 0
.000
4 +
0.00
02i
− 0
.000
4 −
0.00
02i
− 0
.000
5 +
0.00
00i
− 0
.004
8 +
0.00
90i
− 0
.004
8 −
0.00
90i
− 0
.005
1 +
0.00
91i
− 0
.005
1 −
0.00
91i
− 0
.076
0 +
0.22
50i
− 0
.076
0 −
0.22
50i
− 0
.185
2 +
0.59
90 −
0.1
852
− 0.
5990
i −
1.0
139
+ 0.
0000
i −
1.5
601
+ 0.
0000
i
0.89
030.
8903
1.00
000.
4704
0.47
040.
4881
0.48
810.
3201
0.32
010.
2954
0.29
541.
0000
1.00
00
1.0e
+ 0
2 *
− 0
.000
5 +
0.00
00i
− 0
.000
5 −
0.00
00i
− 0
.007
4 +
0.00
00i
− 0
.003
7 +
0.01
03i
− 0
.003
7 −
0.01
03i
− 0
.003
7 +
0.01
03i
− 0
.003
7 −
0.01
03i
− 0
.060
4 +
0.19
21i
− 0
.060
4 −
0.19
21i
− 0
.098
7 +
0.30
11i
− 0
.098
7 −
0.30
11i
− 0
.995
2 +
0.00
00i
− 1
.070
6 +
0.00
00i
1.00
001.
0000
1.00
000.
3365
0.33
460.
3346
0.30
010.
3001
0.31
150.
3115
1.00
001.
0000
1.0e
+ 0
2 *
− 0
.000
4 +
0.00
02i
− 0
.000
4 −
0.00
02i
− 0
.000
5 +
0.00
00i
− 0
.003
2 +
0.00
75i
− 0
.003
2 −
0.00
75i
− 0
.005
9 +
0.00
75i
− 0
.005
9 −
0.00
75i
− 0
.171
7 +
0.53
60i
− 0
.171
7 −
0.53
60i
− 0
.195
4 +
0.60
82i
− 0
.195
4 −
0.60
82i
− 1
.419
5 +
0.00
00i
− 1
.603
8 +
0.00
00i
0.86
000.
8600
1.00
000.
3904
0.39
040.
6151
0.61
510.
3050
0.30
500.
3058
0.30
581.
0000
1.00
00
J3 −
0.0
501
+ 0.
0000
i −
0.0
504
+ 0.
0000
i −
1.2
935
+ 0.
0000
i −
0.6
587
+ 1.
3899
i −
0.6
587
− 1.
3899
i −
0.7
205
+ 1.
8138
i −
0.7
205
− 1.
8138
i −
2.5
626
+ 5.
4127
i −
2.5
626
− 5.
4127
i −
2.8
694
+ 7.
3899
i −
2.8
694
− 7.
3899
i −
95.
6239
+ 0
.000
0i −
95.
8559
+ 0
.000
0i
1.00
001.
0000
1.00
000.
4282
0.42
820.
3692
0.36
920.
4279
0.42
790.
3620
0.36
201.
0000
1.00
00
− 0
.050
3 +
0.00
00i
− 0
.050
9 +
0.00
00i
− 0
.418
5 +
1.06
16i
− 0
.418
5 −
1.06
16i
− 1
.180
8 +
0.00
00i
− 0
.437
1 +
1.12
38i
− 0
.437
1 −
1.12
38i
− 2
.600
4 +
5.45
52i
− 2
.600
4 −
5.45
52i
− 2
.650
0 +
5.99
59i
− 2
.650
0 −
5.99
59i
− 9
5.62
36 +
0.0
000i
− 9
5.71
34 +
0.0
000i
1.00
001.
0000
0.36
680.
3668
1.00
000.
3625
0.36
250.
4303
0.43
030.
4042
0.40
421.
0000
1.00
00
− 0
.050
2 +
0.00
00i
− 0
.050
7 +
0.00
00i
− 1
.219
9 +
0.00
00i
− 0
.469
9 +
1.16
27i
− 0
.469
9 −
1.16
27i
− 0
.486
5 +
1.22
65i
− 0
.486
5 −
1.22
65i
− 2
.652
6 +
5.66
51i
− 2
.652
6 −
5.66
51i
− 2
.711
0 +
6.16
62i
− 2
.711
0 −
6.16
62i
− 9
5.64
56 +
0.0
000i
− 9
5.72
09 +
0.0
000i
1.00
001.
0000
1.00
000.
3747
0.37
470.
3687
0.36
870.
4240
0.42
400.
4025
0.40
251.
0000
1.00
00
− 0
.050
3 +
0.00
00i
− 0
.050
9 +
0.00
00i
− 0
.410
6 +
1.06
05i
− 0
.410
6 −
1.06
05i
− 0
.422
3 +
1.09
30i
− 0
.422
3 −
1.09
30i
− 1
.182
1 +
0.00
00i
− 2
.694
6 +
5.82
01i
− 2
.694
6 −
5.82
01i
− 2
.646
6 +
6.12
03i
− 2
.646
6 −
6.12
03i
− 9
5.65
92 +
0.0
000i
− 9
5.73
41 +
0.0
000i
1.00
001.
0000
0.36
100.
3610
0.36
040.
3604
1.00
000.
4201
0.42
010.
3969
0.39
691.
0000
1.00
00
Vol.:(0123456789)
SN Applied Sciences (2021) 3:383 | https://doi.org/10.1007/s42452-021-04349-2 Research Article
Tabl
e 3
(con
tinue
d)
WO
AG
WO
PSO
− T
VAC
MVO
Eige
n va
lues
Dam
ping
ratio
Eige
n va
lues
Dam
ping
ratio
Eige
n va
lues
Dam
ping
ratio
Eige
n va
lues
Dam
ping
ratio
Nor
mal
load
J1 −
0.0
500
+ 0.
0000
i −
0.0
500
+ 0.
0000
i −
1.0
952
+ 0.
0000
i −
2.9
471
+ 0.
0000
i −
3.2
981
+ 0.
0000
i −
3.3
125
+ 0.
0000
i −
3.5
328
+ 0.
0000
i −
0.5
460
+ 4.
5980
i −
0.5
460
− 4.
5980
i −
0.1
183
+ 4.
6549
i −
0.1
183
− 4.
6549
i −
95.
5201
+ 0
.000
0i −
95.
6173
+ 0
.000
0i
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
1179
0.11
790.
0254
0.02
541.
0000
1.00
00
− 0
.050
0 +
0.00
00i
− 0
.050
0 −
0.00
00i
− 0
.756
1 +
0.00
00i
− 2
.927
1 +
0.00
00i
− 2
.944
1 +
0.00
00i
− 3
.527
7 +
0.00
00i
− 3
.715
6 +
0.00
00i
− 0
.328
8 +
4.58
85i
− 0
.328
8 −
4.58
85i
− 0
.478
0 +
4.66
99i
− 0
.478
0 −
4.66
99i
− 9
5.51
70 +
0.0
000i
− 9
5.61
80 +
0.0
000i
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
0715
0.07
150.
1018
0.10
181.
0000
1.00
00
− 0
.050
0 +
0.00
00i
− 0
.050
0 −
0.00
00i
− 0
.757
8 +
0.00
00i
− 2
.923
5 +
0.00
00i
− 2
.936
5 +
0.00
00i
− 3
.527
9 +
0.00
00i
− 3
.729
1 +
0.00
00i
− 0
.325
9 +
4.58
73i
− 0
.325
9 −
4.58
73i
− 0
.478
5 +
4.67
04i
− 0
.478
5 −
4.67
04i
− 9
5.51
67 +
0.0
000i
− 9
5.61
80 +
0.0
000i
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
0709
0.07
090.
1019
0.10
191.
0000
1.00
00
− 0
.050
0 +
0.00
00i
− 0
.050
0 −
0.00
00i
− 0
.758
5 +
0.00
00i
− 2
.918
1 +
0.00
00i
− 2
.933
5 +
0.00
00i
− 3
.527
1 +
0.00
00i
− 3
.728
8 +
0.00
00i
− 0
.325
4 +
4.58
75i
− 0
.325
4 −
4.58
75i
− 0
.478
6 +
4.67
00i
− 0
.478
6 −
4.67
00i
− 9
5.51
67 +
0.0
000i
− 9
5.61
80 +
0.0
000i
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
0708
0.07
080.
1020
0.10
201.
0000
1.00
00
J21.
0e +
02
* −
0.0
005
+ 0.
0001
i −
0.0
005
− 0.
0001
i −
0.0
005
+ 0.
0000
i −
0.0
051
+ 0.
0116
i −
0.0
051
− 0.
0116
i −
0.0
044
+ 0.
0141
i −
0.0
044
− 0.
0141
i −
0.1
323
+ 0.
4367
i −
0.1
323
− 0.
4367
i −
0.1
738
+ 0.
5753
i −
0.1
738
− 0.
5753
i −
1.2
276
+ 0.
0000
i −
1.4
928
+ 0.
0000
i
0.97
420.
9742
1.00
000.
4005
0.40
050.
2983
0.29
830.
2900
0.29
000.
2891
0.28
911.
0000
1.00
00
1.0e
+ 0
2 *
− 0
.000
4 +
0.00
02i
− 0
.000
4 −
0.00
02i
− 0
.000
5 +
0.00
00i
− 0
.005
1 +
0.01
01i
− 0
.005
1 −
0.01
01i
− 0
.006
8 +
0.01
11i
− 0
.006
8 −
0.01
11i
− 0
.166
2 +
0.49
76i
− 0
.166
2 −
0.49
76i
− 0
.152
4 +
0.70
04i
− 0
.152
4 −
0.70
04i
− 1
.355
1 +
0.00
00i
− 1
.685
5 +
0.00
00i
0.89
930.
8993
1.00
000.
4462
0.44
620.
5200
0.52
000.
3167
0.31
670.
2127
0.21
271.
0000
1.00
00
1.0e
+ 0
2 *
− 0
.000
4 +
0.00
02i
− 0
.000
4 −
0.00
02i
− 0
.000
5 +
0.00
00i
− 0
.005
5 +
0.01
00i
− 0
.005
5 −
0.01
00i
− 0
.005
3 +
0.01
17i
− 0
.005
3 −
0.01
17i
− 0
.171
9 +
0.53
55i
− 0
.171
9 −
0.53
55i
− 0
.158
0 +
0.69
12i
− 0
.158
0 −
0.69
12i
− 1
.423
9 +
0.00
00i
− 1
.678
2 +
0.00
00i
0.93
170.
9317
1.00
000.
4841
0.48
410.
4139
0.41
390.
3057
0.30
570.
2228
0.22
281.
0000
1.00
00
1.0e
+ 0
2 *
− 0
.000
4 +
0.00
02i
− 0
.000
4 −
0.00
02i
− 0
.000
5 +
0.00
00i
− 0
.005
5 +
0.00
97i
− 0
.005
5 −
0.00
97i
− 0
.005
1 +
0.01
12i
− 0
.005
1 −
0.01
12i
− 0
.187
8 +
0.58
54i
− 0
.187
8 −
0.58
54i
− 0
.147
8 +
0.71
38i
− 0
.147
8 −
0.71
38i
− 1
.546
0 +
0.00
00i
− 1
.698
9 +
0.00
00i
0.92
650.
9265
1.00
000.
4933
0.49
330.
4142
0.41
420.
3055
0.30
550.
2027
0.20
271.
0000
1.00
00
J3 −
0.0
501
+ 0.
0000
i −
0.0
502
+ 0.
0000
i −
0.2
141
+ 0.
0000
i −
0.7
818
+ 1.
9588
i −
0.7
818
− 1.
9588
i −
1.1
581
+ 2.
4264
i −
1.1
581
− 2.
4264
i −
2.8
975
+ 6.
0227
i −
2.8
975
− 6.
0227
i −
3.0
146
+ 7.
1023
i −
3.0
146
− 7.
1023
i −
95.
9489
+ 0
.000
0i −
96.
1369
+ 0
.000
0i
1.00
001.
0000
1.00
000.
3707
0.37
070.
4307
0.43
070.
4335
0.43
350.
3907
0.39
071.
0000
1.00
00
− 0
.050
1 +
0.00
00i
− 0
.050
2 +
0.00
00i
− 1
.059
4 +
0.00
00i
− 1
.192
0 +
1.91
98i
− 1
.192
0 −
1.91
98i
− 1
.198
2 +
1.94
36i
− 1
.198
2 −
1.94
36i
− 2
.965
1 +
7.99
98i
− 2
.965
1 −
7.99
98i
− 3
.415
2 +
8.29
29i
− 3
.415
2 −
8.29
29i
− 9
6.09
36 +
0.0
000i
− 9
6.22
77 +
0.0
000i
1.00
001.
0000
1.00
000.
5275
0.52
750.
5248
0.52
480.
3475
0.34
750.
3808
0.38
081.
0000
1.00
00
− 0
.050
1 +
0.00
00i
− 0
.050
2 +
0.00
00i
− 0
.213
3 −
0.00
00i
− 0
.748
1 +
1.82
59i
− 0
.748
1 −
1.82
59i
− 1
.280
2 +
2.67
86i
− 1
.280
2 −
2.67
86i
− 2
.819
8 −
6.14
50i
− 2
.819
8 +
6.14
50i
− 2
.973
4 −
6.37
72i
− 2
.973
4 +
6.37
72i
− 9
5.96
79 −
0.0
000i
− 9
6.05
05 −
0.0
000i
1.00
001.
0000
1.00
000.
3791
0.37
910.
4312
0.43
120.
4171
0.41
710.
4226
0.42
261.
0000
1.00
00
− 0
.050
1 +
0.00
00i
− 0
.050
2 +
0.00
00i
− 0
.213
5 −
0.00
00i
− 0
.742
5 +
1.80
45i
− 0
.734
8 −
1.81
92i
− 1
.347
2 +
2.79
55i
− 1
.347
1 −
2.79
60i
− 2
.953
2 +
6.14
49i
− 2
.952
8 −
6.14
98i
− 2
.866
9 −
6.23
11i
− 2
.880
8 +
6.25
12i
− 9
5.97
83 −
0.0
000i
− 9
6.03
28 −
0.0
000i
1.00
001.
0000
1.00
000.
3805
0.37
450.
4341
0.43
400.
4332
0.43
280.
4180
0.41
851.
0000
1.00
00
Vol:.(1234567890)
Research Article SN Applied Sciences (2021) 3:383 | https://doi.org/10.1007/s42452-021-04349-2
Tabl
e 3
(con
tinue
d)
WO
AG
WO
PSO
− T
VAC
MVO
Eige
n va
lues
Dam
ping
ratio
Eige
n va
lues
Dam
ping
ratio
Eige
n va
lues
Dam
ping
ratio
Eige
n va
lues
Dam
ping
ratio
Hea
vy lo
adJ1
− 0
.050
0 +
0.00
00i
− 0
.050
0 +
0.00
00i
− 0
.427
5 +
0.00
00i
− 2
.954
9 +
0.00
00i
− 2
.980
4 +
0.00
00i
− 3
.868
1 +
0.00
00i
− 4
.019
2 +
0.00
00i
− 0
.293
1 +
6.02
02i
− 0
.293
1 −
6.02
02i
− 0
.279
1 +
6.40
83i
− 0
.279
1 −
6.40
83i
− 9
5.57
32 +
0.0
000i
− 9
5.64
90 +
0.0
000i
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
0486
0.04
860.
0435
0.04
351.
0000
1.00
00
− 0
.050
0 +
0.00
00i
− 0
.050
0 +
0.00
00i
− 0
.426
5 +
0.00
00i
− 2
.961
4 +
0.00
00i
− 2
.971
4 +
0.00
00i
− 3
.871
1 +
0.00
00i
− 4
.019
5 +
0.00
00i
− 0
.293
4 +
6.02
01i
− 0
.293
4 −
6.02
01i
− 0
.278
9 +
6.40
81i
− 0
.278
9 −
6.40
81i
− 9
5.57
32 +
0.0
000i
− 9
5.64
89 +
0.0
000i
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
0487
0.04
870.
0435
0.04
351.
0000
1.00
00
− 0
.050
0 +
0.00
00i
− 0
.050
0 +
0.00
00i
− 0
.421
0 +
0.00
00i
− 2
.961
6 +
0.00
00i
− 2
.969
5 +
0.00
00i
− 3
.871
4 +
0.00
00i
− 4
.019
5 +
0.00
00i
− 0
.294
5 +
6.01
95i
− 0
.294
5 −
6.01
95i
− 0
.280
6 +
6.40
90i
− 0
.280
6 −
6.40
90i
− 9
5.57
32 +
0.0
000i
− 9
5.64
89 +
0.0
000i
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
0489
0.04
890.
0437
0.04
371.
0000
1.00
00
− 0
.050
0 +
0.00
00i
− 0
.050
0 +
0.00
00i
− 0
.417
5 +
0.00
00i
− 2
.961
4 +
0.00
00i
− 2
.971
1 +
0.00
00i
− 3
.871
2 +
0.00
00i
− 4
.019
6 +
0.00
00i
− 0
.295
2 +
6.01
91i
− 0
.295
2 −
6.01
91i
− 0
.281
8 +
6.40
97i
− 0
.281
8 −
6.40
97i
− 9
5.57
32 +
0.0
000i
− 9
5.64
89 +
0.0
000i
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
0490
0.04
900.
0439
0.04
391.
0000
1.00
00
J21.
0e +
02
* −
0.0
004
+ 0.
0002
i −
0.0
004
− 0.
0002
i −
0.0
005
+ 0.
0000
i −
0.0
072
+ 0.
0161
i −
0.0
072
− 0.
0161
i −
0.0
111
+ 0.
0163
i −
0.0
111
− 0.
0163
i −
0.0
689
+ 0.
2512
i −
0.0
689
− 0.
2512
i −
0.1
605
+ 0.
5560
i −
0.1
605
− 0.
5560
i −
1.0
288
+ 0.
0000
i −
1.4
409
+ 0.
0000
i
0.93
740.
9374
1.00
000.
4090
0.40
900.
5624
0.56
240.
2646
0.26
460.
2774
0.27
741.
0000
1.00
00
1.0e
+ 0
2 *
− 0
.000
4 +
0.00
02i
− 0
.000
4 −
0.00
02i
− 0
.000
5 +
0.00
00i
− 0
.008
0 +
0.01
38i
− 0
.008
0 −
0.01
38i
− 0
.008
9 +
0.01
48i
− 0
.008
9 −
0.01
48i
− 0
.146
0 +
0.44
21i
− 0
.146
0 −
0.44
21i
− 0
.153
7 +
0.69
12i
− 0
.153
7 −
0.69
12i
− 1
.253
1 +
0.00
00i
− 1
.675
0 +
0.00
00i
0.89
890.
8989
1.00
000.
5002
0.50
020.
5140
0.51
400.
3135
0.31
350.
2170
0.21
701.
0000
1.00
00
0.01
40 −
0.0
251i
− 0
.050
0 +
0.00
00i
− 0
.050
0 +
0.00
01i
− 1
.762
7 +
1.44
15i
− 1
.827
9 +
1.46
79i
− 0
.799
5 −
6.31
57i
− 0
.876
5 −
6.82
94i
− 0
.889
6 −
6.90
35i
− 0
.893
2 −
7.15
44i
− 5
.408
1 +
11.9
227i
− 5
.580
2 +
12.3
899i
− 9
6.85
65 +
0.0
025i
− 9
6.87
88 +
0.0
035i
− 0
.487
71.
0000
1.00
000.
7741
0.77
970.
1256
0.12
730.
1278
0.12
390.
4131
0.41
071.
0000
1.00
00
1.0e
+ 0
2 *
− 0
.000
4 +
0.00
02i
− 0
.000
4 −
0.00
02i
− 0
.000
5 +
0.00
00i
− 0
.007
1 +
0.01
31i
− 0
.007
1 −
0.01
31i
− 0
.008
8 +
0.01
42i
− 0
.008
8 −
0.01
42i
− 0
.165
2 +
0.49
77i
− 0
.165
2 −
0.49
77i
− 0
.137
7 +
0.73
00i
− 0
.137
7 −
0.73
00i
− 1
.357
0 +
0.00
00i
− 1
.713
4 +
0.00
00i
0.89
620.
8962
1.00
000.
4774
0.47
740.
5267
0.52
670.
3151
0.31
510.
1853
0.18
531.
0000
1.00
00
J3 −
0.0
492
+ 0.
0068
i −
0.0
492
− 0.
0068
i −
0.0
500
+ 0.
0000
i −
1.3
048
+ 0.
0000
i −
2.0
356
+ 2.
3587
i −
2.0
356
− 2.
3587
i −
4.1
040
+ 0.
0000
i −
0.3
207
+ 6.
1223
i −
0.3
207
− 6.
1223
i −
3.3
479
+ 10
.657
3i −
3.3
479
− 10
.657
3i −
95.
5727
+ 0
.000
0i −
96.
6257
+ 0
.000
0i
0.99
060.
9906
1.00
001.
0000
0.65
340.
6534
1.00
000.
0523
0.05
230.
2997
0.29
971.
0000
1.00
00
− 0
.050
1 +
0.00
00i
− 0
.050
2 +
0.00
00i
− 0
.587
7 +
0.00
00i
− 1
.907
9 +
2.46
15i
− 1
.907
9 −
2.46
15i
− 2
.072
1 +
2.49
40i
− 2
.072
1 −
2.49
40i
− 3
.631
2 +
9.29
56i
− 3
.631
2 −
9.29
56i
− 3
.270
5 +
10.3
508i
− 3
.270
5 −
10.3
508i
− 9
6.36
66 +
0.0
000i
− 9
6.58
65 +
0.0
000i
1.00
001.
0000
1.00
000.
6126
0.61
260.
6391
0.63
910.
3639
0.36
390.
3013
0.30
131.
0000
1.00
00
− 0
.050
0 −
0.00
00i
− 0
.050
2 +
0.00
02i
0.09
04 −
0.1
394i
− 2
.009
5 +
1.42
11i
− 2
.110
1 +
1.42
46i
− 1
.015
8 −
5.22
87i
− 1
.339
9 −
6.28
23i
− 1
.314
1 −
6.33
86i
− 1
.291
3 −
6.45
75i
− 4
.881
2 +
10.4
753i
− 5
.126
5 +
11.1
228i
− 9
6.63
69 +
0.0
008i
− 9
6.67
40 +
0.0
017i
1.00
001.
0000
− 0
.544
30.
8165
0.82
880.
1907
0.20
860.
2030
0.19
610.
4224
0.41
861.
0000
1.00
00
− 0
.050
1 +
0.00
00i
− 0
.050
2 +
0.00
00i
− 0
.589
9 +
0.00
00i
− 1
.918
7 +
2.47
59i
− 1
.918
7 −
2.47
59i
− 2
.084
0 +
2.50
55i
− 2
.084
0 −
2.50
55i
− 3
.642
1 +
9.26
96i
− 3
.642
1 −
9.26
96i
− 3
.290
9 +
10.3
297i
− 3
.290
9 −
10.3
297i
− 9
6.36
44 +
0.0
000i
− 9
6.58
55 +
0.0
000i
1.00
001.
0000
1.00
000.
6126
0.61
260.
6395
0.63
950.
3657
0.36
570.
3036
0.30
361.
0000
1.00
00
Vol.:(0123456789)
SN Applied Sciences (2021) 3:383 | https://doi.org/10.1007/s42452-021-04349-2 Research Article
ratio as 1. The oscillating eigenvalues can be classified as positively damped or negatively damped poles based on the location of eigenvalue either in RH-side or in LH-side to the imaginary axis. The system with negative damping ratio eigenvalues is highly unstable for any perturbations in the system operating conditions. However, the weekly positive damped eigenvalues also to be taken care for the effectual system operation. Hence, the proposed objec-tive functions will contribute toward the improvement of system performance, which can be analyzed as below.
In Table 3, the system eigenvalues comparison of the proposed optimization algorithms has been presented for different loading conditions. As explained in Sect. 2, the functioning of objective function J1 is to drift all the critical or oscillatory mode of eigenvalues toward the prescribed value of real part (σ = −3), and with objective function J2, the system eigenvalues should have prescribed value of damping ratio (ξ = 0.3). The pooled performance charac-teristics of functions J1 and J2 should be resulted by the multi-objective function J3. From Table 3, the proposed algorithms in the majority of system operating conditions resulted in locating the eigenvalues in the left half of the s-plane with positive damping ratio, which represents the stable system operating conditions. But in case of heavy loading condition, the optimization algorithm PSO-TVAC with objective function J2 and multi-objective function J3 has unable to locate the eigenvalues with satisfactory
damping ratio, and in result, the eigenvalue for a par-ticular system state has been situated in the right half of the s-plane with negative damping ratio. In the previous section, the similar performance for the stated algorithm and loading conditions has been observed with relatively high magnitude in the minimized value of the objective function in contrast with the other optimization methods. However, for the modeled power network with challeng-ing system constants, the proposed optimization algo-rithms are not completely succeeded to place the eigen-values with the deserved characteristics as estimated as considered objective functions. Still, the eigenvalue anal-ysis shown in Table 3 expresses the stable system mode of operation for the different loading conditions under a steady-state with the proposed optimization algorithms in tuning the system control parameters. As the practical power network is prone to handle the complex operat-ing conditions in critical cases, the system study under the adverse operating conditions will recommend the better optimization method with a suitable objective function.
Figures 5, 6, 7 and 8 gives a rigorous analysis of the behavior of system states by considering perturbation in it. The system analysis under perturbation will give a summary on the considered objective function impact in stabilizing the system under various system loading condi-tions. For the purpose of analysis and design, the study has been made on system states with 10% perturbations and
Fig. 5 Damping nature of angular speed deviation under different loading conditions
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Fig. 6 Damping nature of torque angle deviation under different loading conditions
Fig. 7 Damping nature of generators internal voltage deviation under different loading conditions
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presented as, Fig. 5 illustrates the variation in the angular velocities of both machines, Fig. 6 shows the variation in delta angle of both the areas, Fig. 7 shows the variation in internal voltage of both alternators and Fig. 8 shows the variation in the field excitation voltage of both machines. All the above characteristics are presented as a compari-son of optimization algorithms under different system loading conditions.
The functioning of PSS is to control the field excitation in response to the variations in the rotor angular velocity corresponding to the load variations. The tuned param-eters with the proposed optimization algorithms should preserve the stable system operating environment for the predictable abnormalities/fluctuations. From Figs. 5, 6, 7 and 8, the response of system states under perturba-tion can be observed and it has been observed that the response corresponding to the objective function J1 is leading to the more oscillations but its characteristics are analogous to the underdamped system and settling to the zero-deviation value after a time period depending on the algorithm employed and loading condition. Whereas cor-responding to the objective function J2, the system states are taking comparatively more time to get the zero varia-tions in its states for some operating conditions. But, the multi-objective function leading to the much more satis-factory responses by resulting in less settling time and less number of oscillations, which leads to the higher possible
stability level of the system in abnormal system operating conditions. Though the system analysis is satisfactory in all these cases with respect to the eigenvalue analysis, system study during abnormalities will suggest multi-objective function rather than the single objective function that have been explained in this paper. Based on the all the analysis made on the system for the proposed optimiza-tion methods (Tables 2 and 3, Figs. 4, 5, 6, 7 and 8), MVO and GWO have been showing better performance charac-teristics, and in detail, MVO with multi-objective function leads to the efficient performance characteristics under different loading conditions.
6 Conclusion and future scope
In this paper, a rigorous stability analysis has been pre-sented on a sample power system model connected with STATCOM. The coordination among the auxiliary device STATCOM and supplementary controller PSS has been achieved by implementing four meta-heuristic algorithms namely WOA, GWO, MVO and PSO-TVAC on the modeled system and the same has been analyzed by consider-ing different objective functions for the optimal system operating conditions. For choosing the best combination of objective function and algorithm, a detailed analysis has been performed on the sample system based on the
Fig. 8 Damping nature of field voltages deviation under different loading conditions
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minimized value of objective function achieved, system eigenvalues for the corresponding combination. The robustness of the system controller parameters has been determined by conducting the whole analysis under dif-ferent loading conditions. The assessments have been derived from the analysis made by observing the system parameter stabilization under perturbation in its states, and the suitable algorithm has been suggested. Based on the rigorous analysis made on the considered sample system model, MVO with the multi-objective function has been suggested for the robust system operating condition.
The inferences derived from the present work highly suggests the application of MVO in the stability enhance-ment of a multi-machine power system. The system analysis can also be performed under the contingency conditions that suits the practical implementation of the suggested technique.
Compliance with ethical standards
Conflict of interest The authors declare that they have no conflict of interest.
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Appendix A
Power system parameters
M1 = M2 = 6.0; D1 = D2 = 0;Vt1 = 1.0; Vt2 = 0.89;Ka1 = Ka2 = 50; Ta1 = Ta2 = 0.01; Td011 = Td012 = 6.3;xe = 0.15; x1L = 0.3; x2L = 0.3;
Load parameters
Light Load: Pe1 = Pe2 = 0.3; Qe1 = Qe2 = 0.1;Nominal Load: Pe1 = Pe2 = 0.8; Qe1 = Qe2 = 0.6;Heavy Load: Pe1 = Pe2 = 1.3; Qe1 = Qe2 = 1.0;
Software specifications
The study has been conducted in Matlab 2016A platform with the computer having specifications of windows 10 operating system, 6-bit, Intel core-i7 8th gen. processor 16 GB DDR4 RAM.
Optimization algorithm parameters
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Appendix B
System constants
K11
=(Vt1d−x
�
d1I1Lq)(xde1−xdt1)Vt2 sin(�1−�2)
xdee1+
(Vt1q+xq1 I1Ld )(xqt1−xqe1)Vt2 cos(�1−�2)
xqee1 ;
K21 = (I1Lq +Vt1d (x2L+xe)
xdee1)
K31 = 1 +(x
�
d1−xd1)(x2L+xe)
xdee1;
K41 =−(x
�
d1−xd1)(xde1−xdt1)Vt2 sin(�1−�2)
xdee1
Kpe1 =(Vt1d−x
�
d1I1Lq)(xbd1−xde1)Vdc sin de
2xdee1+
(Vt1q+xq1 I1Ld )(xbq1−xqe1)Vdc cos de
2xqee1
Kpde1 =(Vt1d−x
�
d1I1Lq)(xbd1−xde1)meVdc cos de
2xdee1+
(Vt1q+xq1 I1Ld )(−xbq1+xqe1)meVdc sin de
2xqee1
Kpd1 =(Vt1d−x
�
d1I1Lq)(xbd1−xde1)me sin de
2xdee1+
(Vt1q+xq1 I1Ld )(xbq1−xqe1)me cos de
2xqee1
K51
=(Vt1d∕Vt1)xq1(xqt1−xqe1)Vt2 cos(�1−�2)
xqee1−
(Vt1q∕Vt1)x�
d1(xde1−xdt1)Vt2 cos(�1−�2)
xdee1 ;
K61 =(Vt1q∕Vt1)(xdee1+x
�
d1(x2L−xe))
xdee1
K71 = (3
4Cdc){
meVt2 sin(�1−�2)(cos de)xde1
xdee1−
meVt2 cos(�1−�2)(sin de)xqe1
xqee1
};
K81 = (−3
4Cdc)x2Lme cos de
xdee1 similarly constants with respective
to generator 2 can also be written.
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K9 = (3
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⎧⎪⎪⎨⎪⎪⎩
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