Abstract
This paper presents an adaptive multiobjective algorithm based Unified Power Flow Controller (UPFC) tuned for damping oscillations in twoarea multimachine system formulated as multi objective optimization problem. The algorithms such as, Nondominated Sorting Genetic AlgorithmII (NSGAII) and Modified Nondominated Sorting Genetic AlgorithmII (MNSGAII) are proposed for tuning the damping controller with speed deviation and control input as conflicting objectives. The proposed algorithm is implemented in the two area multimachine system using MATLAB Simulink model, and the simulation results were obtained with respect to the characteristics of damping oscillations and the dynamic stability of power systems. The performance measures such as Integral Time Squared Error (ITSE) and Integral Squared Error (ISE) are considered as the objective functions. The results of the two proposed algorithm has been compared and the outcome shows that the MNSGAII algorithm performs better compared to the NSGAII algorithm.
1. Introduction
Power systems are under growing stress as deregulation presents a few new economic objectives for operation. Since control frameworks are being worked near their limits, shabby associations, unexpected occasions, sudden cause of failure in protection system, human blunders, and a large group of different elements may make the system to lose security and even cause system failure. Along these lines, the requirement for enhanced damping control method in a more extensive working is increasing more consideration. Among the accessible damping control techniques, each approach has favorable circumstances and hindrances in various frameworks. The viability of damping control relies upon the device picked, the system modular element, and the connected controller outline technique. The interconnected power systems may create electromechanical oscillations within the low frequency range of 0.2 Hz to 3.0 Hz.
A novel control of PVSTATCOM coordinated with power system stabilizers are used to damp out the electromechanical oscillations in a power system [1]. The author demonstrates that the coordinated control of the proposed method effectively enhances the damping oscillations which leads to improved power transfer in transmission lines. A unified model of a power system is employed with three FACTS devices (SVC, CSC, and PS) for suppressing oscillation and improving power system stability [2]. The author investigated the performance by analyzing the damping torque contribution to the power system framework. The UPFC is one of the most adaptable FACTS devices which can provide a sufficient control of real as well as reactive power flow and voltage regulation for the power system [3]. The author illustrated that the proposed UPFC power frequency model and UPFC network interface method improves the dynamic performance of the system and fairly control the damping oscillations.
Stability enhancement in multimachine power system has been investigated for the best possible location of the UPFC based on small signal, voltage and transient stability using power system analysis toolbox (PSAT) software [4]. This small signal linearized model of UPFC is not suitable for damping controller design. Therefore, a linearized PhillipsHeffron model with STATCOM is proposed and demonstrates that the STATCOM stabilizer improves the power system oscillation stability [5]. An approach to the design of UPFC controllers has been presented and reveals that the damping is adversely affected by the incorporation of the DC voltage regulators [6]. Among the FACTS family, the UPFC is considered as the most powerful device which increase the power system stability [7]. The development of nonlinear dynamic approximation using UPFC augmenting with nonlinear adaptive control based on backstepping for damping oscillation. A multimachine system with long transmission line is compensated with UPFCSMES. The author implemented DCDC chopper to modulate the power of SMES to improve the transient stability [8].
The transient power system stability performance has been improved using the radial basis function neural network design for UPFC [9]. The author develops single neuron and multi neuron systems and implemented in the single machine infinite bus systems and three machine power system networks. The proposed RBFNN exhibits superior damping compared to the conventional PI controllers. A mixed sensitivity design of damping device with UPFC is presented [10]. The system implemented in the two areas four machine system gives satisfactory results both in frequency domain and through nonlinear simulations.
The problem of robustly UPFC based damping controller is formulated as an optimization problem considering the PhillipsHeffron model according to the eigen valuebased multiobjective function comprising the damping factor, and the damping ratio of the undamped electromechanical modes to be solved using particle swarm optimization technique (PSO) that has a strong ability to find the most optimistic results [11]. A novel method for the design of output feedback controller for unified power flow controller (UPFC) is developed. The selection of output feedback gains for UPFC controllers is converted as an optimization problem with the time domainbased objective function which is solved by a particle swarm optimization technique (PSO) that has a strong ability to find the most optimistic results. Only local and available state variables are adopted as the input signals of each controller for the decentralized design [12]. Single objective optimization algorithms are not efficient in solving problems with more than one objective because optimizing one objective may lead to increase the conflicting objectives. Hence multiobjective optimization is proposed for tuning UPFC damping controllers installed in multimachine system. The multiobjective algorithms such as NSGAII and MNSGAII are used for tuning the parameters of PI controller. Robust control methods are implemented to tune the PI controllerbased UPFC for stability enhancement of a twoarea fourmachine and for SISO power system using evolutionary algorithms [1320].
In this paper, a control mechanism of the PI tuned UPFC considering the two area multimachine system using the NSGAII and MNSGAII algorithm to find the optimal values and to minimize the Integral Time Squared Error (ITSE) as the main objective functions. The rest of the paper is organized as follows. Section 2 briefly describes the UPFC multimachine system. Section 3 narrates the nonlinear dynamic model of UPFC. Section 4 discusses about the implementation of the proposed NSGAII algorithm and Section 5 presents the implementation of the MNSGAII algorithm on the two area multimachine system. Section 6 describes the problem formulation of proposed work. Section 7 the simulation results and discussion are presented. Section 8 concludes the paper.
2. UPFC with Multimachine system
UPFC consists of two static converters basically series and shunt converter with a common DC link, coupling transformer connected to the AC system. The shunt converter operates as a static synchronous compensator (STATCOM) controls the AC voltage at its terminals and the voltage of the DC bus. It uses a dual voltage regulation loop: an inner current control loop and an outer loop regulating AC and DC voltages. Control of the series branch is different from the Static Synchronous Series Compensator (SSSC) where the two degrees of freedom of the series converter are used to control the DC voltage and the reactive power. In case of a UPFC the two degrees of freedom are used to control the active power and the reactive power [21]. The twoarea four machine system used in this paper is an 12 Bus multimachine system as shown in Fig. 1. The system contains twelve buses and two areas, connected by a weak tie between bus 7 and 9. Totally two loads are applied to the system at bus 7 and 9. UPFCbased damping controller is connected between buses 7 and 10 with the main functions of power oscillation damping and power flow control carried out by four PI controllers.
Fig. 1Single line diagram of two area four machine system
Fig. 2 depict the MATLAB simulation diagram of the proposed two area four machine system. The simulation test system consists of two areas linked by two 230 kV lines of 220 km length. Two identical generators in each area of 13.8 kV/1000 MVA is connected in the network. This system is used in this work to study about the low frequency oscillations of power system. UPFCbased damping controllers designed for twoareafour machine is also shown in Fig. 3.
Fig. 2Simulation diagram of the proposed system
3. Nonlinear dynamic model of the UPFC
To enhance the small signal stability of the system, a dynamic model of the UPFC is very much essential in the power system network. The modeling of UPFC can be done using park’s transformation neglecting resistance of UPFC transformers. The complete multimachine system with the UPFC located near the load side is discussed in Section 2. The single line diagram of $n$machine UPFC power system is shown in Fig. 4. The four control parameters of UPFC are ${m}_{B}$ – reactive power regulator, ${\delta}_{B}$ – active power regulator, ${m}_{E}$ – AC voltage regulator, ${\delta}_{E}$ – DC voltage regulator.
Fig. 3Simulink diagram of UPFC damping controller
These control inputs provide power compensation in series and shunt line [15].
The $d$$q$ transformation of excitation and boost systems denoted as $E$ and $B$ in Eqs. (13) respectively are:
where ${C}_{dc}$ – DC link capacitance, ${V}_{dc}$ – DC link voltage.
The nonlinear equations Eqs. (49) of the power system under consideration are:
where ${i}_{t}$_{}– armature current, ${v}_{b}$ – bus voltage, ${v}_{Et}$, ${v}_{Bt}$, ${i}_{B}$ and ${i}_{E}$ are the excitation transformer voltage, boosting transformer voltage, boosting transformer current and excitation transformer current respectively.
The state space matrix of the variables under consideration is represented in Eq. (10):
$+\left[\begin{array}{cccc}0& 0& 0& 0\\ \frac{{k}_{pe}}{M}& \frac{{k}_{p\delta e}}{M}& \frac{{k}_{pb}}{M}& \frac{{k}_{p\delta b}}{M}\\ \frac{{k}_{qe}}{{T}_{d0}^{\text{'}}}& \frac{{k}_{q\delta e}}{{T}_{d0}^{\text{'}}}& \frac{{k}_{qb}}{{T}_{d0}^{\text{'}}}& \frac{{k}_{q\delta b}}{{T}_{d0}^{\text{'}}}\\ \frac{{k}_{A}{k}_{ve}}{{T}_{A}}& \frac{{k}_{A}{k}_{v\delta e}}{{T}_{A}}& \frac{{k}_{A}{k}_{vb}}{{T}_{A}}& \frac{{k}_{A}{k}_{v\delta b}}{{T}_{A}}\\ {k}_{ce}& {k}_{c\delta e}& {k}_{cb}& {k}_{c\delta b}\end{array}\right]\left[\begin{array}{c}\mathrm{\Delta}{m}_{E}\\ \mathrm{\Delta}{\delta}_{E}\\ \mathrm{\Delta}{m}_{B}\\ \mathrm{\Delta}{\delta}_{B}\end{array}\right].$
Fig. 4Single line diagram of nmachine UPFC system
4. Implementation of Nondominated Sorting Genetic Algorithm II (NSGAII)
NSGAII is a multiobjective optimization problem working on Paretooptimal solutions. It uses nondominated sorting, particles which are not dominated by other particles are assigned as ${F}_{1}$ (front 1), and particles which are dominated by other particles in ${F}_{1}$ are assigned as ${F}_{2}$ (front 2) [16, 17]. The selection is made based on the tournament i.e. the particles with lowest crowding distance are selected if the two particles are from the same front. $O\left(MN\right)$ comparisons are required for finding the first nondominated solution front. $O\left(M{N}^{2}\right)$ comparisons are required for finding the second nondominated solution front where $M$ is the number of objectives, and $N$ is the population size.
4.1. Algorithm for NSGAII
The step by step procedure algorithm of NSGAII is explained and the parameters taken for NSGAII are shown in Table 1.
Step 1: Initialize population size $N$, crossover and mutation probability, crossover and mutation index, maximum number of generations. Initialize ${n}_{k}=$0 and ${S}_{k}=\phi $, $k\in P$ where ${n}_{k}$ is the number of solutions that dominate $k$, and ${S}_{k}$ be the set of $k$ dominated solutions. $P$ is the set of populations.
Step 2: If $k$ dominates l, ($k<$ l), then add $q$ to ${S}_{k}$, else ${n}_{k}={n}_{k}+1$.
Step 3: If $k$ belongs to ${F}_{1}$, then rank of $k=$1. Add $k$ to ${F}_{1}$.
Step 4: Initialize the front counter $j=$1.
Step 5: While front counter is not empty, go to Step 6.
Step 6: $Q$ be the counter used to store subsequent front which is initialized as empty set.
Step 7: If $k\in {F}_{j}$ and $l\in {S}_{k}$, then decrement ${n}_{l}$.
Step 8: If ${n}_{l}=0$, l belongs to ${F}_{2}$, and rank of $l=j+1$, add $l$ to $Q$.
Step 9: $j=j+1$ and $F=Q$.
Step 10: Calculate crowding distance for $F$.
Step 11: Sort $F$ in the descending order based on rank.
Step 12: Create new population using selection and mutation.
Step 13: Continue the above steps until maximum generation is reached
Table 1NSGAII Parameters
Parameters  Value 
Number of population  300 
Number of generation  500 
Cross over probability  0.8 
Mutation probability  1/8 
Cross over index  2 
Mutation index  20 
Elitism  0.55 
5. Implementation of Modified Nondominated Sorting Genetic Algorithm II (MNSGAII)
NSGAII algorithm is a good optimization technique for multiobjective problems. However, the problem with NSGAII is maintaining the diversity and uniformity. Hence in NSGAII, Dynamic Crowding Distance (DCD) is incorporated to enhance the diversity and uniformity to refine solutions further called as Modified Nondominated Sorting Genetic Algorithm II [18, 19].
5.1. Algorithm for MNSGAII
The step by step procedure algorithm of NSGAII is explained and the parameters taken for NSGAII shown in Table 1 is utilized for the MNSGAII.
Step 1: Initialize population size ${N}_{p}$, crossover and mutation probability, crossover and mutation index, maximum number of generations
Step 2: Generate initial population randomly and set iteration count $i=$0.
Step 3: Calculate objective function ISE for each particle in the population.
Step 4: Using selection, single point crossover and mutation generate offspring from the parent population.
Step 5: Perform the nondominated sorting.
Step 7: By using DCD approach Eqs. (1113), eradicate $R$$N$ particles from nondominated set, if $R>N$, else go to Step 4, where $R$ – size of nondominated set, $N$ – population size, and:
where:
where $M$ – number of objectives, $f$ – objective function, ${V}_{i}$ – variance of crowding distance.
Step 8: Continue the above steps until maximum generation is reached.
6. Problem formation
NSGAII and MNSGAII algorithms are implemented to tune the PI controllerbased UPFC for stability enhancement of a twoarea fourmachine power system. Tuning of PI controllers is done by optimizing the error signal and control input values, formulated as multiobjective optimization problem. Integral Time Squared Error (ITSE) is the performance measure taken in this work. The objective functions are formulated as given in Eqs. (1416) to reduce the peak overshoot and the settling time:
where:
where $e$ is the error signal i.e, speed deviation of four generators for local and interarea modes and $u$ is the input control signal, $t$ is the time of simulation.
The parameters of the system with the proposed MNSGAIIbased UPFC controller shifts to the left by a significant value in the splane indicating that the proposed UPFC controller provides better dynamic stability to the power system. The negative real and complex eigen values confirm that the system is in fast decaying damped oscillatory mode. The eigen value analysis shown in Table 2 reveals that the proposed controller enhances the dynamic stability of the power system greatly. The system eigen values are in the left half of the splane which assures the stability of the system.
NSGA suffers from computational complexity, nonelitist approach and the need to specify a sharing parameter. An improved version of NSGA known as NSGAII. The crowding distance operator will ensure diversity along the nondominated front, lateral diversity will be lost. NSGAII is a fast and elitist MOEA and implements elitism for multiobjective search, using an elitism preserving approach. Diversity and spread of solutions are guaranteed without the use of sharing parameters. When two solutions belong to the same Paretooptimal front, the one located in a lesser crowded region of the front is preferred. However, the lacking of NSGAII in lateral diversity is one of the major shortcomings which guides towards false Paretofront. The important properties involved in MNSGA II algorithm are controlled elitism and Dynamic Crowding Distance. Controlled elitism prevents the number of individuals in the current best nondominated front. It maintains number of individuals in each front distributed predefined manner. Dynamic crowding distance is mainly used to maintain horizontal diversity and, it removes one individual with smaller DCD value at every time and recalculates DCD for the remaining individuals.
Table 2Eigen values for NSGAII and MNSGAII
Eigen values  
NSGAII  MNSGAII 
–1.210 + j5.899  –1.206 + j 5.989 
–1.210 – j5.899  –1.206 – j5.989 
–1.5775 + j6.778  –1.5775 + j6.978 
–1.5775 – j6.778  –1.5775 – j6.978 
–1.861 + j7.253  –1.854 + j7.253 
–1.861 – j7.253  –1.854 – j7.253 
7. Results and discussion
Both the NSGAII and MNGAII algorithms were implemented in the power system network discussed in Section 4 and 5 and the simulation results obtained are presented and discussed in this section. In this work, both the local area and interarea oscillation modes have been considered to calculate the Integral Time Squared Error (ITSE) based on the values of the PI controller tuning parameter ${K}_{p}$ and ${K}_{i}$. The selection of the Paretofront for both the algorithms is shown in Fig. 5.
Fig. 5a) Paretofront for NSGAII, b) Paretofront for MNSGAII
a)
b)
Initially the local area mode oscillations between two different generators have been studied and the results have been analyzed for damping of oscillations. It was inferred from the results shown in Fig. 6, the MNSGAII algorithm yields better results in damping oscillations quickly compared with the other two algorithms.
From the results yielded for the local area mode oscillation, the implementation of algorithm has been studied for the inter area mode oscillations between the generators and the results obtained has been shown in Fig. 7. It clearly depicts that once again the MNSGAII tuned UPFC controller produces very lesser time to settle and quickly damps out the interarea mode oscillations between generators which improves the stability of the power systems.
The gain values of ${K}_{p}$, ${K}_{i}$ of the four PI controllers with ITSE and ISE are tabulated in Table 3 and Table 4. It reveals that the MNSGAII based PI tuned controller gives lesser ITSE compared to other algorithms with improved damping performance with minimum overshoot and undershoots. Fig. 8(a) shows the ITSE convergence for MNSGAII, Fig. 8(b) shows the comparison results of ITSE and Fig. 8(c) shows the comparison results of ISE.
Fig. 6Local area mode oscillations between generator 1 and 2
Fig. 7Inter area mode oscillations
a) Between generator 1 and 3
b) Between generator 2 and 3
c) Between generator 2 and 4
d) Between generator 1 and 4
Table 3Kp, Ki gain values of the four PI tuned controllers with ITSE
Algorithm  ${K}_{p1}$  ${K}_{i1}$  ${K}_{p2}$  ${K}_{i2}$  ${K}_{p3}$  ${K}_{i3}$  ${K}_{p4}$  ${K}_{i4}$  ITSE 
NTVACPSO  0.2154  1.2485  3.2145  0.2154  1.8794  0.0257  1.3965  1.5485  211.05 
NSGAII  1.3313  0.1231  0.9323  0.3211  2.3432  0.0122  2.1313  0.4221  198.34 
MNSGAII  2.1334  0.3423  3.1341  0.0021  0.3424  0.3423  0.2341  0.4232  136.32 
Table 4Kp, Ki gain values of the four PI tuned controllers with ISE
Algorithm  ${K}_{p1}$  ${K}_{i1}$  ${K}_{p2}$  ${K}_{i2}$  ${K}_{p3}$  ${K}_{i3}$  ${K}_{p4}$  ${K}_{i4}$  ISE 
NTVACPSO  1.024  0.1562  0.29  0.48  2.04  0.2476  1.3400  0.464  1.389 
NSGAII  1.0635  0.1751  0.9873  0.3011  2.1232  0.2122  1.1156  0.4221  0.9917 
MNSGAII  1.1356  0.3823  3.1981  0.0231  0.3421  0.3113  0.3113  1.4422  0.6811 
The control energy results obtained for the four PI tuned controllers are shown in Fig. 9(a) to 9(d) where the control energy is the input error functions of the controller. From the results, it is inferred that MNSGAIIbased PI tuned controllers damp out the oscillations by controlling the control energy with less overshoot and minimum settling time compared to NSGAII based PI tuned controller.
Fig. 8a) ITSE convergence for MNSGAII, b) ITSE convergence comparison results, c) ISE convergence comparison results
a)
b)
c)
Fig. 9Control energy
a) PI Controller 1
b) PI Controller 2
c) PI Controller 3
d) PI Controller 4
8. Conclusions
In this work, coordinated designs of the NSGAII and MNSGAIIbased PItuned UPFC controllers have been carried out in two areas four machine systems to enhance damping of oscillations. The results obtained and presented establish that the designed PItuned UPFC damping controllers are more competent to increase the damping coefficient and damping ratio of the electromechanical modes. The results of time domain simulations on the multimachine system have revealed that MNSGAIIbased PItuned UPFC controllers damped out oscillations quickly and enhance power system dynamic stability with minimum undershoot and overshoot compared to NSGAII. Apart from the above, the eigen values were obtained and proves that the stability of the system gets improved. Also, the ISE parameter has also been considered for stability analysis and the comparison characteristics chart has been depicted. The proposed method also provides enhanced performance compared to NTVACPSOtuned UPFC damping controller.
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