HFAGC based on MOPSO technique: Optimal design,comparison,verification

Abstract

Good recovery of power system’s synchronism in the affected eclectic power system caused by the loading condition changes is the main target of Automatic Generation Control (AGC). Accomplished endeavors are led to suggest Hierarchical Fuzzy AGC (HFAGC) in multi-area interconnected power system aimed at alleviating both prominent issues i.e.: the low frequency power oscillations and the tie-line power exchange deviations. In this regard, application of multi-objective optimization technique isn’t avoidable. Due to high performance of Multi Objective Particle Swarm Optimization (MOPSO) in solving non-linear objectives, it has been engaged to unravel the optimization problem and optimal tune the HFAGC. To confirm the high efficiency and robustness of suggested controller, two different multi-area interconnected power systems have been taken into account for this study. Meantime, the potency of HFAGC has been thoroughly appraised and compared with Conventional AGC (CAGC) via occurrence of the Step Load Perturbation (SLP) in both these test power systems. To sum up, the simulations results have transparently corroborated the high performance of HFAGC as compared with CAGC in both the test power systems.

Keywords

HFAGC CAGC MOPSO HFRBS SLP thermal power system hydro-thermal power system

1 Introduction

In the past decades, the sustaining growth of electrical power systems in size and complexity as well as increase of the interconnected system have become a prominent issue in the field of electric quality and dynamic stability [1 –4]. The dynamic behavior of power systems is highly affected by operating conditions and load perturbation that might lead to growing the amplitude of oscillations and consequently loss of synchronism [5]. Unraveling this problem is main responsibility of Automatic Generation Control (AGC). This importance is due to its accurate and safe operation in interconnected power systems to alleviate the system frequency and tie-line power exchange deviations during the occurrence of load perturbations [6]. In other words, AGC system pursues three prominent targets in the multi-machine power systems:

Zeroing of the steady state errors of frequency deviations

Acquiring good tracking performance without sustained oscillations during the load disturbances.

Minimizing the oscillations of the tie-line power and frequency deviations.

Over the years, AGC has been one of the emphasized subjects in the field of interconnected power systems. Likewise, many different techniques and controllers have been proposed for controlling the AGC systems [7 –9]. Naturally, the occurrence of different disturbances in power system irrefutably has particular responses that may lead to instability. But then, it is essential to employ the high efficiency controller in order to retrieve the power system to the stable status. In recent years, fuzzy controllers have been successfully and widely implemented in many different nonlinear systems whose behavior is extremely intricate to model [10 –12]. Power systems are such the nonlinear systems, thus occupy the Fuzzy Logic Controller (FLC) could be a good solution. Currently, one of the most prominent problems in the field of fuzzy systems is how to decrease the number of included rules along with the relevant computational requirements.

In the conventional fuzzy controllers, the number of rules has exponentially growth while the elicited results are hardly precise or explainable. As an example for n input variables and m membership functions for each variable, mⁿ rules would be essential to set out a fuzzy controller. According to the mentioned relationship, by increasing the n the rule base will rapidly occupied the memory that may lead to difficult application of fuzzy controller. One practicable strategy to unravel this problem is set out the input variables hierarchically [13, 14]. Hierarchical Fuzzy Rule Base System (HFRBS) is constructionally composed of a number of low-dimensional hierarchical fuzzy systems. This strategy will lead to: linear growth of the total number of rules, less computational effort and high reliability of fuzzy controller [15, 16]. It is because the hierarchical fuzzy rule base system has been engaged to design an effective controller i.e.: Hierarchal Fuzzy Automatic Generation Control (HFAGC) to enhance dynamic stability of power system. Furthermore, being multi object of problem, It is imperative that an effective multi objective optimization technique to be applied. In this regard, Multi Objective Particle Swarm Optimization (MOPSO) technique owing to high efficiency in solving non-linear objectives has been employed to solve the optimization problem. For this study, two different multi-area interconnected power systems i.e.: three-area hydro-thermal power system and five-area thermal power system have been selected to test and analyze the performance of HFAGC. These systems have been affected by Step Load Perturbation (SLP) to appraise the deal with the efficiency and robustness of HFAGC. Meanwhile, a Conventional AGC (CAGC) has simultaneously been along side of proposed controller to more and better exhibit the HFAGC performance. Eventually, the simulations results have transparently demonstrated the high performance of HFAGC as compared with CAGC in both the test powersystems.

2 Multi-objective particle swarm optimization technique

2.1 MOPSO review

PSO is a stochastic global optimization method based on the simulation of social behavior of birds and fish into the flock. In PSO, each particle of the swarm regulates its trajectory in accordance with its own flight experiment and the other particles’ flying experiment among its juxtaposition in the search space [17]. At first, the particles’ positions and velocities are randomly produced which are iteratively upgraded based on their prior positions and each particle’s neighbors. In other words, this algorithm act in accordance with two prominent criterions i.e.: parameter selection technique and particles’ trajectory analysis [18]. The importance of trajectory analysis is owing to assist to define the position of each particle at evolutionary step and accordingly assist to explicate the running PSO’s mechanism in order to unravel the optimization problems. Likewise, the importance of parameter selection is due to sensitivity of chosen parameters in performance of PSO. In this study, the procedure of optimization has more than one objective, thus enforcement of the multi-objective optimization is obligatory. Generally, the multi-objective alternative of a problem is more difficult than the single objective problem. The interaction of these objectives yields a set of effective solutions known as Pareto optimal solutions which provide a determinant more flexibility in the engagement of an appropriate alternative.

2.2 Basic concepts of multi objective functions

The problem’s solution can be mathematically presented as follows:

Find X which optimizes: $f (X) = [f_{1} (X), f_{2} (X), \dots, f_{k} (X)]$ (1)

Subject to: $g_{i} (x) \leq 0; i = 1, 2, \dots, n$ (2) $h_{i} (x) \leq 0; i = 1, 2, \dots, p$ (3)

Where X = [X₁, X₂, …, X_n]^T Which is the decision variables’ vector, f_i, i = 1, 2, …, k are the objective functions and g_i, h_j, i = 1, 2, …, m, j = 1, 2, …, p are the problem’s constraint functions.

In fact, multi-objective optimization problems are scheduled aimed at is attainment of good compromises. To perceive the concept of Pareto optimality, some definitions have been introduced.

Definition 1. Provide two vectors $X, Y \in R^{n}$ , can be cited that X ≤ Y if x_i ≤ y_i for i = 1, 2, …, k, and that X dominates Y (indicated by X ≺ Y) if X ≤ Y and X ≠ Y.

Definition 2. Stated that a decision variables’ vector $X \in χ \subset R^{n}$ is non-dominated with respect to χ, if there is n’t X′ ∈ χ such that f (X′) ≺ f (X).

Definition 3. Stated that a decision variables’ vector $X^{*} \in F \subset R^{n}$ (F is the feasible region) is Pareto optimal if it is non-dominated with respect to F.

Definition 4. The Pareto Optimal Set P^* is determined by: $P^{*} = {X \in F | X is Pareto optimal}$ (4)

Definition 5. The Pareto Front PF^* is determined by: ${PF}^{*} = {f (X) \in R^{k} | X \in P^{*}}$ (5)

2.3 Particle swarm optimization construction

For proposed approach the velocity and the position of a particle are computed as follows [19]: $\begin{matrix} υ_{i, d}^{(t + 1)} = ψ [w \times υ_{i, d}^{(t)} + c_{1} + r_{1} \times \\ ({pbest}_{i, d} - x_{i, d}^{(t)}) + c_{2} + r_{2} \times ({gbest}_{d} - x_{i, d}^{(t)})] \end{matrix}$ (6) $\begin{matrix} x_{i, d}^{(t + 1)} = x_{i, d}^{(t)} + υ_{i, d}^{(t + 1)}, \\ w ith i = 1, 2, \dots n and d = 1, 2, \dots, m \end{matrix}$ (7) $υ_{d}^{min} \leq υ_{i, d}^{(t)} \leq υ_{d}^{max}$ (8) $ψ = \frac{2}{2 - φ - \sqrt{φ^{2} - 4 φ}}$ (9) $φ = φ_{1} + φ_{2} where φ > 4$ (10) $w = w_{max} - \frac{w_{max} - w_{min}}{{iter}_{max}} \times iter$ (11)

Where c₁ and c₂ are both positive constants, called personal and global learning coefficients respectively. r₁ and r₂, are random numbers produced from a uniform distribution in the range (0, 1). $x_{i, d}^{(t)}$ is the dth component of the position of particle i at iteration t; p_best,i is the p_best of particle i; gbest is the gbest of the group. φ₁ and φ₂ are acceleration constants, ψ is constriction factor extracted from the stability analysis of Equation 6 to confirm the system to be converged but not precipitately. ψ is mathematically function of φ₁ and φ₂. Are acceleration constants, ψ is the inertia weight for the PSO’s convergence which controls the impact of the prior history of velocities on the present velocity to find a suitably optimal solution. Likewise, iter_max, iter, w_min and w_max are maximum iteration number, current iteration number, final weight and initial weight, respectfully.

2.4 Non-dominated sorting particle swarm optimization principle

The proposed MOPSO applies another independent or auxiliary archive in order to store the found solutions along the search process. Likewise, the neighbor is considered in the swarm for this algorithm. The swarm has been stored into the auxiliary archive in each iteration subsequent to updating the positions of particles. Whenever, the maximum bound imposed on archive’s size is reached, the algorithm carries out non-dominated sorting for retaining the solutions located into the first five-fronts, given in Fig. 1. The procedure figures out the first particles of non-dominated front for all of the archive’s members. If the front length is less than the maximum bound, the front is retained into the archive. Therefore, in order to figure out the individuals in the next front, the first front’s solutions are provisionally overlooked, and also the above procedure is replicated till five fronts to be found (Fig. 1a). Whenever, the front is retained into the archive then it is greater than admissible bound, a crowding length has been calculated to filter out solutions, and the following fronts will be omitted (Fig. 1b).

Ultimately the flowchart of proposed optimization algorithm is shown in Fig. 2.

3 Hierarchical mamdani-type fuzzy strategy to design AGC

3.1 Fuzzy logic guideline

The main portion of the FLC is regulation of linguistic control rules concerned to two concepts of fuzzy implication and the organizational rule of inference. The FLC performs converting the linguistic control system in accordance with a drastic arithmetical technique, into an automatic control system. There are several alternatives to engage FLC aimed at the control system. As shown in Fig. 3, FLC structure is concluded all controller plan [20]. A fuzzy system has been set up based on four main structures: fuzzifier, inference engine, knowledge base and defuzzifier. At the first, the fuzzy system computations have been transformed from the numeric into the fuzzy sets which is called fuzzification. In viewpoint of fuzzy set theory, the inference engine is assigned as the heart of the fuzzy system that carries out all logic operations in a fuzzy system. The knowledge base system has been made up of membership functions and IF–THEN rules [20]. The inference procedure result will be an output represented by a fuzzy set, even so the fuzzy system output turn to a numeric value by defuzzification process. Furthermore, it’s essential that input and output scaling factors to be amended into the universe of discourse.

3.2 Fuzzy controller structure

The triangle type membership function presented in Fig. 4 has been used to determine the membership grade. The fuzzy quantities, such as PB (Positive Big), PM (Positive Medium), PS (Positive Small), ZE (Zero), NS (Negative Small), NM (Negative Medium) and NB (Negative Big) are with regard to membership functions. This paper applies the fuzzy control rules of state evaluation, which are similar to the institutional thinking of humans [21]. $R^{i} : if C is A_{1 i} Δ e is B_{1 i} then u is C_{1 i}$

Where, e, Δe and u indicate the system variables error, error deviation and output, respectively. Also, A_1i, B_1i and C_1i are the linguistic value of the fuzzy variable to express the universe of discourse of the fuzzy set. For the inference mechanism, madmani’s sum-product technique has been applied. The fuzzy sets must be defuzzified to acquire the appropriate control output for the control system. Among the different defuzzification methods, centroid method is more common, which conforms to the following Equation [18]. $\frac{\int (u (x) x) dx}{\int u (x) dx}$ (12)

Where, μ is the membership grade of output x.

The rule table and the membership functions boundaries for both subsystems presented in Table 1 are similarly selected.

3.3 Hierarchical fuzzy rule base system

In hierarchical fuzzy system, the total number of rules is declined via reconstructing the fuzzy rules into hierarchical simple fuzzy subsystems. In this strategy, the modulation process has been hierarchically performed from initial level to the final level [13 –16]. In other words, the first level renders an unpurified output, and adjusted by the second level in the next that the process can be repeated in following levels. Figure 5 displays the general structure of HFRBS with N input variables (a), accompanied by classical FRBS with N input variables (b). Presumed that M membership functions are used to fuzzify each input variable, the total number of rules can be presented as follows:

$M^{N} rules in a conventional FRBS$ (13) $M^{2} . (N - 1) rules in a hierarchical FRBS$ (14)

3.4 Hierarchical fuzzy automatic generation control structure

A HFRBS strategy is engaged to construct an effective AGC i.e. HFAGC which is presented in Fig. 6. Using this hierarchical structure, the controller performance is enhanced without exponential growth in the rule table. As can be seen, HFAGC is constructed by two fuzzy subsystems that each subsystem is fed by two inputs. Seven triangle membership functions are used for the inputs and outputs. Meantime, Mamdani inference system and centroid defuzzification are chosen for thisstudy.

4 Presence of HFAGC in both interconnected power systems

4.1 Linearized model of three-area hydro-thermal power system

The studied power system is comprised of three equal generating areas; two reheat thermal system for area 1 and area 2, and also hydro system for area 3. The linearized model of hydro-thermal system which is shown in Fig. 7, has been simulated in MATLAB/SIMULINK environment. Generation Rate Constraint (GRC) is considered 3% /min in thermal areas, and also 270% /min (4.5% /s) for raising and 360% /min (6% /s) for lowering generation in hydro area. A bias setting of B_i is chosen for both thermal and hydro areas. The other relevant parameters’ data are given in Appendix A.

4.2 Linearized model of five area thermal power system

On the contrary, this system is carried out of five unequal generating areas. The linearized model of aforesaid system is presented in Fig. 8. The thermal systems are made up with single reheat turbine and GRC of 3% /minute in each area. Meantime, the other relevant parameters’ data are given in Appendix B.

4.3 Multi area interconnected power systems with presence of HFAGC & CAGC

Zeroing of the Area Control Error (ACE) of each area of interconnected power system is scheduled as the main target of AGC system. The system frequency and tie-line power exchange deviations are the two prominent benchmarks related to ACE that can be presented by following equation [22, 23]: ${ACE}_{i} = \sum_{j} Δ P_{tie, i, j} + B_{i} Δ f_{i}$ (15)

Where, B_i is frequency bias coefficient of ith area, Δf_i is frequency error of ith area, ΔP_tie,i,j is tie-line power deviation between ith area and jth area.

The output of AGC controller in Laplace domain will be obtained by: $- G_{AGC, i} (s) . {ACE}_{i} (s)$ (16)

Meantime, PID structure is considered as conventional AGC for this study. G_AGC (s) of this structure can be presented by following equations: $G_{CAGC, i} (s) = K_{P, i} + K_{I, i} * S^{- 1} + K_{D, i} * S$ (17)

The parameters of CAGC and HFAGC which should be determined by MOPSO technique, including:

HFAGC: K_ef, K_def, K_uf and K_dep and K_u

CAGC: K_P, K_I and K_D

4.4 Objective functions

There are many different methods to appraise the response performance of a control system, namely: Integral of Time weighted Absolute value of Error (ITAE), Integrated Absolute Error (IAE), Integral of Squared Error (ISE), and Integral of Time weighted Squared Error (ITSE). In this study, ITAE criterion is considered for both the objective functions aimed at optimizing by MOPSO technique: $J_{f} = \int_{t = 0}^{t = t_{sim}} (\sum | Δ f_{i} |) . t . dt$ (18) $J_{P} = \int_{t = 0}^{t = t_{sim}} (\sum | Δ p_{tie, i, j} |) . t . dt$ (19)

Where, t_sim is simulation time period, and also J_f and J_P are two prominent benchmarks in line with control the system stability. The time-domain simulation of the non-linear system model is performed for the simulation period. It has been scheduled to minimize this fitness function. The problem constraints are the optimized parameter bounds. Therefore, the design problem is formulated as the following optimization problem: $Minimize J_{f} & J_{P}$ (20)

Subject to: $\begin{matrix} K_{P, i}^{min} \leq K_{P, i} \leq K_{P, i}^{max} \\ K_{I, i}^{min} \leq K_{I, i} \leq K_{I, i}^{max} \\ K_{D, i}^{min} \leq K_{D, i} \leq K_{D, i}^{max} \\ K_{ef, i}^{min} \leq K_{ef, i} \leq K_{ef, i}^{max} \\ K_{def, i}^{min} \leq K_{def, i} \leq K_{def, i}^{max} \\ K_{uf, i}^{min} \leq K_{uf, i} \leq K_{uf, i}^{max} \\ K_{dep, i}^{min} \leq K_{dep, i} \leq K_{dep, i}^{max} \\ K_{u, i}^{min} \leq K_{u, i} \leq K_{u, i}^{max} \end{matrix}$ (21)

5 Simulation results

5.1 In three-area hydro-thermal power system

To appraise the efficiency of suggested controller, it has been tested under 3% SLP in area 1 which is triggered at t = 10 s. The response of affected power system has been analyzed and inspected with presence of HFAGC and CAGC through evaluation of suggested fitness function. After unraveling the optimization problem, the optimal parameters of HFAGC and CAGC have been extracted that are tabulated in Table 2, Table 3. Likewise, the Pareto solution front of the optimization problem is depicted by Fig. 9.

Figure 10 has transparently corroborated the effectiveness and robustness of HFAGC as compared with CAGC to enhance the dynamic stability of hydro-thermal power system.

5.2 Five-area thermal power system

In this part of the paper, this power system has been affected by 1% SLP in area 3 which is triggered at t = 1 s. Like the previous section, the performance of proposed HFAGC has been evaluated using suggested fitness functions as well as accompanied by CAGC. As consequence of completing the optimization problem, the optimal parameters of HFAGC and CAGC are revealed which are tabulated in Tables 4, 5. Meanwhile, the pareto solution front of the optimization problem is given in Fig. 11. The system response under this disturbance with presence of both the HFAGC and CAGC has been presented by Fig. 12. As expected, the same previous section result has been revealed.

6 Conclusion

In this paper, a robust and high efficient HFAGC is suggested to quickly retrieve the affected power system into stable status. In this regard, mitigation of both the prominent issues i.e.: the low frequency power oscillations and the tie-line power exchange deviations have been formulated as a multi objective optimization problem. Due to high performance of MOPSO technique in solving non-linear objectives, it has been employed to unravel the optimization problem. Two different multi-area interconnected power systems have been considered to appraise and deal with the high efficiency and robustness of suggested HFAGC. Occurrence of a SLP which has severe impacts on dynamic stability of power system has highly affected both the power systems until HFAGC and CAGC exhibit their roles in mitigating the power system oscillations. Eventually, the simulations results drawn out from both the multi area interconnected power systems have transparently corroborated the high performance of HFAGC as compared with CAGC.

Footnotes

Appendix

A. Three-Area Hydro-Thermal Power System

f = 60 Hz; T_gi = 0.08 s; T_ri = 10 s; H_i = 5 s; T_ti = 0.3 s; K_r = 0.5; P_ri = 2000 MW; T_pi = 20 s; K_d = 4.0, K_p = 1.0, K_i = 5.0; D_i = 0.00833p.u.MW/Hz; K_pi = 120 Hz/p.u MW, T_w = 1 s.

B. Five-Area Thermal Power System

Area 1:2000 MW, Area 2:4000 MW, Area 3:8000 MW, Area 4:10000 MW and Area 5:12000 MW. f = 60 Hz, Tgi = 0.08 s, Tri = 10 s, Hi = 5 s, Tti = 0.3 s, Kr = 0.5, Ptie-max=200 MW, Di = 0.00833 p.u.MW/Hz, Tpi = 20 s, Kpi = 120 Hz/p.uMW, loading = 50%, Tij = 0.544, Tw = 1.0 s, Kp = 1.0, Kd = 4.0, Ki = 5.0.

References

Kazemi

M.R.

, Mmotlagh

and Naghshbandi

A.H.

, Application of a new multi-variable feedback linearization method for improvement of power systems transient stability, Electrical Power and Energy Systems29 (2007), 322–328.

Bevarani

, Hiyama

and Mitani

, Power system dynamic stability and voltage regulation enhancement using an optimal gain vector, Control Engineering Practice16 (2008), 1109–1119.

Iracleus

D.P.

and Alexandridis

A.T.

, A multi-task automatic generation control for power regulation, Electric PowerSystems Research73 (2005), 275–285.

Tan

and Xu

, Robust analysis and design of load frequency controller for power systems, Electric PowerSystems Research79 (2009), 846–853.

Falehi

A.D.

, Rostami

, Doroudi

and Ashrafian

, Optimization and coordination of SVC-based supplementary controllers and PSSs to improve the power system stability using genetic algorithm, Turk J Elec Eng & Comp Sci20 (2012), 639–654.

Kundur

, Klein

, Rogers

G.J.

and Zywno

M.S.

, Application of power system stabilizers for enhancement of overall system stability, IEEE Trans PWRS4 (1989), 614–626.

Rebours

, Kirschen

, Trotignon

and Rossignol

, A survey of frequency and voltage control ancillary services. Part I. Technical features, IEEE Transactions on PowerSystems22 (2007), 350–357.

Veluami

and Chidamberam

, Design of decentralized biased dual mode controllers for load frequency control of interconnected power systems, Electric Power Components and Systems34 (2006), 1057–1075.

Bevarani

and Hiyama

, Robust decentralized PI based LFC design for time delay power systems, Energy Conversion and Management49 (2008), 193–204.

10.

Elshafei

A.L.

, El-Metwally

K.A.

and Shaltout

A.A.

, A variable-structure adaptive fuzzy-logic stabilizer for single and multi-machine power systems, Control Eng Pract13 (2005), 413–423.

11.

Karnavas

Y.L.

and Papadopoulos

D.P.

, Excitation control of a power generating system based on fuzzy logic and neural networks, European Transactions on Electrical Power10 (2000), 233–241.

12.

Moradi

, Shirazi

K.H.

, Keshavarz

and Falehi

A.D.

, Mosayeb, Moradi, Smart piezoelectric patch in nonlinear beam: Design, vibration control and optimal location, Transactions of the Institute of Measurement and Control36 (2014), 131–144.

13.

Raju

, Zhou

and Kisner

, Hierarchical fuzzy control, Int J Control54 (1991), 1201–1216.

14.

Benítez

A.D.

and Casillas

, Multi-objective genetic learning of serial hierarchical fuzzy systems for large-scale problems, Soft Comput17 (2013), 165–194.

15.

Kikuchi

, Otake

and Nakanishi

, Functional completeness of hierarchical fuzzy modeling, Information Science110 (1998), 51–60.

16.

Lee

M.-L.

, Chung

H.-Y.

and Yu

F.-M.

, Modeling of hierarchical fuzzy systems, Fuzzy Sets and Systems138 (2003), 343–361.

17.

Kennedy

and Eberhart

, Swarm Intelligence, 1st ed., Academic press, San Diego, CA, 2001.

18.

Hingorani

N.G.

and Gyugyi

, Understanding FACTS: Concepts and Technology of Flexible AC Transmission Systems, IEEE Press, New York, 2000.

19.

Panda

and Padhy

, Comparison of particle swarm optimization and genetic algorithm for FACTS-based controller design, Appl Soft Comput8 (2008), 1418–1427.

20.

Dash

P.K.

, Morris

and Mishra

, Design of a Nonlinear Variable-Gain Fuzzy Controller for FACTS Devices IEEE Transactions on Control System Technology12 (2004), 428–438.

21.

Talaat

H.E.A.

, Abdennour

and Al-Sulaiman

A.A.

, Design and experimental investigation of a decentralized GA-optimized neuro-fuzzy power system stabilizer, Electrical Power and Energy Systems32 (2010), 751–759.

22.

Divya

K.C.

and Nagendra Rao

P.S.

, A simulation model for AGC studies of hydro–hydro systems, Electrical Power and Energy Systems27 (2005), 335–342.

23.

Goshal

S.P.

, Optimization of PID gains by particle swarm optimization in fuzzy based automatic generation control, Electric Power Systems Research72 (2004), 203–212.