4th International Conference on Advances in Control and Optimization of Dynamical Systems 4th in 4th International International Conference Conference on on Advances Advances in Control Control and and Available February 1-5, of 2016. NIT Tiruchirappalli, India online at www.sciencedirect.com Optimization Dynamical Systems Optimization of Dynamical Systems February February 1-5, 1-5, 2016. 2016. NIT NIT Tiruchirappalli, Tiruchirappalli, India India
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Introducing LQR-Fuzzy Technique with Dynamic Demand Response Introducing LQR-Fuzzy Technique with Dynamic Demand Control Loop to Load Frequency Control Model Response Control Loop to Load Frequency Control Model P.Srividya Devi*, Dr. R Vijaya Santhi**, Dr. D.V.Pushpalatha*** P.Srividya Devi*, Dr. Santhi**, Dr. D.V.Pushpalatha*** P.Srividya Devi*, scholar, Dr. R R Vijaya Vijaya Santhi**, Dr.Visakapatanam D.V.Pushpalatha*** *Research Andhra University, (e-mail:
[email protected]) *Research scholar, *Research scholar, Andhra Andhra University, University, Visakapatanam Visakapatanam **Assistant Professor ,Andhra University, Visakapatanam (e-mail:
[email protected]) (e-mail:
[email protected]) ) **Assistant (e-mail:
[email protected] Professor ,Andhra ,Andhra University, University, Visakapatanam Visakapatanam **Assistant Professor ***(e-mail:
[email protected] Professor, EEE, GRIET, Hyderabad (e-mail:
[email protected] )) (e-mail:
[email protected]) *** Professor, Professor, EEE, GRIET, GRIET, Hyderabad Hyderabad *** EEE, (e-mail:
[email protected]) (e-mail:
[email protected])
Abstract: The present paper indicates a novel approach of LQR-Fuzzy controller to stabilize the frequency in allpresent (normalpaper and emergency) under of grid environment. Frequency Abstract: indicates novel LQR-Fuzzy controller to the Abstract: The The present paper indicates aa conditions novel approach approach ofmicro LQR-Fuzzy controller Load to stabilize stabilize the Control (LFC) model plays a vital role in electric power system design and operation. From the literature, frequency in all (normal and emergency) conditions under micro grid environment. Load Frequency frequency in all (normal and emergency) conditions under micro grid environment. Load Frequency it is proven that the implementation (Demand is aand keyoperation. to the future grid. In Control (LFC) model plays in electric power system From the Control (LFC) model plays aa vital vital role roleof in DR electric power Response) system design design and operation. Frommicro the literature, literature, practice, LFC-DR model is tuned by conventional controllers like PI, PD, PID controllers, with their it is proven that the implementation of DR (Demand Response) is a key to the future micro grid. In it is proven that the implementation of DR (Demand Response) is a key to the future micro grid. In fixed gains. However, theyis incapable of obtaining good dynamic performance over a widewith range of LFC-DR model tuned by controllers like PD, their practice, practice, LFC-DR model isare tuned by conventional conventional controllers like PI, PI, PD, PID PID controllers, controllers, with their operating conditions load This paper presents idea ofperformance introducing over a DRaacontrol loop of in fixed However, they are incapable of good dynamic wide fixed gains. gains. However,and they arechanges. incapable of obtaining obtaining good an dynamic performance over wide range range of the traditional LFC and model LFC-DR) using Intelligent for a aa power system. operating conditions load changes. This presents an of DR loop in operating conditions and load(called changes. This paper paper presents an idea ideaController of introducing introducing DR control control loopDR in communication delaymodel latency in theLFC-DR) controllerusing design is considered and for is linearized using Padé the (called Intelligent Controller aa power system. DR DR the traditional traditional LFC LFC model (called LFC-DR) using Intelligent Controller for power system. addition DRthe control loop guarantees of the closed-loop approximation. The delay latency controller design considered and is using Padé communication communication delay latencyof in in the controller design is is stability considered and overall is linearized linearized usingsystem Padé and effectively improves the system dynamic performance. Simulation results show that LQRFuzzy approximation. The addition of DR control loop guarantees stability of the overall closed-loop system approximation. The addition of DR control loop guarantees stability of the overall closed-loop system Logic Controllerimproves based LFC-DR single-area power system have better performance andthat superiority over and the dynamic performance. Simulation results LQRand effectively effectively improves the system system dynamic performance. Simulation results show show that LQR- Fuzzy Fuzzy aLogic classical controller under any operating scenarios. Controller based LFC-DR single-area power system have better performance and superiority over Logic Controller based LFC-DR single-area power system have better performance and superiority over under any operating scenarios. classical controller aa©classical controller under any operating scenarios. 2016, IFAC Load (International Federation of Automatic Control) Hosting Ltd. All rights reserved. Keywords: Frequency control; microgrid; Demandby Elsevier Response; Linear Quadratic Regulator(LQR);Fuzzy Logic Control; padé approximation. Keywords: Load control; microgrid; Keywords: Load Frequency Frequency control; microgrid; Demand Demand Response; Response; Linear Linear Quadratic Quadratic Regulator(LQR);Fuzzy Regulator(LQR);Fuzzy Logic Logic Control; Control; padé padé approximation. approximation. 1. INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION For the stability, sensitivity and security considerations, the frequency a power system a very For the the stability, of sensitivity sensitivity and issecurity security For stability, and important performance signal of system operator [1]. considerations, the frequency frequency ofto aa the power system is aa very very considerations, the power system is Power system frequency signal controlto /regulation haveoperator been one[1]. of important performance signal to the system system operator [1]. important performance the for research, the important control problems been Power system frequency frequency control /regulation /regulation have traditionally Power system control have been one one of of termed as Loadcontrol Frequency Controlfor (LFC). It has traditionally been one of the problems research, the important important control problems for research, traditionally the functions of Automatic Control (AGC)[2].The termed as Frequency Control It been termed as Load Load Frequency Generation Control (LFC). (LFC). It has has been one one of of frequency and tie-line power exchanges are two variables the functions of Automatic Generation Control (AGC)[2].The the functions of Automatic Generation Control (AGC)[2].The which should considered under investigation in load frequency and be tie-line power exchanges exchanges are two two variables variables frequency and tie-line power are frequency control problem [3],[4]. Frequency regulation in which should be considered under investigation in load load which should be considered under investigation in power system is problem achieved [3],[4]. by pondering generation frequency control problem [3],[4]. Frequency regulationand in frequency control Frequency regulation in i.e., spinning and nondemand throughis following, generation and power system system is load achieved by pondering pondering power achieved by generation and spinning The future poweri.e., grid,spinning on the other demand reserves. through load load following, i.e., spinning and hand, nondemand through following, and nonis foreseen to have high incursion of Renewable Energy (RE) spinning reserves. The future power grid, on the other hand, spinning reserves. The future power grid, on the other hand, power generation.[5].The ancillary services like frequency is foreseen to have have high high incursion incursion of Renewable Renewable Energy (RE) Energy (RE) is foreseen to of and voltage control, are essential parts of a power system. power generation.[5].The ancillary services like frequency power generation.[5].The ancillary services like frequency The voltage parameter which is representing balance of and control, are parts system. and voltage control, are essential essential parts of of aathepower power system. generation and consumption a power system frequency The parameter which representing the of The parameter which is is in representing the isbalance balance of demand can be [6]. It is worth noting that generation system is generation and in generation and consumption consumption in aa power power and system is frequency frequency equally the frequency control. and Demand response is [6]. worth that demand can [6]. It It is isapplied worth tonoting noting that generation generation and demand can be be defined as: "changes in electric usage by demand-side equally to control. Demand response equally applied applied to the the frequency frequency control. Demand response is is resources from"changes their normal consumption in response defined in usage by demand-side defined as: as: "changes in electric electric usagepatterns by demand-side to changesfrom in the price of electricity over time, or to resources from their normal consumption patterns in incentive response resources their normal consumption patterns in response payments premeditated to induce lower electricity use at to changes in the price of electricity over time, or to incentive to changes in the price of electricity over time, or to incentive times of high wholesale market prices or when system payments premeditated to induce lower electricity use at at payments premeditated to induce lower electricity use times of of high high wholesale wholesale market times market prices prices or or when when system system
reliability is jeopardized" by Federal Energy Regulatory Commission . In by suchFederal cases, energy and reliability jeopardized" Energy Regulatory reliability is is (FERC)[7] jeopardized" by Federal Energystorage Regulatory responsive loads shows a very great swear for balancing Commission (FERC)[7] . In such cases, energy storage Commission (FERC)[7] . In such cases, energy storage and and generation demand, help to avoid use of responsive loads shows very great swear for balancing swear for the balancing responsive and loads shows asaa they verywill great the traditional schemes, which generation and demand, they to the use of generation and generation demand, as asfollowing they will will help help to avoid avoid the can use be of costly and/or environmentally unfriendly. With the limited the traditional generation schemes, which can the traditional generation following following schemes, which can be be availability, efficiency, andunfriendly. high cost With of large costly environmentally the limited costly and/or and/orlow environmentally unfriendly. With the storage limited real -time smart responsive load participation, devices, and high cost of large availability, low efficiency, availability, low efficiency, and high cost of large storage storage known Demand (DR),it load has actively devices, -time smart participation, devices, asreal real -time Response smart responsive responsive load been participation, considered for power balancing. It can be achieved by active known as Demand Response (DR),it has been actively known as Demand Response (DR),it has been actively consumer involvement in real-time to maintain balance considered for power balancing. It can be achieved by active considered for power balancing. It can be achieved by active between involvement generation in andreal-time demand with two-way consumer involvement in real-time to maintain maintain balance consumer to balance communication [8], [9]. Itand is well known that between generation and demand withDR increases two-way between generation demand with two-way system reliability flexibility, decreases the increases cost of communication [8], and [9]. It It is well well known known that DR DR increases communication [8], [9]. is that system efficiency [10]. The use of operation, and enhances decreases the cost system reliability and flexibility, system reliability and flexibility, decreases the cost of electricity demand response as a new assess for fast reserves operation, and enhances system efficiency [10]. The use operation, and enhances system efficiency [10]. The use of of inside an autonomous (islanded) micro grid for during electricity demand as assess fast reserves electricity demand response response as aa new new assess for fast different reserves operating conditions (islanded) including frequency inside micro grid during duringcontrolled different grid different inside an an autonomous autonomous (islanded) micro disturbance and normal ones[11],[12]. This kind of demand operating conditions including frequency frequency controlled operating conditions including controlled can respondand autonomously to frequency and it disturbance and normal ones[11],[12]. ones[11],[12]. This variation kind of of demand demand disturbance normal This kind provides fast reserve to the system by equipping themand with can respond respond autonomously to frequency frequency variation and it can autonomously to variation it control intelligence frequencyfast system by by equipping equipping them with with provides fastsensors reserve and to the theappropriate provides reserve to system them [13],[14]. results show that, control using thisintelligence approach, frequency Simulation sensors and and appropriate control intelligence frequency sensors appropriate i.e. by introducing Demand Response(DR) control loop to [13],[14]. Simulation results show that, using this approach, [13],[14]. Simulation results show that, using this approach, the traditional LFC model (called LFC-DR), the demand i.e. by introducing Demand Response(DR) control loopside to to i.e. by introducing Demand Response(DR) control loop can make a significant and reliable contribution to major the traditional LFC model (called LFC-DR), the demand side the traditional LFC model (called LFC-DR), the demand side frequency while the advantages, that can make make response significant andpreserving reliable contribution contribution to major major can aa significant and reliable to frequency response response while while preserving preserving the the advantages, advantages, that that frequency
2405-8963 © 2016, IFAC (International Federation of Automatic Control) Copyright 2016 IFAC 567 Hosting by Elsevier Ltd. All rights reserved. Peer review©under responsibility of International Federation of Automatic Control. Copyright 2016 IFAC 567 Copyright © 2016 IFAC 567 10.1016/j.ifacol.2016.03.115
IFAC ACODS 2016 568 P.Srividya Devi et al. / IFAC-PapersOnLine 49-1 (2016) 567–572 February 1-5, 2016. NIT Tiruchirappalli, India
consumers derive from their supply of electric energy. These models are useful in small disturbance studies such as small variations in load and generation, and in controller design. In the past decades, Fuzzy Logic Controllers (FLCs) have been effectively developed for analysis and control of non-linear systems [15], [16]. Power system is a highly non-linear and uncertain system[17],[18].The objective of this study is to investigate the Load Frequency Control with Demand Response (LFCDR) control loop at different operating conditions using the intelligent control techniques associated with LQR(Linear Quadratic Regulator) approach which is already demonstrated[19]. Padé approximation is used which is an important parameter in the system dynamic performance of LFC-DR [20].Taking this feature into consideration, the robust control scheme is designed using LQR-Fuzzy logic. The proposed controller is simulated for a single area power system and was compared with PI (S. Ali Pourmousavi) [19] controller Results of simulation show that the LQR-Fuzzy controllers guarantee the robust performance. 2.
Therefore; the only obstacle for DR is communication delay, known as latency, which could affect the system dynamic performance. There are various methods available for Input-Output Linearization Problem (IOLP) for a class of single-input-single-output nonlinear systems with delays .In order to linearise the communication delay latency, Padé approximation is used which is explained in the next section. 3.
State-space representation of the LFC model is a useful tool for the application of modern/robust control theory. For creating a general framework of LFC in dynamic frequency analysis this type of representation can be conveniently modified and applied to power system of any size. Therefore deriving the dynamic model of the power system, including DR in the state-space representation, in order to study the effect of DR on LFC performance and controller design. The proposed LFC-DR model of Fig.1. is based on a simplified power system model with a non-reheat steam turbine. The state-space realization of a single-area power system with DR [19] (shown in Fig.1.) is given by equations
SYSTEM MODEL
In general the power balance equation in the frequencydomain for low-order linearized power system LFC model for the purpose of frequency control and analysis is given by [1], [6]: ΔPT (s)-ΔPL(s) =2H.s.Δf(s) +D. Δf(s)
.
(3)
X ( t ) x ( t ) u ( t ) w (t ) y (t ) C x ( t ) 1 2H 0 1 RTg 0 0 0 0 0
(1)
The modified block diagram of single area power system(Thermal) with the consideration of Demand Response (DR) control loop for load frequency control, with communication delay latency is shown in Fig.1. Since DR performs like spinning reserve in magnitude and power flow direction, i.e., once frequency deviation becomes negative (positive), it is required to turn OFF (ON) a portion of the responsive loads for ancillary services (i.e., DR), Equation (1) can be modified as: ΔPT (s)-ΔPL(s) +ΔPDR=2H.s.Δf(s) +D. Δf(s)
STATE-SPACE DYNAMICAL MODEL FOR LFC-DR
1 30 0 2H 2H.Td 23 1 1 0 Tt Tt 1 0 0 Tt 30 0 0 Td 0 0 128
0
0
0
0
105 105 Td2 25 Td3 28 0 0
0 0
0 0
64 0
0 32
0
0
0
0
0
T
The power utilization status of controllable loads can be changed instantaneously by the demand signal they receive. Unlike the usual spinning reserve-provider power plants, there is no ramp up and down limitations on the DR resources.
105 0 2HTd3 211
0 0
0 0 1 0 2H
(2)
1 Tg
0
0
0
0
0
16 0
0
0
0 0
CT 1 0 0 0 0 0 0 0 1 T 2H
0
0
0
0
945 2HTd5 220 0 0 0 0 945 945 Td4 214 Td5 217 0 0 0 0 0 0 16 0
0
0
0
0
D= [0] where A - System matrix, B -control input matrix, - Disturbance matrix, X - State vector,u (t) - input vector,C- Observation matrix, - Disturbance variable, w (t)- observation matrix, Y- System output. In order to derive the linear state-space model of the system, it is required to have a linear model of the system under study. From Fig. 1, it can be seen that the system has only one nonlinear element which is the time delay in the DR control loop. Therefore, we need to linearize the time delay for derivation of the linear state-space model. Padé approximation is used for linearizing the DR time delay.
. Fig.1. Block Diagram representation of single area power system 568
IFAC ACODS 2016 February 1-5, 2016. NIT Tiruchirappalli, India P.Srividya Devi et al. / IFAC-PapersOnLine 49-1 (2016) 567–572
4.
controller such that the performance index is minimized for the system given in equation (3). With the modified state space equations, ensuring the system matrix is controllable.
PADÉ APPROXIMATION
In order to linearize systems with time delays in control engineering with very strong and successive convergent results Padé approximation is widely used [20]. Among the many methods Padé approximations are the most frequently used methods to approximate a dead-time by a rational function. It basically approximates time delays by a quotient of polynomials. Classical control system theory provides the basic relation, but usually only for an approximation with equal numerator and denominator degree are most widely recommended.
y C
Fuzzy logic controllers are rule-based systems. The essential part of the FLC system is a set of Fuzzy Control Rules (FCRs) related by means of a fuzzy implication and the compositional rule of inference. Since power system dynamic characteristics are complex and variable, conventional control methods cannot provide the desired results. Intelligent controllers can be replaced with conventional controllers to get fast and good dynamic response in load frequency control problems. If the system robustness and reliability are more important, fuzzy logic controllers can be more useful in solving a wide range of control problems and in nonlinear system applications.The basic configuration of a fuzzy-logic control is composed of four principle components: a fuzzification, a knowledge base, a inference engine, and defuzzification. The fuzzifier maps the input crisp values into fuzzy variables using normalized membership functions and input gains. The fuzzy logic inference engine then infers the proper control action based on the available rule-base. The fuzzy control action is translated to the proper crisp value through the defuzzifier using normalized membership functions and output gains. The block diagram of a fuzzy logic system is shown Fig.2.
(5)
where =
m
( m n k )!m!
(m n)!k!(m k )! ( s.T
d
)k
d
)k
k 0
Qn (e sTd )
=
n
(m n k )!n!
(m n)!k!(n k )! (s.T k 0
(6) (7)
‘P’ and ‘Q’ are the polynomials of order ‘m’ and ‘n’, respectively. It is usually common for the numerator and denominator of the approximation fractional functions to have the same order, and the order usually varies between 1 and 10.The 5th-order Padé approximation is acceptable and is used in this study, Since the cut-off frequency of the low pass filters, i.e., speed-governor and turbine, in the model of the power system are usually less than 15 rad/sec. The magnitudes of all orders of Padé approximation in the frequency domain have also been compared to that of pure time delay. 5.
(8)
B. Fuzzy logic controllers
It is as follows:
Pm (e sTd )
0 w
The implementation of LQR method with required state and control matrices and R (scalar quantity) is well presented in [19].
(4)
Rmn ( s.Td ) Pm (e sTd ) / Qn (e sTd )
^ 0 X ^ u 0 x1 0 X 0 x1
X x1 C
The Padé function for the time delay functions
e sTd Rmn ( s.Td )
569
CONTROLLER DESIGN FOR LFC-DR MODEL
A. General approach using LQR Design for the LFC-DR model Numerous different classical and modern control theories have been utilized for the LFC problem. In this section, a general controller design (LQR) approach for the LFC problem with the DR control loop is presented. The theory of optimal control is concerned with operating a dynamic system at minimum cost, where the system dynamics are described by a set of linear differential equations and the cost is described by a quadratic function is called the LQ problem. The solution is provided by the Linear-Quadratic Regulator (LQR). Optimal control in state space is centered around the Riccati Equation with state variable functions that has to be solved to yield the control law or trajectory. The simplified version of the LQR problem is to design the 569
Fig. 2. Block Diagram representation of a Fuzzy Logic System The two normalized input variables, error e and change in error Δe are inputs of FLC, are first fuzzified by T1 fuzzy sets. Two inputs signals are converted to fuzzy numbers first in fuzzifier using three Triangular membership functions, named as Positive Big (B), Positive Small(S), and Zero (Z)). Finally resultant fuzzy subsets representing the controller output are converted to the crisp values using the Central Of Area (COA) defuzzifier scheme. The rules for the controller design are shown in the Table I with error (e) and
IFAC ACODS 2016 570 P.Srividya Devi et al. / IFAC-PapersOnLine 49-1 (2016) 567–572 February 1-5, 2016. NIT Tiruchirappalli, India
change in error (Δe) are applied and the robust performance for the proposed model can be achieved.
DR control loop of the system are modified and governed by the below equation: (16)
(1-α)G(s)+α.H(s) TABLE I CONTROL RULES FOR FUZZY LOGIC CONTROLLER (e)
S
Z
B
B
S
S
S
Z
S
S
S
B
where 08
0.019
LFC-DR ,α=0.8,LQRFuzzy LFC-DR, α=0.1,LQR LFC-DR, α=0.1,LQRfuzzy control
>6
0.005
>7
0.009
4
0.0025
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From the above simulation studies it is seen that the proposed method is the best and robust. It can also be applied and seen at different perturbations using different classical controllers as well as with proposed controllers for LFC-DR model. 8.
Syst., vol. 20, no. 1, pp. 346–357, Feb. 2005. [12] Karnavas YL, Papadopoulos DP. “AGC for autonomous power system using combined intelligent techniques.” Electr Power Syst Res 2002;62:225–39. [13] Indulkar CS, Raj B. Application of fuzzy controller to automatic generation control. Electr Mach Power Syst 1995;23:209–20. [14] Chown GA, Hartman RC. Design and experiment with a fuzzy controller for automatic generation control (AGC). IEEE Trans Power Syst 1998;13:965–70 [15] D.V.Pushpalatha, K.R.Sudha and A.Satya Devi “Design of Adaptive Fuzzy Controller for a Robot gripper” proceedings of IEEE Advances in Recent Technologies in Communication and Computing, ARTCom '09. International Conference on 27-28 Oct. 2009 , pp 254 – 256. [16] K.R.Sudha, R.VijayaSanthi ”Robust Decentralized Load Frequency Control of Interconnected Power System with Generation Rate Constraint using Type-2 Fuzzy approach” International Journal of Electrical Power and Energy Systems, Vol. 33, pp. 699–707, Feb 2011 [17] S. A. Pourmousavi and M. H. Nehrir, “Real-time central demand response for primary frequency regulation in microgrids,” IEEE Trans. Smart Grid, vol. 3, no. 4, pp. 1988–1996, Dec. 2012. [18] D Jay, KS Swarup, “Dynamic demand response and control in smart grid environment” India Conference (INDICON), 2011 Annual IEEE, 16-18 Dec. 2011 [19] S. Ali Pourmousavi,M. Hashem Nehrir, “Introducing Dynamic Demand Response in the LFC Model” IEEE Trans. Power Syst., vol. 29, no. 4, pp. 1562–1572, June2014 [20] R. C. Dorf and R. H. Bishop, Modern Control Systems, 7th ed. New York, NY, USA: AddisonWesley, 1995, p. 807.
CONCLUSIONS
In this paper, a new methodology for solving LFC problem with DR loop using fuzzy controller and LQR-Fuzzy for a single-area power system has been proposed. While designing the controller, DR communication delay latency is considered. It is linearized using Padé approximation. To demonstrate the robustness of the proposed controller, settling time and undershoot of the system are being considered (shown in Table-IV). In future scope, the simulation studies can be carried out with the proposed controller strategy for different communication latencies and can also be extended to LFC-DR in multi-area power systems under restructured schemes. 9. REFERENCES [1] P. Kundur, Power system Stability and Control. New York, NY, USA: McGraw-Hill, 1994, ch. 11 [2] Rahul Umrao,Sanjeev Kumar, Man Mohan, D.K.Chaturvedi, “Load Frequency Control Methodologies for Power System” International Conference on Power, Control and Embedded Systems [3] K.Okada, R.Yokoyama, G.Shirai, H.Sasaki, "Decentralized Load Frequency Control with Load demand Prediction in Multi-Area Power systems," IEEE 2000 [4] Mohyi el-din Azzam, "An Optimal Approach to Robust Controller for Load-Frequency Control," IEEE 2002 [5] T. Hiyama, “Design of decentralized load-frequency regulators for interconnected power systems,” Proc. IEEE Generation, Transmission, Distribution Conf., vol. 129, no. 1, pp. 17–23, 1982. [6] H. Bevrani, Robust Power System Frequency Control. New York, NY, USA: Springer, 2009, ch. 1–3. [7] Federal Energy Regulatory Commission (FERC):http://www.ferc.gov/industries/electric/indu sact/demand-response/dem-res-adv-metering.asp [8] Mojtaba khederzadeh “Frequency Control of Microgrids by Demand Response” CIRED Workshop - Lisbon 29-30 May 2012 [9] S. A. Pourmousavi and M. H. Nehrir, “Demand response for smart microgrid: Initial results,” in Proc. 2nd IEEE PES Innov. Smart Grid Technol. (ISGT), Anaheim, CA, USA, 2011, pp. 1–6. [10] J. Medina, N. Muller, and I. Roytelman, “Demand response and distribution grid operations: Opportunities and challenges,” IEEE Trans.Smart Grid, vol. 1, no. 2, pp. 193–198, Sep. 2010 [11] Ibraheem, P. Kumar, and D. P. Kothari, “Recent philosophies of automatic generation control strategies in power systems.” IEEE Trans. Power 572
