- Email: [email protected]
Hybrid Consensus-based Algorithm for Distributed Economic Dispatch Problem*
Proceedings of of the the 20th 20th World World Congress Congress Proceedings Proceedings of the 20th World The Federation of Automatic The International International Federation of Congress Automatic Control Control Proceedings of the 20th World The International Federation of Congress Automatic Control Toulouse, France, July 9-14, 2017 Available online at www.sciencedirect.com Toulouse, France, July 9-14, 2017 The International Federation of Automatic Control Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017
ScienceDirect
IFAC PapersOnLine 50-1 (2017) 177–182
Hybrid Consensus-based Algorithm for Hybrid Consensus-based Algorithm for Hybrid Consensus-based Algorithm for ⋆⋆ Distributed Economic Dispatch Problem Distributed Economic Dispatch Problem ⋆ Distributed∗,∗∗Economic Dispatch Problem ∗,∗∗ ∗∗∗,∗∗∗∗ Xiao-Kang Xiao-Kang Xiao-Kang Xiao-Kang
∗,∗∗ Huaicheng Yan ∗∗∗,∗∗∗∗ Liu Yan-Wu Liu ∗,∗∗ Yan-Wu Wang Wang Huaicheng Yan ∗∗∗,∗∗∗∗ ∗,∗∗ ∗,∗∗ ∗,∗∗ Yan-Wu Wang ∗,∗∗ Huaicheng ∗,∗∗ ∗,∗∗ Liu Yan ∗∗∗,∗∗∗∗ ∗,∗∗ Xiaoya ∗,∗∗ Xiaoping Wang Hu ∗,∗∗ ∗,∗∗ Xiaoping WangWang Xiaoya Hu Liu Yan-Wu Huaicheng Yan ∗∗∗,∗∗∗∗ ∗,∗∗ ∗,∗∗ ∗,∗∗ Xiaoping Wang ∗,∗∗ Xiaoya Hu ∗,∗∗ ∗,∗∗ Xiaoping Wang Xiaoya Hu ∗ ∗ School of Automation, Huazhong University of Science and School of Automation, Huazhong University of Science and ∗ ∗ School of Automation, Huazhong University of Science and Technology, Huazhong Wuhan, 430074, 430074, China ∗ Technology, Wuhan, China School of Automation, University of Science and ∗∗ ∗∗ Key Laboratory Technology, Wuhan, 430074, China of Image Processing and Intelligent Key Laboratory of Image Processing and Intelligent Control Control Technology, Wuhan, 430074, ∗∗ ∗∗ Key Laboratory of Image Processing andChina Intelligent Control (Huazhong University of and Ministry of ∗∗ (Huazhong University of Science Science and Technology), Technology), Ministry of Key Laboratory of Image Processing and Intelligent Control (Huazhong University of Science and Technology), Ministry of Education Education (Huazhong University of Science and Technology), Ministry of ∗∗∗ Education ∗∗∗ Key Laboratory Laboratory of of Advanced Advanced Control and and Optimization Optimization for for Key Control Education ∗∗∗ ∗∗∗ of Control Optimization Chemical of Education ∗∗∗ Key Laboratory Chemical Process Process of Ministry Ministry ofand Education Key Laboratory of Advanced Advanced Controlof and Optimization for for ∗∗∗∗ ∗∗∗∗ SchoolChemical Process of Ministry of Education of Information Science and Engineering East China School of Information Science and Engineering East China Chemical Process of Ministry of Education ∗∗∗∗ ∗∗∗∗ School of Information Science and Engineering East China University of ofScience Science and Technology Technology 130Engineering Meilong Road, Road, Shanghai ∗∗∗∗ University and 130 Meilong Schoolof Information Science and EastShanghai China University of Science and Technology 130 Meilong Road, 200237, P. R. China 200237, P. R. China University of Science and Technology 130 Meilong Road, Shanghai Shanghai 200237, 200237, P. P. R. R. China China Abstract: Economic dispatch is one of the fundamental Abstract: Economic dispatch is one of the fundamental problems problems in in the the power power system system research. research. Abstract: Economic dispatch is one of the fundamental problems in the power system research. The existing algorithms are either discrete iterative algorithms or continuous-time dynamical The existingEconomic algorithms are either discrete iterative algorithms orincontinuous-time dynamical Abstract: dispatch is one of the fundamental problems the power system research. The existingBy algorithms are hybrid either discrete iterative algorithms or continuous-time algorithms. virtue of of the the techniques, hybrid consensus-based algorithm is isdynamical proposed algorithms. virtue techniques, aa hybrid consensus-based algorithm proposed The existingBy algorithms are hybrid either discrete iterative algorithms or continuous-time dynamical algorithms. By virtue of the hybrid techniques, a hybrid consensus-based algorithm is proposed in this paper, which consists of a finite-time continuous-time algorithm and a discrete-time in this paper, which consists of a finite-time continuous-time algorithm and a discrete-time algorithms. By virtue of the hybrid techniques, a hybrid consensus-based algorithm is proposed in this which consists a algorithm discrete-time interacted scheme. Moreover, inspired by cluster and the of interacted scheme. Moreover, inspired by the the continuous-time cluster synchronization synchronization andand theaabenefit benefit of the the in this paper, paper, whichMoreover, consists of of a finite-time finite-time continuous-time algorithm and discrete-time interacted scheme. inspired by the cluster synchronization and the benefit of the projection operator, a supervised strategy is carried out to ensure that each generator unit will projection operator, a supervised strategy isthe carried outsynchronization to ensure that each generator unit will interacted scheme. Moreover, inspired by cluster and the benefit of the projection supervisedconstraint. strategy isFor carried out understanding to ensure that of each unit will work within withinoperator, its power powera generation better thegenerator supervised hybrid work its aa better the supervised hybrid projection operator, ageneration supervisedconstraint. strategy isFor carried out understanding to ensure that of each generator unit will work within its power generation constraint. For a better understanding of the supervised hybrid distributed algorithm, the detailed pseudo-code is presented. Finally, a numerical example is distributed the detailed pseudo-code presented. Finally, of a numerical example is work within algorithm, its power generation constraint. For a is better understanding the supervised hybrid distributed algorithm, the detailed pseudo-code is presented. Finally, a numerical example is given to illustrate the effectiveness of the algorithm. given to illustrate the effectiveness ofpseudo-code the algorithm. distributed algorithm, the detailed is presented. Finally, a numerical example is given to illustrate the effectiveness of the algorithm. given illustrate the effectiveness algorithm. © 2017,toIFAC (International Federation of of the Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: distributed optimization, Keywords: economic economic dispatch, dispatch, hybrid hybrid system, system, consensus consensus protocol, protocol, distributed optimization, Keywords: economic dispatch, hybrid system, consensus protocol, distributed smart grid. smart grid.economic dispatch, hybrid system, consensus protocol, distributed optimization, Keywords: optimization, smart grid. smart grid. 1. incremental 1. INTRODUCTION INTRODUCTION incremental cost cost consensus consensus algorithm algorithm which which selected selected the the 1. INTRODUCTION consensus algorithm which selected the incremental cost of each generation unit as the consensus cost of each generation unit as the consensus 1. INTRODUCTION incremental consensus algorithm which selected the cost of each generation unit as the consensus Economic variable and the EDP was solved in a discrete manner. Economic Dispatch Dispatch Problem Problem (EDP) (EDP) has has received received considconsid- incremental variable andcost the of EDP was solved in a discrete manner. incremental each generation unit as the consensus Economic Dispatch (EDP) receivedof and the EDP was solved in a discrete manner. erable in recent The objective EDP They subsequently extended the result result to the the case case that erable attention attention in Problem recent years. years. Thehas objective ofconsidEDP variable They subsequently extended the to that Economic Dispatch Problem (EDP) hasobjective receivedof considvariable and the EDP was solved in a discrete manner. erable attention in recent years. The EDP They subsequently extended the result to the case that in power system is to minimize the overall consumption considering the power generation constraints and pointed in power system in is to minimize the overall consumption considering the power generation constraints and pointed erable attention recent years. The objective of EDP They subsequently extended the constraints result to the case that in system is to minimize the overall consumption considering the power generation and pointed of power generation while meeting the power demand of out the convergence rate relies on the topology of of power generation while meeting the power demand of considering out the convergence relies on the topology of the the in system is to minimize thethe overall consumption the power rate generation constraints and pointed of power generation while meeting power demand of out the convergence rate relies on the topology of the loads and the constraints. Generally, EDP is (Zhang Chow (2012)). The work loads andgeneration the operational operational constraints. Generally, EDP of is network network (Zhang and and rate Chowrelies (2012)). Thetopology work in inofYang Yang of power while meeting the power demand out the convergence on the the loads and the operational constraints. Generally, EDP is network (Zhang and Chow (2012)). The work in Yang formulated as a nonlinear optimization problem. To solve et al. (2013) (2013) proposed fully (2012)). distributed algorithm where formulated as aoperational nonlinear optimization problem. To solve et al. proposed aa fully distributed algorithm where loads and the constraints. Generally, EDP is network (Zhang and Chow The work in Yang formulated a nonlinear optimization problem. Tobased solve generators et al. (2013)were proposed fully distributed algorithm where the various algorithms have been proposed able learn the the problem, problem,as various algorithms have been proposed based generators able to toaa cooperatively cooperatively learn the mismatch mismatch formulated asvarious a nonlinear optimization problem. Tobased solve et al. (2013)were proposed fully distributed algorithm where the problem, algorithms have been proposed generators were able to cooperatively learn the mismatch on convex optimization techniques or heuristic techniques, between demand and total power generation. Loia on convex optimization techniques or been heuristic techniques, between demand andtototal power generation. Loia and and the problem, various algorithms have proposed based generators were able cooperatively learn the mismatch on convex optimization techniques or heuristic techniques, demand and total power generation. Loia and such as method and (2003)), Vaccaro designed aa distributed and such as Newton-Raphson Newton-Raphson method (Chen (Chen and Chen Chen (2003)), between Vaccaro (2014) (2014) designed distributed and self-organizing self-organizing on convex optimization techniques or heuristic techniques, between demand and total power generation. Loia and such as Newton-Raphson method (Chen and Chen (2003)), (2014) a distributed andconsensus self-organizing genetic algorithm and (2009)), parframework baseddesigned on the the weighted weighted average consensus algogenetic algorithm (Amjady (Amjady and Nasiri-Rad Nasiri-Rad (2009)), par- Vaccaro framework based on average algosuch as Newton-Raphson method (Chen and Chen (2003)), Vaccaro (2014) designed a distributed and self-organizing genetic algorithm (Amjady and et Nasiri-Rad (2009)), ticle optimization (Park al. and on. framework based on weighted average algonetwork. light the ticle swarm swarm optimization (Park et al. (2010)), (2010)), and so soparon. rithm rithm of of agent agent network. Similarly in light of ofconsensus the consensus consensus genetic algorithm (Amjady and et Nasiri-Rad (2009)), parframework based on the the Similarly weighted in average consensus algoticle swarm optimization (Park al. (2010)), and so on. However, these conventional methods for EDP commonly rithm of agent network. Similarly in light of the consensus method, a distributed consensus-like iterative bisection However, these conventional methods for EDP and commonly method, a distributed consensus-like iterative bisection ticle swarm optimization (Park et al. (2010)), so on. rithm of agent network. Similarly in light of the consensus However, conventional methods for EDP commonly awas distributed iterative require control center has to global algorithm proposed consensus-like in Xing Xing et et al. al. (2015) (2015) withbisection no need need require aa these control center that that has access access to the the global method, algorithm proposed in with no However, these conventional methods for EDP commonly method, awas distributed consensus-like iterative bisection require a control center that has access to the global algorithm was proposed in Xing et al. (2015) with no(2014) need information of all the generator units. Unfortunately, the for a data fusion center or a leader. Binetti et al. (2014) information of all the generator units. Unfortunately, the for a data fusion center or a leader. Binetti et al. require a control center that has access to the global algorithm was proposed in Xing et al. Binetti (2015) with no(2014) need information of all the generator units. Unfortunately, the for a data fusion center or a leader. et al. centralized structure could have some drawbacks that limit proposed a framework of two consensus algorithms running centralized structure could have some drawbacks that limit proposed a fusion framework of two consensus algorithms running information of all thecould generator units.drawbacks Unfortunately, the for a data center or a leader. Binetti et al. (2014) centralized structure have some that limit proposed a framework two consensus algorithms running its especially when aa large scale of where first one the its wider wider application, application, especially when largethat scale of in in parallel, parallel, where the theof first one ensured ensured the generationgenerationcentralized structure could have some drawbacks limit a framework two consensus algorithms running its wider application, especially when a large scale of proposed in parallel, where and theofthe first one ensured the generationdistributed energy resources are taken into consideration demand equality and the second one compensated the distributed energy resources are taken into consideration demand equality second one compensated the its wider application, especially when a large scale of in parallel, where the first one ensured the generationdistributed energy resources arecenter takendevices. into consideration power demand equality Based and the second one compensated the or failure occurs mismatch. on the finite-time consensus techor when when some some failure occurs on on center devices. power mismatch. Based on the finite-time consensus techdistributed energy resources are taken into consideration demand equality Based and the second one compensated the or when some failure occurs on center devices. power mismatch. on the finite-time consensus techniques, Guo et al. (2016) designed a iterative scheme niques,mismatch. Guo et al. (2016) designed a iterative scheme or when some failure occurs on center Based on the finite-time consensus techRecently, distributed techniques have been Recently, distributed techniques have devices. been developed developed to to power niques, et al. unit (2016) designed a iterative scheme generator which decomposed the for each eachGuo generator which decomposed the centralcentralRecently, distributed have networks been developed to for niques, Guo et al. unit (2016) designed a iterative scheme solve in light lighttechniques of and solve the the EDP EDP in of multi-agent multi-agent networks and conconfor each generator unit which decomposed the centralRecently, distributed techniques have been developed to ized optimization into optimizations at local agents, as ized optimization into optimizations at local agents, as solve the EDP in light of multi-agent networks and confor each generator unit which decomposed the centralsensus methods. Zhang and Chow (2011) proposed the sensusthe methods. Zhang and Chow (2011) proposed the aized solve EDP in Zhang light ofand multi-agent networks and conoptimization into optimizations at local agents, result the finite-time convergence to the optimal value a result the finite-time convergence to the optimal value sensus methods. Chow (2011) proposed the ized optimization into optimizations at local agents, as as ⋆ ⋆ This work is supported by the National Natural Science Founsensus methods. Zhang and Chow (2011) proposed the aawas result the convergence to was achieved. Event-triggered communication scheme was This work is supported by the National Natural Science Founachieved. Event-triggered communication schemevalue was result the finite-time finite-time convergence to the the optimal optimal value ⋆ This work is supported by the National Natural Science Foundation of China under Grants 61374171, 61572210, 51537003 and was achieved. Event-triggered communication scheme was carried out Li which the dation China under Grants 61374171, 51537003Founand ⋆ carried out in in Event-triggered Li et et al. al. (2016), (2016),communication which greatly greatly reduced reduced the This of work is supported by the National61572210, Natural Science was achieved. scheme was 61673178, the Research Funds for Univerdation of China under Grants 61374171, 51537003 and carried out in Li et al.requirements (2016), whichin greatly reduced the 61673178, the Fundamental Fundamental Research Funds61572210, for the the Central Central Univerinformation exchange requirements in the process of dation of China under Grants 61374171, 61572210, 51537003 and information exchange the process of the carried out in Li et al. (2016), which greatly reduced 61673178, the Fundamental Research Funds for the Central Universities (2015TS030). sities (2015TS030). information exchange requirements in the process of cooperative optimization. Of note is that the above liter61673178, the Fundamental Research Funds for the Central Univercooperative optimization. Of note is that the above literinformation exchange requirements in the process of the the sities (2015TS030). E-mail addresses: [email protected] [email protected] (Yan-Wu (Yan-Wu Wang). Wang). E-mail addresses: cooperative sities (2015TS030). E-mail addresses: [email protected] (Yan-Wu Wang). cooperative optimization. optimization. Of Of note note is is that that the the above above literliter-
E-mail addresses: [email protected] (Yan-Wu Wang). Copyright © 2017, 2017 179 Copyright 2017 IFAC IFAC 179 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright ©under 2017 responsibility IFAC 179Control. Peer review of International Federation of Automatic Copyright © 2017 IFAC 179 10.1016/j.ifacol.2017.08.030
Proceedings of the 20th IFAC World Congress 178 Xiao-Kang Liu et al. / IFAC PapersOnLine 50-1 (2017) 177–182 Toulouse, France, July 9-14, 2017
atures solve the EDP in a discrete-time iterative manner, besides, distributed continuous-time dynamical algorithms have been developed (Yu et al. (2015); Chen et al. (2015); Chen et al. (2017)), which analytically solve the EDP in a continuous-time manner with the rigorous stability analysis and optimality analysis. In light of the discrete-time and continuous-time algorithms for EDPs, a supervised hybrid distributed consensus-based algorithm is proposed in this paper. The hybrid algorithm integrates a finite-time continuoustime dynamical algorithm with a discrete-time interacted scheme. Specifically, each generator unit adopts the continuous-time algorithm locally, which enables the incremental cost to meet the first-order optimality condition at the settling time. As for the discrete-time scheme, generator units will exchange information at time spots and make consensus. Motivated by the cluster synchronization behavior and the benefit of the projection operator, a supervised strategy is designed at the discrete-time level to maintain power generation constraints. Once a generator unit breaks out the boundary of the constraint, the projection will work to restore its power generation to the boundary value. Subsequently, the generator unit will be treated as an isolated node in the network, where it still serves as an intermediary communication link, but no longer takes any influence from its neighbors. With the help of analysis methods for hybrid systems, a rigorous stability analysis and optimality analysis is given in the case that no power generation constraints are taken into consideration. In aid of the supervised strategy, a detailed pseudo-code for the EDP with power constraints is presented. Finally, a simulation example illustrates the effectiveness of our algorithms. The rest of the paper is organized as follows. Some preliminaries of algebraic graph theory and matrix theory, as well as problem statement are given in Section 2. The hybrid distributed algorithms for EDP with or without power generation constraints are proposed in Section 3. The illustrative numerical example is given in Section 4. Finally, conclusions are drawn in Section 5. 2. PRELIMINARIES AND PROBLEM STATEMENT In this section, some preliminaries of algebra graph theory are given, as well as description and definition of the economic dispatch problem. 2.1 Graph theory A graph G = (V, E, A) is described with a set of nodes V = {1, 2, . . . , N}, a set of edges E ∈ V × V and a weighted adjacency matrix A = (aij )N ×N with nonnegative adjacency elements. Node i represents the ith system, where an edge eji in graph is denoted by the ordered pair nodes {j, i}. {j, i} ∈ E if and only if the ith system can obtain the information from the jth system. Define the adjacency matrix A = (aij )N ×N associated with G such that aij > 0 if {j, i} ∈ E, and aij = 0 otherwise. We assume there is no self-loop in the graph G, i.e., aii = 0. Define Ni = {j|aij > 0}. The Laplacian matrix L = (lij )N ×N of graph G is defined as lij = −aij if � i �= j, otherwise lij = N k=1,k�=i aik .
180
2.2 Economic dispatch problem Consider a power grid with N generators labeled by IG = {1, 2 · · · , N }, where the cost function of ith power generation is modelled as (Wood and Wollenberg (2012)) (1) Ci (Pi ) = γi Pi2 + βi Pi + αi , where Pi is the output of the ith power generator, and αi , βi , γi are the corresponded parameters. The objective of economic dispatch is to minimize the total cost of the power grid while meeting the demand and power generators constraints, which is formulated as an optimization problem: N �
min
Pi , i∈IG
Ci (Pi ),
(2)
Pi = PD ,
(3)
i=1
N �
s.t.
i=1
(4) Pimin ≤ Pi ≤ Pimax , i ∈ IG max min and Pi are the where PD is the demand power, Pi upper and lower bounds of the ith generator. Define the lagrangian operator L(P1 , . . . , PN ) such that � � N N � � L(P1 , . . . , PN ) = Pi , (5) Ci (Pi ) + λ PD − i=1
i=1
where λ is the Lagrange multiplier.
For the EDP without power generation constraints, based on the first-order optimality condition, it can be obtained that the incremental cost is equal to λ, i.e., ∂Ci λ= = 2γi Pi + βi . (6) ∂Pi By (3), the values of λ and optimal power output Pi∗ can be calculated in a centralized way, � βi PD + ni=1 2γ λ − βi i �n λ= , Pi∗ = . (7) 1 2γi i=1 2γi
For the EDP with power generation constraints, there exists the optimal incremental cost λ∗ satisfies the following relationship: 2γi Pi + βi = λ∗ , for Pimin < Pi < Pimax 2γi Pi + βi < λ∗ , for Pi = Pimax (8) 2γi Pi + βi > λ∗ , for Pi = Pimin 3. DISTRIBUTED HYBRID ALGORITHM FOR ECONOMIC DISPATCH
In this section, a distributed hybrid algorithm is first presented to solve the EDP without power generator constraints, and the optimal property as well as convergence is strictly proved. Then, based on the distributed hybrid algorithm, a projection operator is introduced to solve the EDP with power generator constraints. 3.1 EDP without power generator constraints Define the discrete time sequence σ = {t1 , t2 , . . .} such that 0 = t0 < t1 < t2 < · · · , and tk+1 − tk > τ , k = 0, 1, 2, . . .,
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Xiao-Kang Liu et al. / IFAC PapersOnLine 50-1 (2017) 177–182
where τ > 0 is the smallest time interval of σ. Then, the distributed hybrid algorithm is designed as Pi (t) = ηi (t) + Pi0 , � �1 � βi − λi �� 2 ˙ � λi (t) = µγi �Pi + 2γi � βi − λi · sign(Pi + ), t ∈ [tk , tk+1 ), (9) � 2γi λi (t+ ) = λi (tk ) + 2ǫγi aij (λj (tk ) − λi (tk )) , k j∈Ni tk+1 ), η˙i (t) = 0, t ∈ [tk , � + aij (λj (tk ) − λi (tk )) , ηi (tk ) = ηi (tk ) + ε j∈Ni ηi (0) = 0, i ∈ IG , where µ and ǫ are positive gain parameters, ηi is the auxiliary variable and λi denotes the estimate of the optimal incremental cost�for ith generator, Pi0 is a virtual initial value satisfying ni=1 Pi0 = PD . Note that limt→t− λi (t) = k λi (tk ), i.e., λi (t) is left continuous at time tk . Remark 1. The distributed hybrid algorithm contains two parts, continuous dynamical algorithm and discrete iterative algorithm, in which the continuous part takes finitetime dynamical algorithm while the discrete part takes consensus algorithm. In some sense, controller (9) is kind of like the impulsive controller for the multi-agent network (Wang et al. (2014)). Compared with Yu et al. (2015), Chen et al. (2015) and Chen et al. (2017), our distributed hybrid algorithm only requires information exchange at time spots instead of on the whole time interval. �N Remark 2. It is worthy noting that i=1 ηi (t) = 0, where �N �N it follows i=1 Pi (t) = i=1 (Pi0 ) = PD , t > 0. Thus, our distributed algorithm will not break the demand equality constraint (3) during the process. Lemma 1. (Bhat and Bernstein (2000)). Consider the continuous function x˙ = f (x) with f (0) = 0. Suppose that there are C 1 function V (x) defined on a neighborhood of the origin, and real numbers c > 0 and 0 < α < 1, such that (1)V (x) is position definite; (2)V˙ (x) + cV α (x) ≤ 0. Then, the origin is locally finite-time stable, and the settling time, depending on the initial state x(0) = x0 , satisfies T (x0 ) ≤ (V 1−α (x0 )/c(1 − α)). Lemma 2. (Olfati-Saber et al. (2007)). Consider a network with the topology G applying the distributed consensus algorithm x(k + 1) = (I − ǫL)x(k), where 0 < ǫ < 1/∆ and ∆ is the maximum degree of network. Let G be a strongly connected digraph. Then, a consensus is asymptotically reached for all initial states. Furthermore, the final consensus value is given by rˆx(0), where rˆ is the normalized left eigenvector of (I − ǫL) associated with the one eigenvalue. Theorem 3. Let the communication network G is an undirect connected graph, then the distributed hybrid algorithm (9) solves the EDP (2-3) if 0 < ǫ < 1/(maxi {2γi ∆i }), where ∆i is the in-degree of node i. Proof. Motivated by Wang et al. (2005) and Chen et al. (2015), select the following Lyapunov function �2 � βi − λi Vi (t) = Pi + . (10) 2γi Note that the dynamic of λi is piecewise continuous, where it jumps at time instants tk , k = 0, 1, 2, · · · . Hence, take the derivative of V (t) on the time interval [tk , tk+1 ), 181
179
�� ˙ � � βi − λi −λi ˙ Vi (t) = 2 Pi + 2γi 2γi �1 � �� βi − λi �� βi − λi �� 2 = −µ Pi + �Pi + 2γi � 2γi � � βi − λi · sign Pi + 2γi �3 � � 3 βi − λi �� 2 = −µV 4 (t). = −µ ��Pi + 2γi �
(11)
At the time spots tk , k = 0, 1, 2, · · · , by (9), one has + Pi (t+ k ) − Pi (tk ) = ηi (tk ) − ηi (tk ) � aij (λj (tk ) − λi (tk )) =ǫ j∈Ni
1 = (λi (t+ k ) − λi (tk )), 2γi
(12)
hence, �2 � βi − λi (t+ + k) Pi (tk ) + 2γi �2 � βi − λi (tk ) = Vi (tk ). = Pi (tk ) + 2γi
Vi (t+ k)=
(13)
3
Combine (11) and (13), it follows that V˙ i (t) + µVi 4 (t) ≤ 0 for t ∈ [0, +∞). By Lemma 1, it can be obtained that Vi (t) convergences to zero in finite time with the settling time �1 � � βi − λi0 �� 2 Ti0 ≤ µ ��Pi0 + , (14) 2γi � where λi0 is the initial values of the incremental cost estimation for the ith generator. Because σ = {t1 , t2 , . . .} is a strictly increasing sequence, there always exists some K > 0 such that tK > maxi {Ti0 }, and it has λi = 2γi Pi + βi , i ∈ IG when t > tK . Thus, it can be obtained that � λ˙ i (t) = 0, t ∈ [tk , tk+1 ) � (15) λi (t+ ) = λ (t ) + 2ǫγ aij (λj (tk ) − λi (tk )) i k i k j∈Ni
Clearly, λi keeps constant on time interval [tk , tk+1 ), k > ¯i (k) as the constant value of K. For convenience, denote λ λi (t) on time interval [tk , tk+1 ), then (15) is converted to a discrete system which is ¯ i (k) + 2ǫγi ¯i (k + 1) = λ λ
�
j∈Ni
� � ¯ j (k) − λ ¯ i (k) . (16) aij λ
� � ¯ N (k) T , (16) can be rewritten ¯ 1 (k), . . . , λ ¯ Denote Λ(k) = λ in a compact form, ¯ + 1) = (I − ǫDL)Λ(k), ¯ Λ(k (17) where D = diag(2γ1 , . . . , 2γN ). It is noted that DL is also a Laplacian matrix, of which the row sum equals to zero. In light of Lemma 2, because 0 < ǫ < 1/(maxi {2γi ∆i }), maxi {2γi ∆i } is the maximum degree of the DL. Thus, a ¯ i (k) is asymptotically reached, i.e., λ ¯ i (k) → consensus of λ λf as k → +∞, where it follows that λi (t) → λf as t → +∞. Furthermore the final consensus value λf is
Proceedings of the 20th IFAC World Congress 180 Xiao-Kang Liu et al. / IFAC PapersOnLine 50-1 (2017) 177–182 Toulouse, France, July 9-14, 2017
derived by calculating the left eigenvector of (I − ǫDL) which is 1/2γN 1/2γ1 . (18) rˆ = N , · · · , N i=1 (1/2γ1 ) i=1 (1/2γN ) N Due to λi (tK ) = 2γi Pi (tK ) + βi and i=1 (Pi (tK )) = N i=1 (Pi0 ) = PD , it can be obtained that
thus,
λf = rˆλ(tK ) N i=1 (λi (tK )/2γi ) = N i=1 (1/2γi ) PD + N i=1 (βi /2γi ) = , N i=1 (1/2γi )
(19)
(20)
(21) λ1 = λ2 = · · · = λN = λf as t → +∞. Note that (19) is the optimal condition of (7), which means the estimate λi converges to the optimal incremental cost value. Consequently, the EDP without power constraints is solved by the distributed hybrid algorithm (9). Remark 3. Based on the results in Olfati-Saber et al. ¯ (2007), the Λ(k) in the discrete system (17) exponentially vanishes with a speed faster or equal to µ2 , where µ2 = 1 − ǫλsec (DL) and λsec (·) is the operator calculating the second smallest eigenvalue of the matrix. For the variable λ(t), the exponential convergence rate can be obtained by introducing the definition of average impulsive interval in Lu et al. (2010). 3.2 EDP with power generator constraints For the EDP with power generation constraints, some supervised strategies are required, which is to ensure the obtained optimal power outputs to meet the upper and lower bounded constraints. For a better understanding of the supervised hybrid distributed algorithm, the definition of projection operator is introduced. Let’s define Proj(·) as a projection operator from R to Ω ⊆ R: Proj(x) = arg min �υ − x� (22) ν∈Ω
where Ω is a closed convex set. The power generation constraints (3) can be described as Ωi = {Pi ∈ R|Pimin ≤ Pi ≤ Pimax } for the ith generators. As seen in (8), the optimal cost λ∗ can be deemed as the agreement for generators who are not working at their boundary power outputs. Thus, a supervised strategy is required to check which generator breaks the constraint condition. Sepecifically, the intrinsic idea of the supervised strategy is based on the relationship between λi and Pi , i.e., λi = 2γi Pi +βi , which is guaranteed by a finite-time continuoustime algorithm in (9). Because Pimin ≤ Pi ≤ Pimax , obviously, λi also subjects to the constraint λmin ≤ λi ≤ i λmax , where i λmin = 2γi Pimin + βi , λmax = 2γi Pimax + βi . (23) i i Thus, when t = tk , k = 1, 2, . . ., the supervised strategy works and takes three steps: 182
Algorithm 1 Graph Transformation Algorithm. Input: Adjacency matrix A and node i. 1: for l ∈ Ni do 2: for m ∈ Ni do 3: A[l,m] = 1 4: end for 5: end for 6: A[i,:] = 0, A[:,i] = 0 7: Calculating Laplacian matrix L corresponded to A. Output: Laplacian matrix L Algorithm 2 Supervised Hybrid Distributed Algorithm. 1: Initialize the Pi0 for the ith generator. Set discrete time sequence σ = {t1 , t2 , . . .} and the accessible set Ωi = {λi ∈ R|λmin ≤ λi ≤ λmax }, where λmax = i i i min 2γi Pimax + βi and λmin = 2γ P + βi . Set λi (0) ∈ Ωi i i i and ǫ < 1/(maxi {2γi ∆i }). 2: while t < Tf inal do 3: if t = tk then 4: for i = 1 : N do 5: Pit = Pi (tk ) + ǫ j∈Ni aij (λj (tk ) − λi (tk )); λti = λi (tk ) + 2ǫγi j∈Ni aij (λj (tk ) − λi (tk )) 6: if λti > λmax or λti < λmin then i i 7: for j ∈ Ni do (Proj(λti )−λti ) 8: λtj = λtj − |Ni | (P t −P max or P min )
9: 10: 11: 12: 13: 14: 15: 16: 17: 18:
Pjt = Pjt − i i |Ni | i end for λti = Proj(λti ), Pi = Pimax or Pimin Change the neighbors of node j, j ∈ Ni , (execute Algorithm 1) end if λti → λi (tk ), Pit → Pi (tk ). end for else 12 λ˙ i (t) = µγi Pi + βi −λi sign(Pi + βi −λi ) 2γi
2γi
end if end while
(1) Check each generator unit whether the λi lies on the constraint set (23). If it is true, then go straight to the third step, otherwise, a projection operator is imposed on λi and make the deviation caused by the projection , (λi − Proj(λi )), evenly shared on its neighbor generators; (2) If λi does not lie on the constraint set, then the graph G will be transformed. Figure 1 illustrates the graph transformation with taking node 1 as example, where the node 1 pretends to be isolated and no longer takes any influence from its neighbors. However, it still serves as an intermediary communication link that permits the information exchange among the node 2,3,4. In detail, Graph Transformation Algorithm (GTA) is presented as seen in Algorithm 1; (3) Restore the value of λi (tk ) and Pi (tk ) after checking all generator units. In light of the projection operator (22) and GTA, a supervised hybrid distributed algorithm is designed for the EDP with power generation constraints, as seen in Algorithm 2.
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Xiao-Kang Liu et al. / IFAC PapersOnLine 50-1 (2017) 177–182
Remark 4. In Algorithm 2, the line 6-15 show the three steps of the supervised strategy. Note that the statements in line 5 and 16 are exact the hybrid distributed algorithm (9). In other words, by setting Pimax = +∞ and Pimin = 0, the Algorithm 2 is reduced to solve the EDP without power generation constrains.
181
12 11 10 9 8
1
λ
1 4
4
2
6 5
2 3
5
3
(a)
λ1 λ2 λ3 λ4 λ5
7
4
5
3
(b)
0
5
10
15
Time t (sec)
Fig. 1. The schematic of the graph transformation. Take node 1 as example, (a) the original graph topology; (b) the transformed graph topology.
Fig. 2. The evolution of incremental cost estimator λi without generation constraints. 90
4. SIMULATION
80
Consider a power system with 14 buses and 5 generators, where the communication graph of generators is shown in Figure 1(a). The corresponding Laplacian matrix is 3 −1 −1 −1 0 −1 1 0 0 0 L = −1 0 2 0 −1 . −1 0 0 2 −1 0 0 −1 −1 2
The parameters of cost functions and the generator output power constraints are given in Table 1.
P1 P2 P3 P4 P5
70
Output Power
In this section, an example of 14-bus system is presented to show the effectiveness of the proposed distributed hybrid algorithm in the context of EDP with and without power generation constraints, respectively.
60 50 40 30 20 10
0
5
10
15
Time t (sec)
Fig. 3. The evolution of output power Pi without generation constraints. 12
Table 1. The generator parameters γi 0.081 0.091 0.063 0.071 0.043
βi 4.623 6.143 5.357 3.736 2.715
αi 68 53 50 62 38
Pimin (kW ) 25 30 17 20 22
11
Pimax (kW ) 80 60 50 80 45
10 9 8
λ
Generator 1 2 3 4 5
7
The total power demand is PD = 200 kW , and the initialization of the Pi0 is set as P10 = 35 kW , P20 = 40 kW , P30 = 50 kW , P40 = 45 kW , P50 = 30 kW . The discrete sequence σ is selected as a average time interval sequence, where tk+1 − tk = 0.2 (sec), k = 0, 1, 2, . . .. With setting µ = 2 and ǫ = 1.5 such that ǫ < 1/(maxi {2γi ∆i }) = 2.0576, EDP without power generation constraints is solved by the hybrid distributed algorithm (9). Fig. 2 shows the incremental cost estimate of each generator unit, λi , converges to the optimal value λ∗ = 9.476. Fig. 3 shows the output power of each generator unit finally stabilizes to the optimal value, where P1∗ = 29.96 kW , P2∗ = 18.31 kW , P3∗ = 32.69 kW , P4∗ = 40.42 kW , P5∗ = 78.62 kW . 183
6
λ1 λ2 λ3 λ4 λ5
5 4 3
0
1
2
3
4
5 6 Time t (sec)
7
8
9
10
Fig. 4. The evolution of incremental cost estimator λi with generation constraints. When consider the generator power constraints, clearly, the optimal power obtained by hybrid distributed algorithm (9) does not satisfy the generator power constraints. In this case, the supervised strategy is applied, where it final-
Proceedings of the 20th IFAC World Congress 182 Xiao-Kang Liu et al. / IFAC PapersOnLine 50-1 (2017) 177–182 Toulouse, France, July 9-14, 2017
60
P1 P2 P3 P4 P5
55 Generator 5
Output Power
50
45
40
35
30 Generator 2 25
0
1
2
3
4
5 6 Time t (sec)
7
8
9
10
Fig. 5. The evolution of output power Pi with generation constraints. ly obtains P1∗ = 36.36 kW , P2∗ = 30 kW , P3∗ = 40.92 kW , P4∗ = 47.72 kW , P5∗ = 45 kW . Fig. 4 and Fig. 5 show the trajectories of incremental cost estimator and output power of each generator unit, respectively. It can be seen in Fig. 4, λ2 and λ5 are isolated and others are eventually making consensus to λ∗ = 10.51. Accordingly, in Fig. 5, P2 = P2min and P5 = P5max . Obviously, generator 2 and 5 are running on the boundaries of their power constraints, while others are not. 5. CONCLUSION This paper has investigated the economic dispatch problem by a hybrid consensus-based algorithm, which has integrated a finite-time continuous-time dynamical algorithm with a discrete-time interacted scheme. Inspired by the cluster synchronization behavior and the benefit of the projection operator, the supervised strategy has been carried out on each generator unit to ensure the power generation constraints. The stability analysis and optimality analysis of the hybrid algorithm has been given for the EDP without the power generation constraints. Moreover, the detailed pseudo-code has been presented for the EDP with power generation constraints. However, it is still in need of rigorous analysis. Future work will focus on this issue. REFERENCES Amjady, N. and Nasiri-Rad, H. (2009). Economic dispatch using an efficient real-coded genetic algorithm. IET Generation, Transmission & Distribution, 3(3), 266– 278. Bhat, S.P. and Bernstein, D.S. (2000). Finite-time stability of continuous autonomous systems. SIAM Journal on Control & Optimization, 38(3), 751–766. Binetti, G., Davoudi, A., Lewis, F.L., Naso, D., and Turchiano, B. (2014). Distributed consensus-based economic dispatch with transmission losses. IEEE Transactions on Power Systems, 29(4), 1711–172. Chen, G., Lewis, F.L., Feng, E.N., and Song, Y. (2015). Distributed optimal active power control of multiple generation systems. IEEE Transactions on Industrial Electronics, 62(11), 7079–7090. 184
Chen, G., Ren, J., and Feng, E.N. (2017). Distributed finite-time economic dispatch of a network of energy resources. IEEE Transactions on Smart Grid, 8(2), 822 – 832. Chen, S.D. and Chen, J.F. (2003). A direct newtoncraphson economic emission dispatch. International Journal of Electrical Power & Energy Systems, 25(5), 411–417. Guo, F., Wen, C., Mao, J., and Song, Y.D. (2016). Distributed economic dispatch for smart grids with random wind power. IEEE Transactions on Smart Grid, 7(3), 1572 – 1583. Li, C., Yu, X., Yu, W., and Huang, T. (2016). Distributed event-triggered scheme for economic dispatch in smart grids. IEEE Transactions on Industrial Informatics, 12(5), 1775 – 1785. Loia, V. and Vaccaro, A. (2014). Decentralized economic dispatch in smart grids by self-organizing dynamic agents. IEEE Transactions on Systems Man & Cybernetics Systems, 44(4), 397–408. Lu, J., Ho, D.W.C., and Cao, J. (2010). A unified synchronization criterion for impulsive dynamical networks. Automatica, 46(7), 1215–1221. Olfati-Saber, R., Fax, A., and Murray, R.M. (2007). Consensus and cooperation in networked multi-agent systems. Proceedings of the IEEE, 95(1), 215–233. Park, J.B., Jeong, Y.W., Shin, J.R., and Lee, K.Y. (2010). An improved particle swarm optimization for nonconvex economic dispatch problems. IEEE Transactions on Power Systems, 25(1), 156–166. Wang, Y.W., Guan, Z.H., and Wang, H.O. (2005). Impulsive synchronization for takagi-sugeno fuzzy model and its application to continuous chaotic system. Physics Letters A, 339(3-5), 325–332. Wang, Y.W., Liu, M., Liu, Z.W., and Yi, J.W. (2014). Formation tracking of the second-order multi-agent systems using position-only information via impulsive control with input delays. Applied Mathematics & Computation, 246(1), 572–585. Wood, A.J. and Wollenberg, B.F. (2012). Power generation, operation, and control. John Wiley & Sons. Xing, H., Mou, Y., Fu, M., and Lin, Z. (2015). Distributed bisection method for economic power dispatch in smart grid. IEEE Transactions on Power Systems, 30(6), 3024–3035. Yang, S., Tan, S., and Xu, J.X. (2013). Consensus based approach for economic dispatch problem in a smart grid. IEEE Transactions on Power Systems, 28(11), 4416– 4426. Yu, W., Li, C., Yu, X., Wen, G., and L, J. (2015). Distributed consensus strategy for economic power dispatch in a smart grid. In Control Conference (ASCC), 2015 10th Asian, 1–6. Zhang, Z. and Chow, M.Y. (2011). Incremental cost consensus algorithm in a smart grid environment. In 2011 IEEE Power and Energy Society General Meeting, 1–6. Zhang, Z. and Chow, M.Y. (2012). Convergence analysis of the incremental cost consensus algorithm under different communication network topologies in a smart grid. IEEE Transactions on Power Systems, 27(4), 1761– 1768.
ScienceDirect
IFAC PapersOnLine 50-1 (2017) 177–182
Hybrid Consensus-based Algorithm for Hybrid Consensus-based Algorithm for Hybrid Consensus-based Algorithm for ⋆⋆ Distributed Economic Dispatch Problem Distributed Economic Dispatch Problem ⋆ Distributed∗,∗∗Economic Dispatch Problem ∗,∗∗ ∗∗∗,∗∗∗∗ Xiao-Kang Xiao-Kang Xiao-Kang Xiao-Kang
∗,∗∗ Huaicheng Yan ∗∗∗,∗∗∗∗ Liu Yan-Wu Liu ∗,∗∗ Yan-Wu Wang Wang Huaicheng Yan ∗∗∗,∗∗∗∗ ∗,∗∗ ∗,∗∗ ∗,∗∗ Yan-Wu Wang ∗,∗∗ Huaicheng ∗,∗∗ ∗,∗∗ Liu Yan ∗∗∗,∗∗∗∗ ∗,∗∗ Xiaoya ∗,∗∗ Xiaoping Wang Hu ∗,∗∗ ∗,∗∗ Xiaoping WangWang Xiaoya Hu Liu Yan-Wu Huaicheng Yan ∗∗∗,∗∗∗∗ ∗,∗∗ ∗,∗∗ ∗,∗∗ Xiaoping Wang ∗,∗∗ Xiaoya Hu ∗,∗∗ ∗,∗∗ Xiaoping Wang Xiaoya Hu ∗ ∗ School of Automation, Huazhong University of Science and School of Automation, Huazhong University of Science and ∗ ∗ School of Automation, Huazhong University of Science and Technology, Huazhong Wuhan, 430074, 430074, China ∗ Technology, Wuhan, China School of Automation, University of Science and ∗∗ ∗∗ Key Laboratory Technology, Wuhan, 430074, China of Image Processing and Intelligent Key Laboratory of Image Processing and Intelligent Control Control Technology, Wuhan, 430074, ∗∗ ∗∗ Key Laboratory of Image Processing andChina Intelligent Control (Huazhong University of and Ministry of ∗∗ (Huazhong University of Science Science and Technology), Technology), Ministry of Key Laboratory of Image Processing and Intelligent Control (Huazhong University of Science and Technology), Ministry of Education Education (Huazhong University of Science and Technology), Ministry of ∗∗∗ Education ∗∗∗ Key Laboratory Laboratory of of Advanced Advanced Control and and Optimization Optimization for for Key Control Education ∗∗∗ ∗∗∗ of Control Optimization Chemical of Education ∗∗∗ Key Laboratory Chemical Process Process of Ministry Ministry ofand Education Key Laboratory of Advanced Advanced Controlof and Optimization for for ∗∗∗∗ ∗∗∗∗ SchoolChemical Process of Ministry of Education of Information Science and Engineering East China School of Information Science and Engineering East China Chemical Process of Ministry of Education ∗∗∗∗ ∗∗∗∗ School of Information Science and Engineering East China University of ofScience Science and Technology Technology 130Engineering Meilong Road, Road, Shanghai ∗∗∗∗ University and 130 Meilong Schoolof Information Science and EastShanghai China University of Science and Technology 130 Meilong Road, 200237, P. R. China 200237, P. R. China University of Science and Technology 130 Meilong Road, Shanghai Shanghai 200237, 200237, P. P. R. R. China China Abstract: Economic dispatch is one of the fundamental Abstract: Economic dispatch is one of the fundamental problems problems in in the the power power system system research. research. Abstract: Economic dispatch is one of the fundamental problems in the power system research. The existing algorithms are either discrete iterative algorithms or continuous-time dynamical The existingEconomic algorithms are either discrete iterative algorithms orincontinuous-time dynamical Abstract: dispatch is one of the fundamental problems the power system research. The existingBy algorithms are hybrid either discrete iterative algorithms or continuous-time algorithms. virtue of of the the techniques, hybrid consensus-based algorithm is isdynamical proposed algorithms. virtue techniques, aa hybrid consensus-based algorithm proposed The existingBy algorithms are hybrid either discrete iterative algorithms or continuous-time dynamical algorithms. By virtue of the hybrid techniques, a hybrid consensus-based algorithm is proposed in this paper, which consists of a finite-time continuous-time algorithm and a discrete-time in this paper, which consists of a finite-time continuous-time algorithm and a discrete-time algorithms. By virtue of the hybrid techniques, a hybrid consensus-based algorithm is proposed in this which consists a algorithm discrete-time interacted scheme. Moreover, inspired by cluster and the of interacted scheme. Moreover, inspired by the the continuous-time cluster synchronization synchronization andand theaabenefit benefit of the the in this paper, paper, whichMoreover, consists of of a finite-time finite-time continuous-time algorithm and discrete-time interacted scheme. inspired by the cluster synchronization and the benefit of the projection operator, a supervised strategy is carried out to ensure that each generator unit will projection operator, a supervised strategy isthe carried outsynchronization to ensure that each generator unit will interacted scheme. Moreover, inspired by cluster and the benefit of the projection supervisedconstraint. strategy isFor carried out understanding to ensure that of each unit will work within withinoperator, its power powera generation better thegenerator supervised hybrid work its aa better the supervised hybrid projection operator, ageneration supervisedconstraint. strategy isFor carried out understanding to ensure that of each generator unit will work within its power generation constraint. For a better understanding of the supervised hybrid distributed algorithm, the detailed pseudo-code is presented. Finally, a numerical example is distributed the detailed pseudo-code presented. Finally, of a numerical example is work within algorithm, its power generation constraint. For a is better understanding the supervised hybrid distributed algorithm, the detailed pseudo-code is presented. Finally, a numerical example is given to illustrate the effectiveness of the algorithm. given to illustrate the effectiveness ofpseudo-code the algorithm. distributed algorithm, the detailed is presented. Finally, a numerical example is given to illustrate the effectiveness of the algorithm. given illustrate the effectiveness algorithm. © 2017,toIFAC (International Federation of of the Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: distributed optimization, Keywords: economic economic dispatch, dispatch, hybrid hybrid system, system, consensus consensus protocol, protocol, distributed optimization, Keywords: economic dispatch, hybrid system, consensus protocol, distributed smart grid. smart grid.economic dispatch, hybrid system, consensus protocol, distributed optimization, Keywords: optimization, smart grid. smart grid. 1. incremental 1. INTRODUCTION INTRODUCTION incremental cost cost consensus consensus algorithm algorithm which which selected selected the the 1. INTRODUCTION consensus algorithm which selected the incremental cost of each generation unit as the consensus cost of each generation unit as the consensus 1. INTRODUCTION incremental consensus algorithm which selected the cost of each generation unit as the consensus Economic variable and the EDP was solved in a discrete manner. Economic Dispatch Dispatch Problem Problem (EDP) (EDP) has has received received considconsid- incremental variable andcost the of EDP was solved in a discrete manner. incremental each generation unit as the consensus Economic Dispatch (EDP) receivedof and the EDP was solved in a discrete manner. erable in recent The objective EDP They subsequently extended the result result to the the case case that erable attention attention in Problem recent years. years. Thehas objective ofconsidEDP variable They subsequently extended the to that Economic Dispatch Problem (EDP) hasobjective receivedof considvariable and the EDP was solved in a discrete manner. erable attention in recent years. The EDP They subsequently extended the result to the case that in power system is to minimize the overall consumption considering the power generation constraints and pointed in power system in is to minimize the overall consumption considering the power generation constraints and pointed erable attention recent years. The objective of EDP They subsequently extended the constraints result to the case that in system is to minimize the overall consumption considering the power generation and pointed of power generation while meeting the power demand of out the convergence rate relies on the topology of of power generation while meeting the power demand of considering out the convergence relies on the topology of the the in system is to minimize thethe overall consumption the power rate generation constraints and pointed of power generation while meeting power demand of out the convergence rate relies on the topology of the loads and the constraints. Generally, EDP is (Zhang Chow (2012)). The work loads andgeneration the operational operational constraints. Generally, EDP of is network network (Zhang and and rate Chowrelies (2012)). Thetopology work in inofYang Yang of power while meeting the power demand out the convergence on the the loads and the operational constraints. Generally, EDP is network (Zhang and Chow (2012)). The work in Yang formulated as a nonlinear optimization problem. To solve et al. (2013) (2013) proposed fully (2012)). distributed algorithm where formulated as aoperational nonlinear optimization problem. To solve et al. proposed aa fully distributed algorithm where loads and the constraints. Generally, EDP is network (Zhang and Chow The work in Yang formulated a nonlinear optimization problem. Tobased solve generators et al. (2013)were proposed fully distributed algorithm where the various algorithms have been proposed able learn the the problem, problem,as various algorithms have been proposed based generators able to toaa cooperatively cooperatively learn the mismatch mismatch formulated asvarious a nonlinear optimization problem. Tobased solve et al. (2013)were proposed fully distributed algorithm where the problem, algorithms have been proposed generators were able to cooperatively learn the mismatch on convex optimization techniques or heuristic techniques, between demand and total power generation. Loia on convex optimization techniques or been heuristic techniques, between demand andtototal power generation. Loia and and the problem, various algorithms have proposed based generators were able cooperatively learn the mismatch on convex optimization techniques or heuristic techniques, demand and total power generation. Loia and such as method and (2003)), Vaccaro designed aa distributed and such as Newton-Raphson Newton-Raphson method (Chen (Chen and Chen Chen (2003)), between Vaccaro (2014) (2014) designed distributed and self-organizing self-organizing on convex optimization techniques or heuristic techniques, between demand and total power generation. Loia and such as Newton-Raphson method (Chen and Chen (2003)), (2014) a distributed andconsensus self-organizing genetic algorithm and (2009)), parframework baseddesigned on the the weighted weighted average consensus algogenetic algorithm (Amjady (Amjady and Nasiri-Rad Nasiri-Rad (2009)), par- Vaccaro framework based on average algosuch as Newton-Raphson method (Chen and Chen (2003)), Vaccaro (2014) designed a distributed and self-organizing genetic algorithm (Amjady and et Nasiri-Rad (2009)), ticle optimization (Park al. and on. framework based on weighted average algonetwork. light the ticle swarm swarm optimization (Park et al. (2010)), (2010)), and so soparon. rithm rithm of of agent agent network. Similarly in light of ofconsensus the consensus consensus genetic algorithm (Amjady and et Nasiri-Rad (2009)), parframework based on the the Similarly weighted in average consensus algoticle swarm optimization (Park al. (2010)), and so on. However, these conventional methods for EDP commonly rithm of agent network. Similarly in light of the consensus method, a distributed consensus-like iterative bisection However, these conventional methods for EDP and commonly method, a distributed consensus-like iterative bisection ticle swarm optimization (Park et al. (2010)), so on. rithm of agent network. Similarly in light of the consensus However, conventional methods for EDP commonly awas distributed iterative require control center has to global algorithm proposed consensus-like in Xing Xing et et al. al. (2015) (2015) withbisection no need need require aa these control center that that has access access to the the global method, algorithm proposed in with no However, these conventional methods for EDP commonly method, awas distributed consensus-like iterative bisection require a control center that has access to the global algorithm was proposed in Xing et al. (2015) with no(2014) need information of all the generator units. Unfortunately, the for a data fusion center or a leader. Binetti et al. (2014) information of all the generator units. Unfortunately, the for a data fusion center or a leader. Binetti et al. require a control center that has access to the global algorithm was proposed in Xing et al. Binetti (2015) with no(2014) need information of all the generator units. Unfortunately, the for a data fusion center or a leader. et al. centralized structure could have some drawbacks that limit proposed a framework of two consensus algorithms running centralized structure could have some drawbacks that limit proposed a fusion framework of two consensus algorithms running information of all thecould generator units.drawbacks Unfortunately, the for a data center or a leader. Binetti et al. (2014) centralized structure have some that limit proposed a framework two consensus algorithms running its especially when aa large scale of where first one the its wider wider application, application, especially when largethat scale of in in parallel, parallel, where the theof first one ensured ensured the generationgenerationcentralized structure could have some drawbacks limit a framework two consensus algorithms running its wider application, especially when a large scale of proposed in parallel, where and theofthe first one ensured the generationdistributed energy resources are taken into consideration demand equality and the second one compensated the distributed energy resources are taken into consideration demand equality second one compensated the its wider application, especially when a large scale of in parallel, where the first one ensured the generationdistributed energy resources arecenter takendevices. into consideration power demand equality Based and the second one compensated the or failure occurs mismatch. on the finite-time consensus techor when when some some failure occurs on on center devices. power mismatch. Based on the finite-time consensus techdistributed energy resources are taken into consideration demand equality Based and the second one compensated the or when some failure occurs on center devices. power mismatch. on the finite-time consensus techniques, Guo et al. (2016) designed a iterative scheme niques,mismatch. Guo et al. (2016) designed a iterative scheme or when some failure occurs on center Based on the finite-time consensus techRecently, distributed techniques have been Recently, distributed techniques have devices. been developed developed to to power niques, et al. unit (2016) designed a iterative scheme generator which decomposed the for each eachGuo generator which decomposed the centralcentralRecently, distributed have networks been developed to for niques, Guo et al. unit (2016) designed a iterative scheme solve in light lighttechniques of and solve the the EDP EDP in of multi-agent multi-agent networks and conconfor each generator unit which decomposed the centralRecently, distributed techniques have been developed to ized optimization into optimizations at local agents, as ized optimization into optimizations at local agents, as solve the EDP in light of multi-agent networks and confor each generator unit which decomposed the centralsensus methods. Zhang and Chow (2011) proposed the sensusthe methods. Zhang and Chow (2011) proposed the aized solve EDP in Zhang light ofand multi-agent networks and conoptimization into optimizations at local agents, result the finite-time convergence to the optimal value a result the finite-time convergence to the optimal value sensus methods. Chow (2011) proposed the ized optimization into optimizations at local agents, as as ⋆ ⋆ This work is supported by the National Natural Science Founsensus methods. Zhang and Chow (2011) proposed the aawas result the convergence to was achieved. Event-triggered communication scheme was This work is supported by the National Natural Science Founachieved. Event-triggered communication schemevalue was result the finite-time finite-time convergence to the the optimal optimal value ⋆ This work is supported by the National Natural Science Foundation of China under Grants 61374171, 61572210, 51537003 and was achieved. Event-triggered communication scheme was carried out Li which the dation China under Grants 61374171, 51537003Founand ⋆ carried out in in Event-triggered Li et et al. al. (2016), (2016),communication which greatly greatly reduced reduced the This of work is supported by the National61572210, Natural Science was achieved. scheme was 61673178, the Research Funds for Univerdation of China under Grants 61374171, 51537003 and carried out in Li et al.requirements (2016), whichin greatly reduced the 61673178, the Fundamental Fundamental Research Funds61572210, for the the Central Central Univerinformation exchange requirements in the process of dation of China under Grants 61374171, 61572210, 51537003 and information exchange the process of the carried out in Li et al. (2016), which greatly reduced 61673178, the Fundamental Research Funds for the Central Universities (2015TS030). sities (2015TS030). information exchange requirements in the process of cooperative optimization. Of note is that the above liter61673178, the Fundamental Research Funds for the Central Univercooperative optimization. Of note is that the above literinformation exchange requirements in the process of the the sities (2015TS030). E-mail addresses: [email protected] [email protected] (Yan-Wu (Yan-Wu Wang). Wang). E-mail addresses: cooperative sities (2015TS030). E-mail addresses: [email protected] (Yan-Wu Wang). cooperative optimization. optimization. Of Of note note is is that that the the above above literliter-
E-mail addresses: [email protected] (Yan-Wu Wang). Copyright © 2017, 2017 179 Copyright 2017 IFAC IFAC 179 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright ©under 2017 responsibility IFAC 179Control. Peer review of International Federation of Automatic Copyright © 2017 IFAC 179 10.1016/j.ifacol.2017.08.030
Proceedings of the 20th IFAC World Congress 178 Xiao-Kang Liu et al. / IFAC PapersOnLine 50-1 (2017) 177–182 Toulouse, France, July 9-14, 2017
atures solve the EDP in a discrete-time iterative manner, besides, distributed continuous-time dynamical algorithms have been developed (Yu et al. (2015); Chen et al. (2015); Chen et al. (2017)), which analytically solve the EDP in a continuous-time manner with the rigorous stability analysis and optimality analysis. In light of the discrete-time and continuous-time algorithms for EDPs, a supervised hybrid distributed consensus-based algorithm is proposed in this paper. The hybrid algorithm integrates a finite-time continuoustime dynamical algorithm with a discrete-time interacted scheme. Specifically, each generator unit adopts the continuous-time algorithm locally, which enables the incremental cost to meet the first-order optimality condition at the settling time. As for the discrete-time scheme, generator units will exchange information at time spots and make consensus. Motivated by the cluster synchronization behavior and the benefit of the projection operator, a supervised strategy is designed at the discrete-time level to maintain power generation constraints. Once a generator unit breaks out the boundary of the constraint, the projection will work to restore its power generation to the boundary value. Subsequently, the generator unit will be treated as an isolated node in the network, where it still serves as an intermediary communication link, but no longer takes any influence from its neighbors. With the help of analysis methods for hybrid systems, a rigorous stability analysis and optimality analysis is given in the case that no power generation constraints are taken into consideration. In aid of the supervised strategy, a detailed pseudo-code for the EDP with power constraints is presented. Finally, a simulation example illustrates the effectiveness of our algorithms. The rest of the paper is organized as follows. Some preliminaries of algebraic graph theory and matrix theory, as well as problem statement are given in Section 2. The hybrid distributed algorithms for EDP with or without power generation constraints are proposed in Section 3. The illustrative numerical example is given in Section 4. Finally, conclusions are drawn in Section 5. 2. PRELIMINARIES AND PROBLEM STATEMENT In this section, some preliminaries of algebra graph theory are given, as well as description and definition of the economic dispatch problem. 2.1 Graph theory A graph G = (V, E, A) is described with a set of nodes V = {1, 2, . . . , N}, a set of edges E ∈ V × V and a weighted adjacency matrix A = (aij )N ×N with nonnegative adjacency elements. Node i represents the ith system, where an edge eji in graph is denoted by the ordered pair nodes {j, i}. {j, i} ∈ E if and only if the ith system can obtain the information from the jth system. Define the adjacency matrix A = (aij )N ×N associated with G such that aij > 0 if {j, i} ∈ E, and aij = 0 otherwise. We assume there is no self-loop in the graph G, i.e., aii = 0. Define Ni = {j|aij > 0}. The Laplacian matrix L = (lij )N ×N of graph G is defined as lij = −aij if � i �= j, otherwise lij = N k=1,k�=i aik .
180
2.2 Economic dispatch problem Consider a power grid with N generators labeled by IG = {1, 2 · · · , N }, where the cost function of ith power generation is modelled as (Wood and Wollenberg (2012)) (1) Ci (Pi ) = γi Pi2 + βi Pi + αi , where Pi is the output of the ith power generator, and αi , βi , γi are the corresponded parameters. The objective of economic dispatch is to minimize the total cost of the power grid while meeting the demand and power generators constraints, which is formulated as an optimization problem: N �
min
Pi , i∈IG
Ci (Pi ),
(2)
Pi = PD ,
(3)
i=1
N �
s.t.
i=1
(4) Pimin ≤ Pi ≤ Pimax , i ∈ IG max min and Pi are the where PD is the demand power, Pi upper and lower bounds of the ith generator. Define the lagrangian operator L(P1 , . . . , PN ) such that � � N N � � L(P1 , . . . , PN ) = Pi , (5) Ci (Pi ) + λ PD − i=1
i=1
where λ is the Lagrange multiplier.
For the EDP without power generation constraints, based on the first-order optimality condition, it can be obtained that the incremental cost is equal to λ, i.e., ∂Ci λ= = 2γi Pi + βi . (6) ∂Pi By (3), the values of λ and optimal power output Pi∗ can be calculated in a centralized way, � βi PD + ni=1 2γ λ − βi i �n λ= , Pi∗ = . (7) 1 2γi i=1 2γi
For the EDP with power generation constraints, there exists the optimal incremental cost λ∗ satisfies the following relationship: 2γi Pi + βi = λ∗ , for Pimin < Pi < Pimax 2γi Pi + βi < λ∗ , for Pi = Pimax (8) 2γi Pi + βi > λ∗ , for Pi = Pimin 3. DISTRIBUTED HYBRID ALGORITHM FOR ECONOMIC DISPATCH
In this section, a distributed hybrid algorithm is first presented to solve the EDP without power generator constraints, and the optimal property as well as convergence is strictly proved. Then, based on the distributed hybrid algorithm, a projection operator is introduced to solve the EDP with power generator constraints. 3.1 EDP without power generator constraints Define the discrete time sequence σ = {t1 , t2 , . . .} such that 0 = t0 < t1 < t2 < · · · , and tk+1 − tk > τ , k = 0, 1, 2, . . .,
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Xiao-Kang Liu et al. / IFAC PapersOnLine 50-1 (2017) 177–182
where τ > 0 is the smallest time interval of σ. Then, the distributed hybrid algorithm is designed as Pi (t) = ηi (t) + Pi0 , � �1 � βi − λi �� 2 ˙ � λi (t) = µγi �Pi + 2γi � βi − λi · sign(Pi + ), t ∈ [tk , tk+1 ), (9) � 2γi λi (t+ ) = λi (tk ) + 2ǫγi aij (λj (tk ) − λi (tk )) , k j∈Ni tk+1 ), η˙i (t) = 0, t ∈ [tk , � + aij (λj (tk ) − λi (tk )) , ηi (tk ) = ηi (tk ) + ε j∈Ni ηi (0) = 0, i ∈ IG , where µ and ǫ are positive gain parameters, ηi is the auxiliary variable and λi denotes the estimate of the optimal incremental cost�for ith generator, Pi0 is a virtual initial value satisfying ni=1 Pi0 = PD . Note that limt→t− λi (t) = k λi (tk ), i.e., λi (t) is left continuous at time tk . Remark 1. The distributed hybrid algorithm contains two parts, continuous dynamical algorithm and discrete iterative algorithm, in which the continuous part takes finitetime dynamical algorithm while the discrete part takes consensus algorithm. In some sense, controller (9) is kind of like the impulsive controller for the multi-agent network (Wang et al. (2014)). Compared with Yu et al. (2015), Chen et al. (2015) and Chen et al. (2017), our distributed hybrid algorithm only requires information exchange at time spots instead of on the whole time interval. �N Remark 2. It is worthy noting that i=1 ηi (t) = 0, where �N �N it follows i=1 Pi (t) = i=1 (Pi0 ) = PD , t > 0. Thus, our distributed algorithm will not break the demand equality constraint (3) during the process. Lemma 1. (Bhat and Bernstein (2000)). Consider the continuous function x˙ = f (x) with f (0) = 0. Suppose that there are C 1 function V (x) defined on a neighborhood of the origin, and real numbers c > 0 and 0 < α < 1, such that (1)V (x) is position definite; (2)V˙ (x) + cV α (x) ≤ 0. Then, the origin is locally finite-time stable, and the settling time, depending on the initial state x(0) = x0 , satisfies T (x0 ) ≤ (V 1−α (x0 )/c(1 − α)). Lemma 2. (Olfati-Saber et al. (2007)). Consider a network with the topology G applying the distributed consensus algorithm x(k + 1) = (I − ǫL)x(k), where 0 < ǫ < 1/∆ and ∆ is the maximum degree of network. Let G be a strongly connected digraph. Then, a consensus is asymptotically reached for all initial states. Furthermore, the final consensus value is given by rˆx(0), where rˆ is the normalized left eigenvector of (I − ǫL) associated with the one eigenvalue. Theorem 3. Let the communication network G is an undirect connected graph, then the distributed hybrid algorithm (9) solves the EDP (2-3) if 0 < ǫ < 1/(maxi {2γi ∆i }), where ∆i is the in-degree of node i. Proof. Motivated by Wang et al. (2005) and Chen et al. (2015), select the following Lyapunov function �2 � βi − λi Vi (t) = Pi + . (10) 2γi Note that the dynamic of λi is piecewise continuous, where it jumps at time instants tk , k = 0, 1, 2, · · · . Hence, take the derivative of V (t) on the time interval [tk , tk+1 ), 181
179
�� ˙ � � βi − λi −λi ˙ Vi (t) = 2 Pi + 2γi 2γi �1 � �� βi − λi �� βi − λi �� 2 = −µ Pi + �Pi + 2γi � 2γi � � βi − λi · sign Pi + 2γi �3 � � 3 βi − λi �� 2 = −µV 4 (t). = −µ ��Pi + 2γi �
(11)
At the time spots tk , k = 0, 1, 2, · · · , by (9), one has + Pi (t+ k ) − Pi (tk ) = ηi (tk ) − ηi (tk ) � aij (λj (tk ) − λi (tk )) =ǫ j∈Ni
1 = (λi (t+ k ) − λi (tk )), 2γi
(12)
hence, �2 � βi − λi (t+ + k) Pi (tk ) + 2γi �2 � βi − λi (tk ) = Vi (tk ). = Pi (tk ) + 2γi
Vi (t+ k)=
(13)
3
Combine (11) and (13), it follows that V˙ i (t) + µVi 4 (t) ≤ 0 for t ∈ [0, +∞). By Lemma 1, it can be obtained that Vi (t) convergences to zero in finite time with the settling time �1 � � βi − λi0 �� 2 Ti0 ≤ µ ��Pi0 + , (14) 2γi � where λi0 is the initial values of the incremental cost estimation for the ith generator. Because σ = {t1 , t2 , . . .} is a strictly increasing sequence, there always exists some K > 0 such that tK > maxi {Ti0 }, and it has λi = 2γi Pi + βi , i ∈ IG when t > tK . Thus, it can be obtained that � λ˙ i (t) = 0, t ∈ [tk , tk+1 ) � (15) λi (t+ ) = λ (t ) + 2ǫγ aij (λj (tk ) − λi (tk )) i k i k j∈Ni
Clearly, λi keeps constant on time interval [tk , tk+1 ), k > ¯i (k) as the constant value of K. For convenience, denote λ λi (t) on time interval [tk , tk+1 ), then (15) is converted to a discrete system which is ¯ i (k) + 2ǫγi ¯i (k + 1) = λ λ
�
j∈Ni
� � ¯ j (k) − λ ¯ i (k) . (16) aij λ
� � ¯ N (k) T , (16) can be rewritten ¯ 1 (k), . . . , λ ¯ Denote Λ(k) = λ in a compact form, ¯ + 1) = (I − ǫDL)Λ(k), ¯ Λ(k (17) where D = diag(2γ1 , . . . , 2γN ). It is noted that DL is also a Laplacian matrix, of which the row sum equals to zero. In light of Lemma 2, because 0 < ǫ < 1/(maxi {2γi ∆i }), maxi {2γi ∆i } is the maximum degree of the DL. Thus, a ¯ i (k) is asymptotically reached, i.e., λ ¯ i (k) → consensus of λ λf as k → +∞, where it follows that λi (t) → λf as t → +∞. Furthermore the final consensus value λf is
Proceedings of the 20th IFAC World Congress 180 Xiao-Kang Liu et al. / IFAC PapersOnLine 50-1 (2017) 177–182 Toulouse, France, July 9-14, 2017
derived by calculating the left eigenvector of (I − ǫDL) which is 1/2γN 1/2γ1 . (18) rˆ = N , · · · , N i=1 (1/2γ1 ) i=1 (1/2γN ) N Due to λi (tK ) = 2γi Pi (tK ) + βi and i=1 (Pi (tK )) = N i=1 (Pi0 ) = PD , it can be obtained that
thus,
λf = rˆλ(tK ) N i=1 (λi (tK )/2γi ) = N i=1 (1/2γi ) PD + N i=1 (βi /2γi ) = , N i=1 (1/2γi )
(19)
(20)
(21) λ1 = λ2 = · · · = λN = λf as t → +∞. Note that (19) is the optimal condition of (7), which means the estimate λi converges to the optimal incremental cost value. Consequently, the EDP without power constraints is solved by the distributed hybrid algorithm (9). Remark 3. Based on the results in Olfati-Saber et al. ¯ (2007), the Λ(k) in the discrete system (17) exponentially vanishes with a speed faster or equal to µ2 , where µ2 = 1 − ǫλsec (DL) and λsec (·) is the operator calculating the second smallest eigenvalue of the matrix. For the variable λ(t), the exponential convergence rate can be obtained by introducing the definition of average impulsive interval in Lu et al. (2010). 3.2 EDP with power generator constraints For the EDP with power generation constraints, some supervised strategies are required, which is to ensure the obtained optimal power outputs to meet the upper and lower bounded constraints. For a better understanding of the supervised hybrid distributed algorithm, the definition of projection operator is introduced. Let’s define Proj(·) as a projection operator from R to Ω ⊆ R: Proj(x) = arg min �υ − x� (22) ν∈Ω
where Ω is a closed convex set. The power generation constraints (3) can be described as Ωi = {Pi ∈ R|Pimin ≤ Pi ≤ Pimax } for the ith generators. As seen in (8), the optimal cost λ∗ can be deemed as the agreement for generators who are not working at their boundary power outputs. Thus, a supervised strategy is required to check which generator breaks the constraint condition. Sepecifically, the intrinsic idea of the supervised strategy is based on the relationship between λi and Pi , i.e., λi = 2γi Pi +βi , which is guaranteed by a finite-time continuoustime algorithm in (9). Because Pimin ≤ Pi ≤ Pimax , obviously, λi also subjects to the constraint λmin ≤ λi ≤ i λmax , where i λmin = 2γi Pimin + βi , λmax = 2γi Pimax + βi . (23) i i Thus, when t = tk , k = 1, 2, . . ., the supervised strategy works and takes three steps: 182
Algorithm 1 Graph Transformation Algorithm. Input: Adjacency matrix A and node i. 1: for l ∈ Ni do 2: for m ∈ Ni do 3: A[l,m] = 1 4: end for 5: end for 6: A[i,:] = 0, A[:,i] = 0 7: Calculating Laplacian matrix L corresponded to A. Output: Laplacian matrix L Algorithm 2 Supervised Hybrid Distributed Algorithm. 1: Initialize the Pi0 for the ith generator. Set discrete time sequence σ = {t1 , t2 , . . .} and the accessible set Ωi = {λi ∈ R|λmin ≤ λi ≤ λmax }, where λmax = i i i min 2γi Pimax + βi and λmin = 2γ P + βi . Set λi (0) ∈ Ωi i i i and ǫ < 1/(maxi {2γi ∆i }). 2: while t < Tf inal do 3: if t = tk then 4: for i = 1 : N do 5: Pit = Pi (tk ) + ǫ j∈Ni aij (λj (tk ) − λi (tk )); λti = λi (tk ) + 2ǫγi j∈Ni aij (λj (tk ) − λi (tk )) 6: if λti > λmax or λti < λmin then i i 7: for j ∈ Ni do (Proj(λti )−λti ) 8: λtj = λtj − |Ni | (P t −P max or P min )
9: 10: 11: 12: 13: 14: 15: 16: 17: 18:
Pjt = Pjt − i i |Ni | i end for λti = Proj(λti ), Pi = Pimax or Pimin Change the neighbors of node j, j ∈ Ni , (execute Algorithm 1) end if λti → λi (tk ), Pit → Pi (tk ). end for else 12 λ˙ i (t) = µγi Pi + βi −λi sign(Pi + βi −λi ) 2γi
2γi
end if end while
(1) Check each generator unit whether the λi lies on the constraint set (23). If it is true, then go straight to the third step, otherwise, a projection operator is imposed on λi and make the deviation caused by the projection , (λi − Proj(λi )), evenly shared on its neighbor generators; (2) If λi does not lie on the constraint set, then the graph G will be transformed. Figure 1 illustrates the graph transformation with taking node 1 as example, where the node 1 pretends to be isolated and no longer takes any influence from its neighbors. However, it still serves as an intermediary communication link that permits the information exchange among the node 2,3,4. In detail, Graph Transformation Algorithm (GTA) is presented as seen in Algorithm 1; (3) Restore the value of λi (tk ) and Pi (tk ) after checking all generator units. In light of the projection operator (22) and GTA, a supervised hybrid distributed algorithm is designed for the EDP with power generation constraints, as seen in Algorithm 2.
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Xiao-Kang Liu et al. / IFAC PapersOnLine 50-1 (2017) 177–182
Remark 4. In Algorithm 2, the line 6-15 show the three steps of the supervised strategy. Note that the statements in line 5 and 16 are exact the hybrid distributed algorithm (9). In other words, by setting Pimax = +∞ and Pimin = 0, the Algorithm 2 is reduced to solve the EDP without power generation constrains.
181
12 11 10 9 8
1
λ
1 4
4
2
6 5
2 3
5
3
(a)
λ1 λ2 λ3 λ4 λ5
7
4
5
3
(b)
0
5
10
15
Time t (sec)
Fig. 1. The schematic of the graph transformation. Take node 1 as example, (a) the original graph topology; (b) the transformed graph topology.
Fig. 2. The evolution of incremental cost estimator λi without generation constraints. 90
4. SIMULATION
80
Consider a power system with 14 buses and 5 generators, where the communication graph of generators is shown in Figure 1(a). The corresponding Laplacian matrix is 3 −1 −1 −1 0 −1 1 0 0 0 L = −1 0 2 0 −1 . −1 0 0 2 −1 0 0 −1 −1 2
The parameters of cost functions and the generator output power constraints are given in Table 1.
P1 P2 P3 P4 P5
70
Output Power
In this section, an example of 14-bus system is presented to show the effectiveness of the proposed distributed hybrid algorithm in the context of EDP with and without power generation constraints, respectively.
60 50 40 30 20 10
0
5
10
15
Time t (sec)
Fig. 3. The evolution of output power Pi without generation constraints. 12
Table 1. The generator parameters γi 0.081 0.091 0.063 0.071 0.043
βi 4.623 6.143 5.357 3.736 2.715
αi 68 53 50 62 38
Pimin (kW ) 25 30 17 20 22
11
Pimax (kW ) 80 60 50 80 45
10 9 8
λ
Generator 1 2 3 4 5
7
The total power demand is PD = 200 kW , and the initialization of the Pi0 is set as P10 = 35 kW , P20 = 40 kW , P30 = 50 kW , P40 = 45 kW , P50 = 30 kW . The discrete sequence σ is selected as a average time interval sequence, where tk+1 − tk = 0.2 (sec), k = 0, 1, 2, . . .. With setting µ = 2 and ǫ = 1.5 such that ǫ < 1/(maxi {2γi ∆i }) = 2.0576, EDP without power generation constraints is solved by the hybrid distributed algorithm (9). Fig. 2 shows the incremental cost estimate of each generator unit, λi , converges to the optimal value λ∗ = 9.476. Fig. 3 shows the output power of each generator unit finally stabilizes to the optimal value, where P1∗ = 29.96 kW , P2∗ = 18.31 kW , P3∗ = 32.69 kW , P4∗ = 40.42 kW , P5∗ = 78.62 kW . 183
6
λ1 λ2 λ3 λ4 λ5
5 4 3
0
1
2
3
4
5 6 Time t (sec)
7
8
9
10
Fig. 4. The evolution of incremental cost estimator λi with generation constraints. When consider the generator power constraints, clearly, the optimal power obtained by hybrid distributed algorithm (9) does not satisfy the generator power constraints. In this case, the supervised strategy is applied, where it final-
Proceedings of the 20th IFAC World Congress 182 Xiao-Kang Liu et al. / IFAC PapersOnLine 50-1 (2017) 177–182 Toulouse, France, July 9-14, 2017
60
P1 P2 P3 P4 P5
55 Generator 5
Output Power
50
45
40
35
30 Generator 2 25
0
1
2
3
4
5 6 Time t (sec)
7
8
9
10
Fig. 5. The evolution of output power Pi with generation constraints. ly obtains P1∗ = 36.36 kW , P2∗ = 30 kW , P3∗ = 40.92 kW , P4∗ = 47.72 kW , P5∗ = 45 kW . Fig. 4 and Fig. 5 show the trajectories of incremental cost estimator and output power of each generator unit, respectively. It can be seen in Fig. 4, λ2 and λ5 are isolated and others are eventually making consensus to λ∗ = 10.51. Accordingly, in Fig. 5, P2 = P2min and P5 = P5max . Obviously, generator 2 and 5 are running on the boundaries of their power constraints, while others are not. 5. CONCLUSION This paper has investigated the economic dispatch problem by a hybrid consensus-based algorithm, which has integrated a finite-time continuous-time dynamical algorithm with a discrete-time interacted scheme. Inspired by the cluster synchronization behavior and the benefit of the projection operator, the supervised strategy has been carried out on each generator unit to ensure the power generation constraints. The stability analysis and optimality analysis of the hybrid algorithm has been given for the EDP without the power generation constraints. Moreover, the detailed pseudo-code has been presented for the EDP with power generation constraints. However, it is still in need of rigorous analysis. Future work will focus on this issue. REFERENCES Amjady, N. and Nasiri-Rad, H. (2009). Economic dispatch using an efficient real-coded genetic algorithm. IET Generation, Transmission & Distribution, 3(3), 266– 278. Bhat, S.P. and Bernstein, D.S. (2000). Finite-time stability of continuous autonomous systems. SIAM Journal on Control & Optimization, 38(3), 751–766. Binetti, G., Davoudi, A., Lewis, F.L., Naso, D., and Turchiano, B. (2014). Distributed consensus-based economic dispatch with transmission losses. IEEE Transactions on Power Systems, 29(4), 1711–172. Chen, G., Lewis, F.L., Feng, E.N., and Song, Y. (2015). Distributed optimal active power control of multiple generation systems. IEEE Transactions on Industrial Electronics, 62(11), 7079–7090. 184
Chen, G., Ren, J., and Feng, E.N. (2017). Distributed finite-time economic dispatch of a network of energy resources. IEEE Transactions on Smart Grid, 8(2), 822 – 832. Chen, S.D. and Chen, J.F. (2003). A direct newtoncraphson economic emission dispatch. International Journal of Electrical Power & Energy Systems, 25(5), 411–417. Guo, F., Wen, C., Mao, J., and Song, Y.D. (2016). Distributed economic dispatch for smart grids with random wind power. IEEE Transactions on Smart Grid, 7(3), 1572 – 1583. Li, C., Yu, X., Yu, W., and Huang, T. (2016). Distributed event-triggered scheme for economic dispatch in smart grids. IEEE Transactions on Industrial Informatics, 12(5), 1775 – 1785. Loia, V. and Vaccaro, A. (2014). Decentralized economic dispatch in smart grids by self-organizing dynamic agents. IEEE Transactions on Systems Man & Cybernetics Systems, 44(4), 397–408. Lu, J., Ho, D.W.C., and Cao, J. (2010). A unified synchronization criterion for impulsive dynamical networks. Automatica, 46(7), 1215–1221. Olfati-Saber, R., Fax, A., and Murray, R.M. (2007). Consensus and cooperation in networked multi-agent systems. Proceedings of the IEEE, 95(1), 215–233. Park, J.B., Jeong, Y.W., Shin, J.R., and Lee, K.Y. (2010). An improved particle swarm optimization for nonconvex economic dispatch problems. IEEE Transactions on Power Systems, 25(1), 156–166. Wang, Y.W., Guan, Z.H., and Wang, H.O. (2005). Impulsive synchronization for takagi-sugeno fuzzy model and its application to continuous chaotic system. Physics Letters A, 339(3-5), 325–332. Wang, Y.W., Liu, M., Liu, Z.W., and Yi, J.W. (2014). Formation tracking of the second-order multi-agent systems using position-only information via impulsive control with input delays. Applied Mathematics & Computation, 246(1), 572–585. Wood, A.J. and Wollenberg, B.F. (2012). Power generation, operation, and control. John Wiley & Sons. Xing, H., Mou, Y., Fu, M., and Lin, Z. (2015). Distributed bisection method for economic power dispatch in smart grid. IEEE Transactions on Power Systems, 30(6), 3024–3035. Yang, S., Tan, S., and Xu, J.X. (2013). Consensus based approach for economic dispatch problem in a smart grid. IEEE Transactions on Power Systems, 28(11), 4416– 4426. Yu, W., Li, C., Yu, X., Wen, G., and L, J. (2015). Distributed consensus strategy for economic power dispatch in a smart grid. In Control Conference (ASCC), 2015 10th Asian, 1–6. Zhang, Z. and Chow, M.Y. (2011). Incremental cost consensus algorithm in a smart grid environment. In 2011 IEEE Power and Energy Society General Meeting, 1–6. Zhang, Z. and Chow, M.Y. (2012). Convergence analysis of the incremental cost consensus algorithm under different communication network topologies in a smart grid. IEEE Transactions on Power Systems, 27(4), 1761– 1768.