Distributed discrete-time event-triggered algorithm for economic dispatch problem

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ContentslistsavailableatScienceDirect

Pattern Recognition Letters

journalhomepage:www.elsevier.com/locate/patrec

Distributed discrete-time event-triggered algorithm for economic dispatch problem

Renyun Jin, Haifeng Qiu

^∗

, Liguo Weng

State Grid Hangzhou Xiaoshan Power Supply Company, Hangzhou 311200, China

a rt i c l e i n f o

Article history:

Received 1 May 2020 Revised 13 August 2020 Accepted 15 August 2020 Available online 16 August 2020 MSC:

41A05 41A10 65D05 65D17 Keywords:

Distributed discrete-time algorithm Economic dispatch problem Convex optimization Equality constraint

Event-triggered communication

a b s t r a c t

Inthispaper, adistributed discrete-timeevent-triggered algorithmis proposed todealwith the eco- nomicdispatchproblemwithequalityconstraint.Thecommunicationcanbereducedandtheenergyof systemscanbesavedbyadoptingevent-triggeredcommunicationmechanism.TheZenobehaviorisnat- urallyexcludedbasedondiscreteiterationscheme.Moreover,theproposedalgorithmhasbeenproved tobeexponentiallyconvergentwiththeaidofconvexoptimizationandLyapunovstabilitytheorythat theoptimalvalueisobtainedwithdiscreteexponentialconvergencerate.Finally,theeffectivenessofthe proposedalgorithmisveriﬁedviaanumericalexample.

1. Introduction

Recently,manyresearchershavedevotedtostudyingeconomic dispatchproblem(EDP)insmart grids,whichaims tohandlethe power allocation among generators under satisfying the condi- tions oftotalload[1,2,5–7,11–15,18].Generally,EDPcan becasted as a constrained optimization problem, that is, the total generation cost is minimizedby allocating the generation powerunder theloadconstraint, i.e., thesumofactivepowergeneratedby all generators equaltothe totalloaddemand.Traditionalcentralized method, such as Lagrange multiplier method, can solve the EDP, whichdemand powerfulcentral processorstoobtain globalinfor- mation and deal with a large number of data, see for example [8,13,16] and references therein. However, with the development of networkscience, renewable resources, clean energyandsmart grids, network scale anddata volumewill continueto grow. The centralized methodwill be restricted,its processingcapacitywill bereduced,andthecommunicationcostwillbegreatlyincreased.

Therefore,distributedalgorithmisagoodchoicetodealwithEDP, whichcanincreaseprocessingcapacity,eﬃciencyandreliability.It

∗ Corresponding author.

E-mail address: [email protected] (H. Qiu).

alsohasgoodapplicationsinlargeimagedataprocessing,machine learningandotherﬁelds[9,10].

Each generator does not need to obtain all variable information based on distributedeconomic dispatch algorithm. The core ideaistotransformthecentralizedoptimizationproblemintosev- erallocal optimization problems. Onthe basis oflocal optimization,theglobaloptimalresultcanbeobtainedbyexchanginglocal informationthroughthecommunicationnetwork.Aclassofimpor- tantdistributed algorithmsforsolving EDP areconstructed based on theidea of consensus, which are consensus-based distributed economic dispatch approaches. Forinstance, by selecting the in- crementalcostofeach generation unit astheconsensus variable, the incremental cost consensus algorithm wasable to solve EDP in a distributed manner in [20]. The authors in [2] studied not onlytheEDPwithgeneratorconstraintsbutalsotheactualsitua- tionoftransmissionlinelossesinthesystem.In[19],anovelcon- sensusbasedalgorithmwithatacticalinitial setupwasproposed to solve EDP in a distributed fashion under strongly connected communicationgraph.Toavoidinitialization,aninitialization-free distributed algorithm was designed over a strongly connected, weight-balanceddigraphin[7],startingfromanyinitialpoweral- location.Tosolvethe rapidityofeconomic dispatch,a distributed andfasteconomicdispatchalgorithmwasprovidedin[5],andthe optimalvalue wasachievedintheﬁnite-time,whichmakes more https://doi.org/10.1016/j.patrec.2020.08.016

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senseinrealapplications.Moreover,considering thattheincrease ofnetworkscale will increasethe communicationburden, event- triggered communication mechanism can effectively reduce this communication burden. A distributed event-triggered algorithm was proposed in [22] to solve a convex quadratic optimization problem under undirected and connected topologies, which had appeared in the economic dispatch, the resources allocation, the optimalportfolio,andsoon.The event-triggeredcommunication- based scheme for EDP had been investigated in [12], where the economicdispatchprocedurewasinvolvedincontinuous-timeop- timization and discrete-time communication. To reduce required capacities for information exchanges in micro-grids, a novel dis- tributedevent-based algorithm was proposed in [21]for optimal energymanagementinamicro-gridwithconsideringpowerlosses.

Continuous-timeevent-triggeredalgorithmmustexcludetheZeno behaviortoavoidcontinuouscommunication,thatis,toavoidinﬁ- nitecommunicationinlimitedtime.Becauseofthis,thealgorithm designedinthispaperadoptsdiscreteiterativescheme.

Motivated by the distributed algorithm in [4], a distributed discrete-timeevent-triggered algorithm is proposed to solve EDP inthispaper.Theburdenofcommunicationisreducedandtheen- ergyofsystemsissavedviaevent-triggeredcommunicationmechanism,andtheZenobehavior isnaturallyexcluded basedon discrete iteration scheme.The exponential convergence of the algorithm is proved with the aid of convex optimization and Lya- punov stability theory. The main contributions of this paper in- clude:(1)Adistributeddiscrete-timeoptimizationalgorithmisde- signedtosolveaclassofEDPbasedonevent-triggeredcommunicationmechanism. Thedesignedalgorithm hastheadvantagesto reducethecommunicationandsavetheenergyofsystems,andthe Zenobehavioris naturallyexcluded;(2)The optimalvalueofthe totalgenerationcostisobtainedwithdiscreteexponentialconver- gencerate.

The paper is organized as follows.Firstly, some preliminaries andproblemformulationaregiveninSection2.Section3provides adistributed discrete-timeevent-triggered algorithm andconver- genceanalysis.Numericalsimulations aregiveninSection4.Sec- tion5concludesthispaper.

Notation: In thispaper,Rⁿ denotes thesetof n× 1realvec- tors;Rⁿ^×ⁿdenotesthesetofn×nrealmatrices;0n/1ndenotesan n×1columnvectorofallzeros/ones;Indenotesann-dimensional identitymatrix;

·

^denotes^the^Euclidean^norm;^[

χ

^]^denotes^the

largestintegernomorethan

χ

^;^Let

λ

2(L)and

λ

^max(^L)^be^the^mini- malpositiveandmaximaleigenvalueofLaplacianmatrixL,respec- tively.C:Rⁿ→Rⁿ is the gradient ofC; ²C:Rⁿ→Rⁿ^×n is the HessianmatrixofC.

2. Preliminariesandproblemformulation 2.1.Algebraicgraphtheory

Acommunicationnetworkofnnodesisdenotedbyaweighted undirectedgraphG=

{

^V,E,A

}

,inwhichV=

{

¹,2,...,n

}

^denoting

the network nodes and the edge set E⊆V × V representing the communicationlinks. Anundirected edge(i, j) ∈E indicates that nodei andnode jcan exchangeinformationwitheachother.De- ﬁnetheweightedadjacencymatrixA=[a_{i j}]∈Rⁿ^×ⁿwitha_{i j}=a_ji>

0if(i,j)∈Eanda_{i j}=0otherwise.NotethatGissaidtobeundi- rectedandconnectedif(j, i)∈E implies(i,j) ∈E forarbitraryi, j ∈V. The set ofcommunicationneighbors ofagent i is denoted by N_i=

{

^j∈V

|

(^j,i)∈E

}

^. ^The ^Laplacian ^matrix ^L[l_{i j}]∈Rⁿ^×ⁿ of graphGis deﬁnedasl_{i j}=−a_{i j} when i = j andl_ii=n

j=1a_{i j}.For an undirected and connected graph, 0 is a simple eigenvalue of theLaplacianmatrixLwiththe eigenvector1n,andall theother eigenvaluesarepositive.

2.2. Usefullemma

Lemma1([17].). Forany vectorsx,y∈Rⁿ andonepositivedeﬁnite matrixH∈Rⁿ^×ⁿ,thefollowinginequalityholds

2x^Ty≤x^THx+y^TH⁻¹y. (1)

2.3. Problemformulation

In this subsection, let us consider a smart grid of n generators. LetP_i denotes the active power generated by the generator i,whereeachgeneratorhasagenerationcostC_i(P_i),whichisonly known by generator i. The total generation cost C(P) is the sum of the all local objective functions, i.e., C(^P)=n

i=1C_i(^Pi), where C_i(^Pi)=

α

iP_i²+

β

iP_i+

γ

i.Thetargetofeconomicdispatchistomini- mizethetotalgenerationcostwhilesatisfyingthebalancebetween demandandsupply,i.e.

MinC

(

^P

)

=_n

i=1

α

ⁱ^Pi²+

β

ⁱ^Pⁱ⁺

γ

ⁱ

, s.t._n

i=1Pi=PD, (2)

whereP=[P₁,P₂,...,P_n]^T∈Rⁿ is adecision vector;

α

i,

β

iand

γ

i

representthecostcoeﬃcients;P_D isaglobalconstraintvariable.

The above economic dispatch problem can be solved by the conventional centralized method, that is, the Lagrange multiplier method. The optimal Lagrangian multiplier is obtained, i.e.,

λ

^∗=

P_D+n i=1 β_i

2α_i n

i=1 1

2α_i ,andtheoptimalsolutionP_i^∗=^λ^∗₂⁻_α^βⁱ

i .Socentralizedal- gorithm can solve this problem easily when all parameters can be processedcentrally. The purposesof thisworkis toprovide a distributed discrete-time algorithm to solve the above economic dispatchproblem via event-triggeredcommunicationmechanism, whichcanreducingthecommunicationburdenonlargescalenet- works.

Assumption 1. The objective function C(P) is twice continu- ously differentiable, strongly convex with convexity parameters ≥

σ

>0 such that

σ

^In≤

∇

²^C⁽^P⁾≤^In.

Assumption 2. The communication topology is undirected and connected.

Assumption3. Thecommunicationenvironmentissecure,reliable andwithoutdelayanddisturbanceincommunicationprocess.

2.4. Event-triggeredcommunicationmechanism

Before proposing distributed event-triggered algorithm, we ﬁrst make a brief introduction to event-triggered communicationmechanism. x_i(^k)(ⁱ=1,2,...,n,k=0,1,2,...) ^represents^the transportable variable in the communication network. xˆ_i(^k)= x_i(^kⁱt_i),kⁱ_t

i≤k<kⁱ_t

i+1 representstheinformationto betransmitted atevent-triggeredtimeinstantkⁱ_t

i.F_i(k,x_i(k))denotestriggerfunc- tion fornode i.Only whenthe triggerfunction F_i(k, x_i(k))> 0is satisﬁed,thenodeitransmitsits currentinformationtoitsneigh- boring nodes immediately. The sequence of event-triggered dis- creteinstants

{

^kⁱt_i

}

^for^nodeⁱ^is^deﬁnediterativelyaskⁱ_t

i+1=inf

{

^k^:

k>k_tⁱ

i,F_i(^k,x_i(^k))>0

}

⁽^ti=0,1,2,...). Itisworthnotingthat the Zeno behavior is naturallyexcluded because ofthe discrete-time scheme.

3. Algorithmdesignandconvergenceanalysis 3.1. Distributeddiscrete-timeevent-triggeredalgorithm

Decentralizedtriggerfunctionsaredeﬁnedas

(3)

F_i

(

^k,^xi

(

^k

))

⁼^e²i

(

^k

)

⁻

γ α

^[^ω^k^]^, ⁽³⁾

G_i

(

^k,^yi

(

^k

))

⁼

ς

i²

(

^k

)

⁻

μβ

^[^σ^k^]^, ⁽⁴⁾

where e_i(^k)=xˆ_i(^k)−x_i(^k)^; ^x^ˆi(^k)=x_i(^kⁱt_i),k_tⁱ

i≤k<kⁱ_t

i+1;

ς

i(^k)=

ˆ

y_i(^k)−y_i(^k)^;^y^ˆi(^k)=y_i(^kⁱs_i),kⁱ_s

i≤k<kⁱ_s

i+1;

α

^,

β

^,

ω

^,

σ

∈(0,1)and

γ

^,

μ

>0aredesignparameters.

Based ontheabove decentralizedtriggerfunctions(3)-(4)and Assumptions 1–3, the following distributed discrete-time event- triggered algorithmisproposed to solveeconomic dispatchproblem(2)

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

y_i

(

^k+1

)

=y_i

(

^k

)

+

ε

j∈Ni

a_{i j}

ˆ

x_i

(

^k

)

−xˆ_j

(

^k

)

, P_i

(

^k

)

⁼

j∈Ni

a_{i j}

ˆ

y_j

(

^k

)

⁻^y^ˆi

(

^k

)

+P_i

(

⁰

)

^, xi

(

^k

)

=

∂

^Cⁱ

(

^Pi

(

^k

))

∂

^Pi

(

^k

)

^,

(5)

where

ε

> 0isconstant.Weseethattheneighboringnodesneed toexchangetheinformationofauxiliaryvariablesx_i(k)andy_i(k).

Suppose that there is one node knows global constraintvari- able P_D, it can start from P_D while the others start from 0,then n

i=1P_i(⁰)=P_D. For undirected and connected graph G, n

i=1P_i(^k)=n i=1

j∈N_i

a_{i j}

ˆ

y_j(^k)−yˆ_i(^k)

+n

i=1P_i(⁰)=P_D. There- fore,the economic dispatchproblem(2)can be transformedinto thefollowingunconstrained convexoptimizationproblemofvari- ableyˆ(^k)

MinC

(

^y^ˆ

(

^k

))

= n

i=1

C_i

j∈Ni

a_{i j}

ˆ

y_j

(

^k

)

−yˆ_i

(

^k

)

+P_i

(

⁰

)

. (6)

Remark 1. Note that the gradient of C(P) can be described as

∇

^C(^P)=x(^k)=[x1(^k),x2(^k),...,xn(^k)^]^T∈Rⁿ. According to algorithm(5),wegetthegradientofC(^y^ˆ)^as

∇

^C(^y^ˆ(^k))=−L^T

∇

^C(^P)=

−L^Tx(^k),whereyˆ(^k)=[yˆ₁(^k),yˆ₂(^k),...,yˆn(^k)^]^T∈Rⁿ.Itisobvious that theHessianmatrixofC(^y^ˆ)^satisﬁes

∇

²^C(^y^ˆ)=L^T

∇

²^C(^P)^L^.^For the unconstrained convex optimization problem (6), the optimal valuey^∗canbeobtainedwhen

∇

^C(^y^ˆ(^k))=0n(^k→∞).

3.2. Convergenceanalysis

Theorem 1. Under Assumptions 1–3, if the parameter

ε

^satisfy-

ing 0<

ε

< ₁ ¹

+2λ^ˆ, the distributed discrete-time event-triggered al- gorithm (5) solves economic dispatch problem (2)with trigger func- tions(3)and(4),andtheZenobehaviorisnaturallyexcluded,where

λ

ˆ=

λ

²max(^L),

λ

max(^L)^denotes^the^largest^eigenvalue^of^the^Laplacian matrixL,and>0istheconvexityparameterinAssumption1.

Proof. Fortheconvenienceofproof,thealgorithm(5)canbewrit- teninmatrixform

_y

₍

_k₊₁

₎

₌_y

₍

_k

₎

₊

ε

^L^x^ˆ

(

^k

)

, P

(

^k

)

⁼^−L^y^ˆ

(

^k

)

⁺^P

(

⁰

)

^, x

(

^k

)

=

∇

^C

(

^P

(

^k

))

,

(7)

where xˆ(^k)=x(^k)+e(^k), yˆ(^k)=y(^k)+

ς

(^k), xˆ(^k)= [xˆ₁(^k),xˆ₂(^k),. . .,xˆn(^k)^]^T, y(^k)=[y₁(^k),y₂(^k),...,yn(^k)^]^T, P(^k)=[P₁(^k),P₂(^k),...,P_n(^k)^]^T^.

Supposethattheuniqueoptimalsolutionfortheunconstrained convex optimization problem (6) is denoted by y^∗, then C(^y^∗)≤ C(^y^ˆ(^k))^.^Thediscrete-timeLyapunovfunctionisselectedas V

(

^k

)

=C

(

^y^ˆ

(

^k

))

−C

(

^y^∗

)

. (8)

The difference of V(k) is given by V(^k+1)=V(^k+1)− V(^k)=C

ˆ y(^k+1)

−C

ˆ y(^k)

.It followsfromthestrongconvexity property[3]that

C

ˆ y

(

^k+1

)

−C

ˆ y

(

^k

)

=

∇

^T^C

(

^y^ˆ

(

^k

))

ˆ

y

(

^k+1

)

−yˆ

(

^k

)

+1 2

yˆ

(

^k⁺¹

)

⁻^y^ˆ

(

^k

)

_T

∇

²^C

(

^y^¯

)

ˆ

y

(

^k⁺¹

)

⁻^y^ˆ

(

^k

)

, (9)

where y¯=yˆ(^k)+

θ

ˆ

y(^k+1)−yˆ(^k)

with

θ

∈ (0, 1). In addition, accordingtoalgorithm(7),wehave

V

(

^k+1

)

=−x^T

(

^k

)

^L

ε

^L

(

^x

(

^k

)

+e

(

^k

))

+

ς (

^k+1

)

−

ς (

^k

)

+1 2

ε

^L

(

^x

(

^k

)

⁺^e

(

^k

))

⁺

ς (

^k⁺¹

)

⁻

ς (

^k

)

TL^T

∇

²^C

(

^P^¯

)

×L

ε

^L

(

^x

(

^k

)

+e

(

^k

))

+

ς (

^k+1

)

−

ς (

^k

)

. (10)

ForundirectedandconnectedgraphG,one gets L^T=L.More- over,byapplyingLemma1,onehas

V

(

^k+1

)

≤ −

ε

^x^T

(

^k

)

^LL^T^x

(

^k

)

+

ε

²

2x^T

(

^k

)

^LL^T^x

(

^k

)

+

λ

ˆ

2e^T

(

^k

)

^e

(

^k

)

+

ε

²

2x^T

(

^k

)

^LL^T^x

(

^k

)

+ 1

2

ε

²

ς (

^k+1

)

−

ς (

^k

)

²

+

^ˆ

λ

2

ε

^L^T

(

^x

(

^k

)

+e

(

^k

))

+

ς (

^k+1

)

−

ς (

^k

)

²

≤ −

ε

⁻

(

¹+2

λ

^ˆ

) ε

²

x^T

(

^k

)

^LL^T^x

(

^k

)

+

λ

^ˆ

2+2

λ

^ˆ²

ε

²

×e^T

(

^k

)

^e

(

^k

)

+

1 2

ε

²⁺

λ

^ˆ

ς (

k+1

)

−

ς (

^k

)

²^, ⁽¹¹⁾

where

λ

ˆ=

λ

²max(^L),

λ

^max(^L) ^denotes^the^largest ^eigenvalue^of^the LaplacianmatrixL,and > 0is theconvexity parameterin As- sumption1. Moreover, accordingto trigger functions(3) and(4), wehavee^T(k)e(k)≤ n

γ α

^[^ω^k] ^and

ς

(^k+1)−

ς

(^k)

²≤4n

μβ

^[^σ^k^]^.

Substitutingthesetwoinequalitiesbackinto(11)yields V

(

^k+1

)

≤−

ξ

^x^T

(

^k

)

^LL^T^x

(

^k

)

+

μ

1

α

^[^ω^k^]+

μ

2

β

^[^σ^k^],

=−

ξ ∇

^C

(

^y^ˆ

(

^k

))

²+

μ

1

α

^[^ω^k^]+

μ

2

β

^[^σ^k^], (12) where

ξ

=

ε

−(¹+2

λ

^ˆ)

ε

²>0,

μ

1=n

γ

ˆλ

2+2

λ

^ˆ²

ε

²

and

μ

2= 4n

μ

₁

2ε² +

λ

^ˆ

.

Accordingto the strongconvexity property [3]andRef. [4], it iseasytoget

∇

^C(^y^ˆ(^k))

²≥¹2

σλ

²2

C(^y^ˆ(^k))−C(^y^∗)

=¹2

σλ

²2V(^k), where

σ

îs^the^convexity^parameterôf^function^CⁱⁿÂssumption^1,

and

λ

2 representsthesmallestpositiveeigenvalueoftheLaplacian matrixL.

Inaddition,[

ω

^k]^denotes ^the^largest^integer ^no^more^than

ω

^k

with

ω

∈(0,1),thus,

α

^[^ω^k^]≤

α

^ω^k⁻¹= ¹_α(

α

^ω)^k^.^Meanwhile,

β

^[^σ^k^]≤

β

^σ^k⁻¹=_β¹(

β

^σ)^k^.^Then,^one^gets V

(

^k+1

)

≤−1

2

ξσλ

²2V

(

^k

)

+

μ

¹

α ⁽ α

^ω

)

^k+

μ

²

β ⁽ β

^σ

)

^k. (13) Let

ν

=1−¹₂

ξσλ

²2∈(⁰,1),onehas

V

(

^k

)

≤

ν

^V

(

^k−1

)

+

μ

¹

α ⁽ α

^ω

)

^k⁻¹+

μ

²

β ⁽ β

^σ

)

^k⁻¹ ..

.

≤

ν

^k^V

(

⁰

)

⁺

μ

1

α

k

i=1

ν

^k⁻ⁱ

( α

^ω

)

ⁱ⁻¹⁺

μ

2

β

k

i=1

ν

^k⁻ⁱ

( β

^σ

)

ⁱ⁻¹

=

ν

^k^V

(

⁰

)

⁺

μ

1

α ν

^k−

( α

^ω

)

^k

ν

⁻

( α

^ω

)

⁺

μ

2

β ν

^k−

( β

^σ

)

^k

ν

⁻

( β

^σ

)

=

V

(

⁰

)

⁻

μ

1

α ( α

^ω⁻

ν )

⁻

μ

2

β ( β

^σ⁻

ν ) ν

^k

+

μ

¹

α ( α

^ω⁻

ν ) ( α

^ω

)

^k+

μ

²

β ( β

^σ⁻

ν ) ( β

^σ

)

^k. (14)

(4)

Fig. 1. Communication network.

Table 1

Parameters αi, βi, γiof C i.

C i 1 2 3 4 5 6

αi 0.096 0.072 0.105 0.082 0.078 0.090 βi 1.22 3.41 2.53 4.02 2.90 2.72

γi 51 31 78 42 57 49

Fig. 2. Trajectories of P i( t ).

Because

ν

^,

α

^ω^,

β

^σ ∈ (0, 1), V(k) exponentially converge to 0. Thus, the function C(^y^ˆ(^k)) exponentially converges to C(y^∗).

In other words, the optimal value y^∗ can be obtained by the designed distributed discrete-time event-triggered algorithm (5). Inthatway,theoptimalvalueP^∗oftheoriginaleconomicdispatch problem(2)canalsobeobtained, i.e.,P^∗=−Ly^∗+P(⁰)^.^Moreover, the Zeno behavior is naturally excluded under discrete iterative

scheme.

4. Simulationresults

Inthissection,an exampleisinvestigatedtoillustrate thedis- tributeddiscrete-time event-triggered algorithm proposed in this paper.The correspondingcommunicationnetworkisa undirected andconnectedgraphas showninFig. 1.The parameters ofeach costfunctionC_i(P_i)arelistedinTable1([22]).

Case1:AlgorithmSimulation

The parameters in ouralgorithm are setas follows:P_D=600,

ε

=0.1,

γ

=50,

μ

=100,

α

=0.85,

β

=0.79,

ω

=0.19,

σ

=0.13. The initial values are set as P(⁰)=[P_D,0,0,0,0,0]^T and y(⁰)= [0,0,0,0,0,0]^T.ForthecommunicationtopologyinFig.1,

λ

2=1 and

λ

^max=5.FortheC(^P)=n

i=1C_i(^Pi),twoconvexityparameters are chosen as=0.105 and

σ

=0.072. The iterativenumber of eachgeneratoris800.Thesimulationresultsunderouralgorithm (5)are shown inFigs. 2–7.

Fig. 2 shows that P_i(^k),i=1,2,...,6, converge to their corresponding optimal values: P₁^∗=97.9219,P₂^∗=115.3539,P₃^∗= 83.2905,P₄^∗=97.5672,P₅^∗=109.7500 and P₆^∗=96.1165.

Fig. 3. Trajectories of C ( P ( k )) and E ( k ).

Fig. 4. Trajectories of ˆ x (k ).

Fig. 5. Trajectories of ˆ y (k ).

Fig. 3 shows the trajectories of the total cost function C(P(k)) and error function E(k), where E(^k)=C(^P(^k))−C(^P^∗), C(^P^∗)=7162.03705. Figs. 4 and 5 show the trajectories of xˆ(^k) and yˆ(^k), which are transmitted in the communication network. They are also the values at the event-triggered instant.

The event instants of each generator, i.e.

{

^kⁱt_i

}

,i=1,2,...,6 and

{

^kⁱs_i

}

,i=1,2,...,6areshowninFigs.6and7.

Moreover, iterations, events and the ratios of communication to iteration for all generators are listed in Table 2. The informa- tiontransmissioninthenetworkdoesnotoccurateveryiteration time,onlyatthemomentoftheevents.Themaximumratioisonly 9.75%, which shows that event-triggered communication mecha-

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Fig. 6. Events of 6 generators with variable x ( k ).

Fig. 7. Events of 6 generators with variable y ( k ).

Table 2

The number of communication events.

Generators 1 2 3 4 5 6

Iterations 800 800 800 800 800 800 Events- x ( k ) 46 44 41 34 53 33 Ratios(%) 5.75 5.5 5.125 4.25 6.625 4.125 Events- y ( k ) 78 52 61 72 59 64 Ratios(%) 9.75 6.5 7.625 9 7.375 8

nism can reduce the burdenof communicationandsave the en- ergyofsystems.

From the above simulation results, the optimal active pow- ers generated by generators can be obtained based on distributeddiscrete-timeevent-triggeredalgorithm(5),andthenum- berofcommunicationissigniﬁcantlyreducedbasedontheevent- triggered communicationmechanism, which isvery beneﬁcial to savenetworkbandwidth.

Case2:AlgorithmComparison

Themainadvantageofthealgorithmproposed inthispaperis to save communication, andit does not need to exchange information in each iteration process. Based on event-triggered communication mechanism, information exchange only occurs when the trigger function is triggered. A excellent consensus basedal- gorithm [19]wasproposed tosolve EDP ina distributedfashion.

For the convenience ofstatement, we use A1 and A2 to respec- tively denotethedistributeddiscrete-timealgorithm[19]andour distributed discrete-timeevent-triggeredalgorithm (5).Algorithm

Fig. 8. Trajectories of x i( t ).

Table 3

The number of communication.

Node 1 2 3 4 5 6

Algorithm A1 200 200 200 200 200 200 Algorithm A2 124 96 102 106 112 97

A1isdescribedas

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

λ

i

(

^k+1

)

=

j∈N_i⁺

pi j

λ

j

(

^k

)

+

^yi

(

^k

)

, x_i

(

^k+1

)

=b_i

λ

i

(

^k+1

)

+a_i, yi

(

^k+1

)

=

j∈N⁺_i

qi jyj

(

^k

)

−

(

^xⁱ

(

^k+1

)

−xi

(

^k

) )

^,

where

=0.02,a_i=−2^βαⁱi,b_i= 2¹αi;p_iandq_isee[19]fordetails.

Here,x_i(k)denotestheactivepowergenerated.Thatis,P_i(k)inour algorithm.Fig.8showsthatthe optimalvaluesare obtainedonly after100iterationsbyusingalgorithmA1.However,thevariables xand y need to be transmitted ineach iteration process. There- fore,thenumber ofcommunicationof each node are200. It can beseenfromTable3thatthenumberofcommunicationbyusing ourdistributed discrete-timeevent-triggeredalgorithm (5) (Algo- rithmA2)arelessthanthatofAlgorithmA1.Itisundeniablethat AlgorithmA1isstillanexcellentalgorithm.

5. Conclusion

Thetarget ofeconomicdispatchin smartgrids istominimize thetotalgenerationcostwhilesatisfyingthebalancebetweende- mandandsupply.Inthispaper,adistributeddiscrete-timeevent- triggered algorithm is designed to solve the economic dispatch problem.Theoptimalvalueofthetotalgenerationcostisobtained with discrete exponential convergence rate. The designed algorithm hastheadvantagesto reduce the communicationandsave the energy of systems, and naturallyexclude the Zeno behavior.

The simulationresults have validatedthe correctnessof the the- oretical resultsandshownthe advantagesofthe algorithm. Con- sideringthe fast convergence ofthe algorithm, the ﬁnite-step or ﬁxed-stepiterativealgorithm willbefurtherstudiedinthefuture work.Moreover,undertheunreliablecommunicationenvironment, howtodesignafastiterativealgorithmisalsoverymeaningful.

DeclarationofCompetingInterest

Wedeclarethat wedonothaveanycommercialorassociative interestthatrepresentsaconﬂictofinterestinconnectionwiththe worksubmitted.

Theauthorsdeclarethattheyhavenoknowncompetingﬁnan- cialinterestsorpersonalrelationshipsthatcouldhaveappearedto inﬂuencetheworkreportedinthispaper.

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