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Electrical Power and Energy Systems

journal homepage: www.elsevier.com/locate/ijepes

Short Communication

Optimal criterion and global/sub-optimal control schemes of decentralizedeconomical dispatch for AC microgrid

Zhangjie Liua, Mei Sua, Yao Suna,⁎, Lang Lia, Hua Hana, Xin Zhangb, Minghui Zhengc

a Central South University, School of Information Science and Engineering, Changsha, ChinabNanyang Technological University, School of Electrical and Electronic Engineering, SingaporecUniversity at Buffalo, The State University of New York, Department of Mechanical and Aerospace Engineering, New York, United States

A R T I C L E I N F O

Keywords:Economical dispatchDecentralized controlCost-based droopMicrogrid

A B S T R A C T

This letter studies the economical dispatch problems (EDPs) in islanded microgrid from three aspects: a) whetherthe global optimal dispatch (GOD) can be achieved through a decentralized manner in a system; (b) if yes, howto achieve the GOD; (c) otherwise, how to obtain a suboptimal alternative. As a result, we arrive a criterion thatGOD can be achieved in decentralized manners if and only if optimal solution functions (OSFs) are all strictlymonotonically increasing. If a system meets this criterion, a decentralized droop method is proposed to help thesystem achieve GOD. Otherwise, a modified decentralized method is presented to guarantee the suboptimaldispatch. Both simulation and experiments are performed to validate the proposed methods.

1. Introduction

ECONOMICAL dispatch is considered as one of the core problems inmicrogrid research [1–3]. Decentralized economical dispatch methodshave drawn increasing attention because of its high reliability and nocommunication.

Some decentralized methods based on droop concept are proposedin [4–12]. The nonlinear droop schemes based on equal cost principleare introduced in [4–7], in which the main idea is to force the cost ofeach distributed generator (DG) equal by droop control. As a result,since the cheaper DGs produce more power, the total system cost isreduced. Moreover, a nonlinear droop construction approach is pro-posed based on polynomial approximation in [8], which can reduce thetotal cost by selecting appropriate coefficients. In order to realize plug-and-play, an improved droop construction approach is proposed in [9],which can function independently for each DG by optimizing each oneagainst hypothetical DGs. The methods in [4–9] are suboptimal. Torealize the global optimal dispatch (GOD), a nonlinear λ-consensus al-gorithm is introduced in [10–12]. The incremental costs are subtlyembedded into the droop control. When the frequency is synchronized,the total cost is minimized in terms of the equal incremental costprinciple (EICP) [10]. The normalization economic scheme based onEICP is proposed for hybrid AC/DC microgrid [11]. Considering thecapacity limitation of DG, a fitting method is applied to balance DG’ssynchronous operation and economy [12].

EICP works only under the convex cost function, which motivates us

to consider the following three problems

1) under what conditions GOD can be realized in a decentralizedmanner

2) How to design the droop-based controller to achieve the optimiza-tion

3) When GOD cannot be realized by decentralized manners, how todesign an appropriate suboptimal controller?

To address these problems, some works have been done and aresummarized as follows:

A criterion is presented to determine whether GOD can be realizedvia a decentralized manner.

An optimal decentralized economical-sharing scheme is proposedwhen the criterion is met.

A suboptimal scheme is proposed when the criterion is not met.

2. Economical dispatch problems formulation

Mathematically speaking, the objective of economical dispatchproblem (EDP) is to minimize the total generation cost subjected to thedemand-supply constraints as well as the generator constraints. Theoptimization problem is formulated as

https://doi.org/10.1016/j.ijepes.2018.06.045Received 2 April 2018; Received in revised form 22 May 2018; Accepted 19 June 2018

⁎ Corresponding author.E-mail address: yaosuncsu@gmail.com (Y. Sun).

Electrical Power and Energy Systems 104 (2019) 38–42

0142-0615/ © 2018 Elsevier Ltd. All rights reserved.

T

∑

∑

⎧

⎨

⎪⎪

⎩⎪⎪

= ⩽ ⩽ ∈ ⋯

=

=

C P

s t P P P P i n

min ( )

. . ; 0 , {1, 2, , }

i

n

i i

i

n

i L i i

1

1,max

(1)

where Pi, Ci (Pi), and Pi, max are the active power, cost function andupper bound of Pi, respectively, and PL is the total load. For an ACmcirogrid with inductive transmission network, PL is equal to the sumof all loads. We use PL,max to denote the upper bound of PL, i.e.,PL≤ PL,max. Because the constraints in (1) are compact and the costfunctions are all continuous, it follows from the well-known extremevalue theorem that the EDP has a global optimal solution (GOS) P∗.

Each PL (PL∈[0,PL,max]) is exactly related to the GOS ⋯∗ ∗ ∗P P P( , , , )n1 2such that the cost is minimal. Thus, the optimal solution ∗Pi can beregarded as the function of the load PL, whose curve can be sketched byoff-line calculation. For convenience, we name it as optimal solutionfunction (OSF). Define = = ∗g P P P( ) ( )i L i L

Δand it inherently yields

∑ ==

g P P( )i

n

i L L1 (2)

In AC microgrid, the economical dispatch is usually achieved via themethods based on droop controls [1–5]. The droop- based controlscheme (P-f) is presented as

= −∗f f φ P( )i i i (3)

where fi and f∗ are the ith DG actual frequency and reference, respec-tively, and φi(Pi) is the droop function. To guarantee the uniqueness ofthe steady operating point (SOP), the droop function φi(Pi) should bemonotonic. When the system frequency achieves synchronization, wehave

= = ⋯=φ P φ P φ P( ) ( ) ( )n n1 1 2 2 (4)

From the balance of power supply-demand, it yields

+ + ⋯+ = ⩽ ⩽P P P P P P(0 )n L L L1 2 ,max (5)

Then, in steady state, the output power of each DG, i.e., SOP, aredetermined by (4) and (5). Given that all the φi(Pi) have the same valuein steady state, we assume they are equal to a common variable y, i.e.,

= = ⋯= =φ P φ P φ P y( ) ( ) ( )n n1 1 2 2 (6)

Then, we obtain = −P φ y( ),i i1 where −φi

1 is the inverse of φi.Substituting it into (5), we have ∑ ==

−φ y P( ) .in

i L11 Denote

= ∑ =−F φ(·) (·)i

ni1

1 . Then, the SOP is obtained as

= = ∘− − −P φ y φ F P( ) ( )i i i L1 1 1 (7)

where ∘f g represents f(g(x)). Likewise, the system SOP also changeswith variation of the load. It can be regarded as the function of PL andwe use = = − −h P φ F P( ) ( ( ))i L i L

Δ 1 1 to denote the SOP function (SOPF).For the problem of droop-based economical dispatch, the key is to

design the droop function φi (Pi) to ensure that SOP is the GOS, ∀ PL∈[0,PL,max]. That is, the curve of hi(PL) completely coincide with gi(PL).Thus, the EDP can be formulated as

∫ −h P g P dPmin ( ( ) ( ))f

Pi L i L L0

2

i

L,max

(8)

3. GOD criterion and decentralized control schemes

3.1. GOD criterion via decentralized manner

To solve the optimization problem (8), the following two questionsare crucial:

I) Under what conditions there exists a series of droop functions φi

such that ≡h P g P( ) ( )i L i L ?

II) How to design a suboptimal fi if there does not exist droop functionssuch that ≡h P g P( ) ( )i L i L ?

Question 1) and 2) are equivalent to I) and 3) becomes II). The mainresults of this letter are listed as follow:

Theorem 1. If all the OSFs are strictly monotonically increasing, theGOD can be achieved by constructing the following droop law

= −∗ −f f mg P( )i i i1 (9)

where m is a positive constant to keep the frequency deviation in theacceptable range, i.e.,

=−∗

−mf fg Pmax{ ( )}

i i L

min1

,max (10)

where fmin is the acceptable minimal value of frequency.

Proof. When the system frequency reaches synchronization, accordingto (7), the SOPF can be obtained as

= ∘ =− − − − −P mg F P g m F P( ) ( ) ( ( ))i i L i L1 1 1 1 1 (11)

Considering = −φ mg ,i i1 F(PL) becomes ∑ =

−g m P( )in

i L11 . According to

(2), we obtain F(PL) = m−1PL. Substituting it into (10), we have

= = =− − −P g m F P g m mP g P( ( )) ( ) ( )i i L i L i L1 1 1 (12)

Thus, ≡h P g P( ) ( )i L i L . That is, for any PL (PL≤ PL,max), the SOP is theGOS. Hence, when the system achieve the steady-state, the GOD will beachieved by taking the droop law as fi= f∗-m gi−1(Pi). □

Theorem 2. If any of the OSFs is not strictly monotonically increasing,GOD cannot be achieved in decentralized manners.

Proof. To illustrate Theorem 2, we first give two typical non–monotonic examples

saturation and dead-zone. Considering a DG with limited capacity,the DG optimal power may become saturated (reaching the limitation)when the load is too heavy. On this occasion, the DG should be set tocurrent/power control mode to maintain the output power at the lim-iting value. However, only when the common frequency is available cancurrent/power control be implemented, which means communication isnecessary. Moreover, without knowing the real-time information ofload, when to switch back to the droop mode cannot be determined.Likewise, in the case that DG OSF shows characteristics of dead-zone,communication is also necessary to generate a command signal.Therefore, in the above two cases, only with centralized communicationcan GOD be realized. Next, we give a generalized proof of Theorem 2.

It is assumed that there exists a series of function φi such that hi(PL)≡ gi(PL), where gi(PL) is nonmonotonic. Then, we obtain F−1(PL) ≡φi(gi(PL)). Given that F−1 is the inverse of function F, F−1 must bemonotonic, which is contradictory with the fact that φi(gi(PL)) is non-monotonic. Therefore, there is no droop function such that the SOP isthe GOS for each load. □

Overall, the questions 1) and 2) raised in section I have been an-swered by Theorem 1 and 2. The Optimization Criterion is summarizedas

Optimization Criterion: GOD can be achieved in decentralized man-ners if and only if OSFs are all strictly monotonically increasing for PL ∈[0, PL,max].

3.2. Proposed decentralized global optimal scheme

When OSFs are all strictly monotonically increasing, the decen-tralized control law of GOD is presented in (9) according to Optimi-zation Criterion. This statement will be verified by Case 1 in the si-mulation section.

Z. Liu et al. Electrical Power and Energy Systems 104 (2019) 38–42

39

3.3. Proposed decentralized suboptimal scheme

The prerequisite in which all OSFs are strictly monotonically in-creasing in microgrid might be too harsh to be realistic. In fact, theOSFs of DGs may have saturation, dead and hysteretic characteristics asshown in Fig. 1 when practical constraints are taken into account. Ac-cording to Theorem 2, GOD cannot be achieved by decentralizedmethods. Then, how to construct a satisfying suboptimal method be-come a problem that needs to be solved. An intuitive method is tomodify the OSF curves and make it monotonous. Define γi(Pi) as themodified curve, and it can also be regarded as suboptimal solutionfunction (SOSF). Thus, according to Theorem 1, the suboptimal droop-based method can be designed as

= −∗ −f f mγ P( )i i i1 (13)

The dotted lines in Fig. 1 imply three typical modified smoothcurves, respectively. In fact, there are numerous functions that satisfythe condition, is there a best one among them? The answer is No. Onthe one hand, the error between OSF and SOSF is smaller, the result ofSOSF is better. For example, the red line is better than green line inFig. 1 (a), but there is not a best monotonically increasing SOSF. On theother hand, the results of SOSF depend on the characteristics of load.For example, when PL∈(0, αi) at most of the time, the red line may bebetter than the green line in Fig. 1 (c)

when PL∈(αi, η) at most of the time, the green line may be betterthan the red line. Therefore, there is no best SOSF.

To ensure satisfactory SOSFs, we design γi(Pi) as:

∑=

⎧

⎨⎪

⎩⎪

+ ∈

∉=

+γa P P α β

g P α β

, [ , ]

, [ , ]i k

N

ik Lk c

P b L i i

i L i i

0

iL i

(14)

where [αi, βi] is the interval in which OSF s curve need to be modifiedby polynomial fitting

the fractional item in (1) is introduced to reduce the order of thepolynomial. The parameters aik, bi, and ci are determined by the fol-lowing optimization problem.

∫⎧

⎨⎩

−

= = >

F γ dP

s t I γ α F α γ β F β II ε

min ( )

. . ( ) ( ) ( ), ( ) ( )( )

αβ

i i L

i i i i i i i idγdP

2i

i

iL (15)

where ε is a positive number. The constraint in (15) are to respectivelyguarantee continuity and monotonicity.

Although γi(Pi) is suboptimal, it remains good performances. WhenPL belongs to the subintervals in which gi(Pi) = γi(Pi) for all DGs, GODcan still be achieved. When PL belongs to the [αi, βi], (14) ensures thatthe operation point is close to the optimal point. Thus, the suboptimaldroop-based method is also practical and economical. □

Remark. The proposed droop strategies need to get the OSF gi(PL),which comes from the off-line calculation, and the overall information(DGs’ cost functions, maximal capacities and the maximal loads) areneeded. Generally, this information is time-invariant. Therefore, insystem operation process after the off-line calculation, the proposedcontrol strategies do not need the other DG’s real-time information, i.e.,communication is not needed.

For an AC mcirogrid with inductive transmission network, PL isequal to the sum of all loads because there are no transmission loss.Thus, the EDP can be formulated as (1) for any network with inductivecable. That is, the proposed methods can be used in network micro-grids.

4. Simulation and experimental results

To verify the correctness and effectiveness of the proposed method,two simulation cases (case 1 and 2) and one experiment case (case 3)are tested. The general operation cost function is

= + + +C P a P b P c P d e P( ) exp( ),i i i i i i i i i i i3 2 and the coefficients and system

parameters are listed in Table 1.

Case 1 (Global optimal case). Firstly, according to (1), sketch thevariation curve of gi(PL) as PL increases from 0 to PL, max by off-linecalculation. As shown in Fig. 2 (a-1), since all OSFs are strictlymonotonically increasing, we could directly find gi−1 and constructthe droop law as = −∗ −f f mg P( )i i i

1 . The f-P droop curves and theirapproximate polynomial are presented in Fig. 2 (b-1) and Table 2,respectively. The curves of the theoretical minimal total cost(∑ = C g P( ( ))i

ni i L1 ) and the actual total cost of the proposed method are

sketched in Fig. 2 (c-1). Fig. 2 (d-1) shows that the relative error( −1actual cost

theoretical minimal cost ) is identically equal to zero over the whole loadrange, which means that the GOD can be achieved for each PL∈[0,PL,max].

Case 2 (Sub-optimal case). Likewise, according to (1), the variationcurve of gi(PL) is sketched as PL increases from 0 to PL, max. As shown inFig. 2 (a-2), the OSFs are not all monotonically increasing, so it isnecessary to fit an alternative monotonically increasing function bycalculating γi from (14) and (15). The curve of SOSF γi is sketched inFig. 2 (a-2), based on which we find the −γi

1 and construct the droop lawas = −∗ −f f mγ P( ).i i i

1 The f-P droop curves and their approximatepolynomial are presented in Fig. 2 (b-2) and Table 2, respectively.Fig. 2 (a-2) shows that ≡γ P g P( ) ( )i L i L when PL∈(16, 25]. According toTheorem 1, GOD can be achieved. After fitting, ≠γ P g P( ) ( )i L i L whilePL∈[0, 16]. According to Theorem 2, only a sub-optimization can berealized. The curves of the theoretical minimal total cost and the actualtotal cost of the proposed method are sketched in Fig. 2 (c-2). Fig. 2 (d-2) shows that the relative error is between 0 and 8% for PL∈(0, 16] andis identically equal to zero for PL∈(16, 25], which verifies Theorem 1

Fig. 1. Construction of suboptimal solution. (The solid and dotted lines re-present OSFs and SOSFs, respectively.)

Table 1Cost Coefficient for Simulation and Experiment.

DGs a/b/c/d/e 10−3/10−3/10−2/10−3/10−1 Pmax PL, PL,max kW kW

DG1 of case 1 0/4/0.4/3/2.86 8 10→ 15→ 20, 25DG1 of case 2 0.4/-5/6/0/0 8 10→ 15→ 20, 25DG2 of case 1,2 0/5.4/0.4/2/2.86 5 10→ 15→ 20, 25DG3 of case 1,2 0/3.3/1.1/1/2.86 8 10→ 15→ 20, 25DG4 of case 1,2 0/2.4/0.8/4/2.86 8 10→ 15→ 20, 25DG1 of case 3 0/800/4/2/28.6 1 0.8→ 1.2→ 1.5, 2DG2 of case 3 0/240/8/2/28.6 1 0.8→ 1.2→ 1.5, 2

Z. Liu et al. Electrical Power and Energy Systems 104 (2019) 38–42

40

and 2. When the load is 10, 15 and 20 kW, the frequency and activepower are presented in Fig. 3 (a-2) and (b-2), respectively. The relativeerror in cost is 4.4%, 5.3% and 0%, respectively.

Case 3 (Sub-optimal case). The curve of gi(PL) in case 3 is shown in Fig. 2(a-3). Here since g1(PL) has saturation characteristics, it is necessary tofit an alternative monotonically increasing function by calculating γifrom (14) and (15). The curve of SOSF γi is also sketched in Fig. 2 (a-3)with dotted lines, based on which we find the −γi

1 and construct thedroop law as = −∗ −f f mγ P( ).i i i

1 The f-P droop curves and theirapproximate polynomial are presented in Fig. 2 (b-3) and Table 2,respectively. Fig. 2 (a-3) shows that ≡γ P g P( ) ( )i L i L when PL∈[0, 1.2).According to Theorem 1, GOD can be achieved. After fitting,

≠γ P g P( ) ( )i L i L while PL∈[0, 16]. According to Theorem 2, only a sub-optimization can be realized. The curves of the theoretical minimaltotal cost, the actual total cost and relative error in cost are sketched inFig. 2 (c-3) and (d-3), respectively. As shown in Fig. 2 (d-3), the relativeerror is identically equal to zero for PL∈[0, 1.2), and is between 0 and2.1% for PL∈[1.2, 2].

Fig. 3 (a-3), (b-3) and Fig. 4 show the curves of frequency, activepower and output voltages when the load is 0.8, 1.2 and 1.5 kW andeach load lasts for 20 s. From Fig. 3 (b-3), the SOPs of DG 1 and 2 are(0.209, 0.339, 0.545) and (0.591, 0.861, 0.955). The theoretical GOSsby solving (1) are (0.213, 0.323, 0.5) and (0.587, 0.877, 1), respec-tively. The SOPs almost coincide with the GOSs when PL is 0.8 kW andare close to the GOSs when PL is 1.2 and 1.5 kW. And the relative errorin cost is 0%, 0.03% and 2%, respectively. Thus, the experiment resultsare consistent with the theoretical analysis.

Therefore, in case 1, the SOPs of the point are identically equal tothe GOS, i.e., GOD can be achieved by the proposed method. In Case 2,when the load PL belongs to (0, 16], the system operation cost can reachthe theoretical global minimum only if the output power of DG1 is zero.When the load PL belongs to (16, 25], DG1 should work under droopcontrol mode. However, when DG1 should switch the control modecannot be determined without knowing the real-time information ofload, i.e., it needs communication. Therefore, under these conditions,the global optimal dispatch cannot be realized via a decentralizedmanner.

Fig. 2. (a) OSFs gi(Pi) and SOSFs γi(Pi), (b) f-P droop, (c) actual total operation costs, (d) relative error in simulation and theoretical minimal total operation cost. In(c), the red and blue numbers are the actual total costs and the theoretical minimal total costs, respectively. (For interpretation of the references to colour in thisfigure legend, the reader is referred to the web version of this article.)

Table 2f-P droop curves.

Cases Approximate polynomial of ω-P

1 = − − +

= − − +

= − += − +

f P P P

f P P P

f P Pf P

0.0058 0.0388 0.39 51

0.006 0.0.042 0.357 51

0.012 0.31 510.2132 51

1 13

12

1

2 23

22

2

3 32

3

4 3

∈∈

∈∈

PP

PP

[0, 4.5][0, 5]

[0, 7.2][0, 8.3]

1

2

3

42

= ⎧⎨⎩

+ +− + − +

= − − +

= − − +

= − − +

−f

PP P P

f P P P

f P P P

f P P P

0.134( 0.1) 50.8660.0044 0.0494 0.3212 50.47

0.025 0.216 0.088 51

0.006 0.0625 0.1562 51

0.002 0.0365 0.107 51

11 1

13

12

1

2 23

22

2

3 33

32

3

4 43

42

3

∈∈

∈∈

∈

PP

PP

P

[0, 4.09](4.09, 7.5]

[0, 4.2][0, 6]

[0, 7.3]

1

1

2

3

4

3= ⎧

⎨⎩

− − +− + − +

= ⎧⎨⎩

− + − +− + +

fP P P

P P P

fP P PP P P

4.634 1.863 1.812 50.52.086 4.46 3.552 50.66

0.15 0.1873 0.7324 50.540.36 135.9 145 0.0352

113

12

1

23

22

2

223

22

2

23

22

2

∈∈∈∈

PPPP

[0, 0.323](0.323, 1][0, 0.877](0.877, 1]

1

2

2

2

Z. Liu et al. Electrical Power and Energy Systems 104 (2019) 38–42

41

In Case 2 and 3, although GOD cannot be achieved, the suboptimalscheme is also acceptable to some extent. So, the simulation and ex-perimental results verify the correctness and effectiveness of the pro-posed method.

5. Conclusions

In this letter, the problem that under what conditions and how canthe system realize global optimal dispatch via decentralized control isstudied. Three contributions are concluded as:

1) The criterion of decentralized GOD is that optimal solution functions(OSFs) are all strictly monotonically increasing;

2) If the system meets this criterion, a decentralized P-f droop methodis proposed to achieve GOD;

3) In particular, the saturation, dead and hysteretic characteristics ofOSFs will not obey the criterion, and then a sub-optimal droopcontrol methods is proposed.

Overall, this letter proposes a new method of decentralized optimaldispatch which provides a design guideline to build economical mi-crogrids.

Acknowledgement

This work was supported by the National Natural ScienceFoundation of China under Grants 51677195 and 61573384, theNatural Science Foundation of Hunan Province of China under Grantno. 2016JJ1019, and the Joint Research Fund of Chinese Ministry ofEducation under Grant no. 6141A02033514.

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Fig. 3. Simulation and experiment results of (a) frequency, (b) active power under three cases.

Fig. 4. Experimental results of the proposed scheme.

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