Optimal Decentralized Economical-sharing Criterion
and Scheme for Microgrid
Zhangjie Liu, Mei Su, Yao Sun, Member, IEEE, Lang Li and Hua Han
Abstract—In order to address the economical dispatch
problem in islanded microgrid, this letter proposes an optimal
criterion and two decentralized economical-sharing schemes. The
criterion is to judge whether global optimal economical-sharing
can be realized via a decentralized manner. On the one hand, if
the system cost functions meet this criterion, the corresponding
decentralized droop method is proposed to achieve the global
optimal dispatch. Otherwise, if the system does not meet this
criterion, a modified method to achieve suboptimal dispatch is
presented. The advantages of these methods are convenient,
effective and communication-less.
Index Terms—Economical dispatch, decentralized control,
cost-based droop, microgrid

I. INTRODUCTION

M

ICROGRID has been identified as a key component of
modern electrical systems to facilitate the integration of
renewable generation units. Economic dispatch, in a way, is
considered as one of the core problems in microgrid research.
Due to the advantages of high reliability and communicationless, decentralized optimal methods have attracted more and
more attention.
To reduce operation cost, some decentralized methods
based on droop concept are proposed in [1]-[5]. The nonlinear
droop schemes based on equal cost principle are introduced in
[1]-[2], and the main idea is to make the cost of each
distributed generator (DG) equal by droop control. As a result,
the cheaper DGs can produce more power, and the system cost
is reduced. Moreover, a nonlinear droop construction approach
is proposed based on polynomial approximation [3], which
can lower the total cost by selecting the appropriate
coefficients. In order to realize plug-and-play, an improved
droop construction approach is proposed in [4], which can
function independently for each DG by optimizing each one
against hypothetical DGs. However, the methods in [1]-[4] are
suboptimal. To realize the global optimization by
decentralized method, a nonlinear λ-consensus algorithm is
introduced in [5]. The incremental costs are subtly embedded
into the droop control. When the frequency is synchronized,
the total cost is minimized in terms of the equal incremental
cost principle (EICP).
Yet, the EICP works only under the convex cost function,
which motivates us to consider some more generalized
problems: A) under what conditions can the global
optimization be realized in decentralized manner? B) How do
we design the droop-based controller to achieve the
optimization? C) When global optimization cannot be realized
by a decentralized manner, how do we design the appropriate
suboptimal controller?

The main work of this letter is summarized as:
An optimal criterion is presented to judge whether global
optimal economical-sharing can be realized via a
decentralized manner.

An optimal decentralized economical-sharing scheme is
proposed when the criterion is met.

A suboptimal scheme is proposed when the criterion is
not met.


II. PROBLEM FORMULATION
Mathematically speaking, the objective of economic
dispatch problem is to minimize the total generation cost
subject to the demand supply constraints as well as the
generator constraints. Its model is formulated as
n

min  Ci  Pi 
i 1

n
 s.t.
 i 1 Pi  PL ; 0  Pi  Pi ,max , i  1, 2,


(1)
, n

where Ci (Pi) is the general comprehensive cost of the ith DG
including maintenance cost, fuel cost, environment cost,
renewable energy subsidies, and so on. So, the cost function
may be not quadratic or convex.
In AC microgrid, the economic dispatch is usually achieved
via the methods based on droop control [1]-[5]. The
droop-based control scheme (P-f) is presented as
fi  f *  Fi  Pi 
(2)
where fi, f*and Pi denote the actual frequency, reference
frequency and output active power of the ith DG, respectively.
Then, when the system achieve frequency synchronization,
the following results is obtained
F1  P1   F2  P2  

 Fn  Pn  ,  0  Pi  Pi ,max 

(3)

th

where Pi, max is the maximal power of the i DG. From the
balance of power supply-demand, it yields
P1  P2 

 Pn  PL

 0  PL  PL,max 

(4)

where PL, Pi, max are the total load and the supremum of load,
respectively. Then, in steady state, the out power of each DG
yields

 F1  P1   F2  P2    Fn  Pn 
(5)
 n

 i 1 Pi  PL ; 0  Pi  Pi ,max , i  1, 2, , n
Normally, the function Fi (Pi) is monotonic to guarantee the
uniqueness of the operation point.
Because the constraint in (1) is compact and the cost
functions are all continuous, it follows from the well-known
extreme value theorem that the economic dispatch problem

has a global optimal solution P

 P1*, P2*,

, Pn*

.

Then, for the problem of droop-based economic dispatch,
the key is how do we design the droop functions Fi (Pi) such
that P* is the solution of problem (5),  PL  [0, PL,max].
III. PROPOSED DECENTRALIZED ECONOMICAL-SHARING
CRITERION AND SCHEME

A. The Decentralized Global Optimization Criterion
For any PL  [0, PL,max], there is a global optimal solution
P1* , P2* , , Pn* such that the cost is minimal. Then the optimal
solution Pi can be regarded as the function of the total load PL.
For convenience, we define it as optimal solution function
(OSF). Note that gi  PL   Pi*  PL  and its implicit condition is
expressed as





 i 1 gi  PL   PL
n

(6)

The main results of this letter is expressed as follow:
Theorem 1. If OSFs are all strictly monotonically increasing,
the global optimization can be achieved by taking Fi(Pi) =
mgi-1(PL), where m is the gain coefficient to keep the
frequency deviation in the acceptable range.
Proof. When the system reaches the steady state, according to
(3) and (4), it yields
 g11  P1   g 21  P2    g n1  Pn 

 n

 i 1 gi  PL   PL ; Pi ,min  Pi  Pi ,max , i  1, 2,

Assume g11  P1   g 21  P2  
gi(k) and

 i 1 gi  k   PL .
n

 g n1  Pn   k

, n

(7)

Although i(Pi) is suboptimal, it has good performances.
When PL belongs to the subintervals such that gi(Pi) =i(Pi) for
all DGs, the global optimization can still be achieved. When
PL belongs to the other subintervals, processing (9) ensures
that the operation point is close to the optimal point. Thus, the
suboptimal droop-based method is also effective and
convenient. Question C is answered.
Pi

Pi

Pi ,max

Since function

 i 1 gi
n

(8)

Since gk(PL,1) = gk(PL,2), namely, xk1= xk2, (9) is obtained
 Fn  xn1  =Fk  xk1   Fk  xk 2  =F1  x12  

 Fn  xn 2  (9)

Because PL,1 ≠ PL,2, according to (9), there must exist xm1 ≠
xm2 such that Fm(xm1) = Fm(xm2), which is contradicted with the
precondition that Fi is monotonic. The proof is accomplished.
Then, question A and B is answered by Theorem 1 and 2.
And the Optimization Criterion is summarized as
The global optimization can be achieved in decentralized
manner if and only if OSFs are all strictly monotonically
increasing for PL  [0, PL,max].
B. The Decentralized Suboptimal Scheme

Pi ,max

PL

is strictly

 F1  x11    Fk  xk1    Fn  xn1 

 F1  x12    Fk  xk 2    Fn  xn 2 

 x11  x21   xk1   xn1  PL ,1
x  x   x   x  P
22
k2
n2
L ,2
 12

Pi
Pi ,max

, then, we have Pi =

monotonically increasing, according to (6), we have k =PL.
Thus, the solution of (7) is the global optimization P*.
Theorem 2. If OSFs are not all strictly monotonically
increasing, then the global optimization can not be achieved in
decentralized manner.
Proof. Assume that there exist PL,1, PL,2  (0, PL,max) such that
gk(PL,1) = gk(PL,2). We define (x11, x21, … , xn1) and (x12, x22, … ,
xn2) as the solution of (5) when the load is PL,1 and PL,2,
respectively. Thus, it yields

F1  x11  

If OSFs are not all strictly monotonically increasing, the
global optimization can not be achieved by decentralized
method. However, how do we construct a satisfied suboptimal
method? According to the criterion, Fig. 1 implies modified
smooth curves instead of the initial non-monotonic curves in
order to make the whole curve continuous and monotonous.
Define i(Pi) as the suboptimal solution function (SOSF). To
ensure satisfactory suboptimal solutions, the dotted line is
designed according to the following principles
min i  F   2 dP
i i i i

(9)
 d
n
 s.t. i   ;   i  PL
 dPi
i 1
where [αi, βi] is the interval to be fitted and ε is an appropriate
positive scalar. Thus, the economical droop law is constructed
as
fi  f *  m i1  Pi 
(10)

0

PL，max  i 

i

0(i)

(a)

i

PL
PL，max 0

(b)

Initial

i

i

PL
PL，max

(c)

Modified

Fig. 1. Construction of suboptimal solution for illustration. (The solid and
dotted lines represent OSFs and SOSFs, respectively.)

IV. SIMULATION AND EXPERIMENT
To verify the correctness and effectiveness of the proposed
method, two simulation cases (case 1, case 2) and one
experiment case (case 3) are tested. The general operation cost
function is Ci  Pi   ai Pi 3  bi Pi 2  ci Pi  di exp  ei Pi  , and the
coefficients and system parameters are listed in Table I.
The OSFs and SOSFs of all cases are obtained by
optimization toolbox of MATLAB, which are shown in Fig.3
(a). The P-f droop curves in Fig. 3 (b) are obtained according
to (10). The number sequences in Fig.3 (a) and (d) are
theoretical optimal and operation points, respectively.
According to the optimization criterion, if the OSFs are all
strictly monotonically increasing, the operation point of the
proposed method is optimal. Otherwise, it is suboptimal. The
simulation and experimental results are shown in Fig. 3(c), (d)
and Fig. 2.
Case 1: Fig 3 (a-1) shows that all the OSFs are strictly
monotonically increasing, which implies that the global
optimization can be achieved. Moreover, Fig 3 (a-1) and (d-1)
show that all the operation points coincide with the theoretical
global optimal point. Therefore, the cost is minimized and the
correctness of Theorem 1 is verified.

Case 2 and 3: Fig. 2 (a-2) implies that the global optimization
can be achieved when load value belong to (16, 25].
Otherwise, the operation point is suboptimal. Comparing the
operation points with the theoretical global points, we can find
that they are completely coincident when load is 20kW and
the former is close to the latter when load is 10kW and 15kW.
The similar conclusions can also be derived from case 3.
Thus, the correctness and effectiveness of the proposed
methods are verified.
Table I
Cost Coefficient for Simulation and Experimental
a/b/c/d/e
Pmax
PL
DG
10-3/10-3/10-2/10-3/10-1
kW
kW
DG1 of case1
DG1 of case2
DG2 of case1,2
DG3 of case1,2
DG4 of case1,2
DG1 of case3
DG2 of case3

0/4/0.4/3/2.86
0.4/-5/6/0/0
0/5.4/0.4/2/2.86
0/3.3/1.1/1/2.86
0/2.4/0.8/4/2.86
0/800/4/2/28.6
0/240/8/2/28.6

10
10
10
8
8
1
1

VI. REFERENCES

In this letter, the problem whether the global optimal
economical dispatch can be realized by P-f droop control is
studied. The criterion and the corresponding optimal droop
control methods are proposed. The salient feature of this
criterion is that the OSFs are obtained by off-line calculation.
Moreover, the optimal and suboptimal schemes are simple and
convenient. The effectiveness of the proposed method has
been verified through the simulations and experiments.
Optimal power

Frequency

f2

49
0
(b-1)

Frequency

f4

f1
1

2

51

4.311
3.151
2.538
0

6.315
5.014
3.671
0

8

49
9 (b-2) 0

Load change

6.027
5.855
4.647
3.471

15
Load (kW)

f1

f3

[1]. I. U. Nutkani, P. C. Loh, W. Peng, and F. Blaabjerg, “Cost-prioritized
droop schemes for autonomous AC microgrids,” IEEE Trans. Power
Electron., vol. 30, no. 2, pp. 1109-1119, Feb. 2015.
[2]. I. U. Nutkani, P. C. Loh, W. Peng, and F. Blaabjerg, “ Linear
Decentralized Power Sharing Schemes for Economic Operation of AC
Microgrids,” IEEE Trans. Industrial Electron., vol.63, no.1, pp.
225-234, Jan. 2016.
[3]. Ali Elrayyah, Faith Cingoz, and Yilmaz Sozer, “Construction of
nonlinear droop relations to optimize Islanded microgrid operation,”
IEEE Trans. Industry Application, vol.51, no.4, pp. 3404-3413, July,
2015.
[4]. Ali Elrayyah, Faith Cingoz, and Yilmaz Sozer, “Plug-and-Play
Nonlinear Droop Construction Scheme to Optimize Islanded Microgrid
Operations,” IEEE Trans. Power Electron., vol.32, no. 4, pp. 27432756, April, 2017.
[5]. F. Chen, M. Chen, Q. Li, K. Meng, Y. Zheng, J. M. Guerrero, and Derek
Abbott, “Cost Based Droop Schemes for Economic Dispatch in Islanded
Microgrids,” IEEE Trans. Smart Grid, vol.8, no.1, pp. 63-74, Jan. 2017.

Case 2: Suboptimal operation

50

3
4
5
6
7
Active power (kW)

Load change
I6:[10A/div]

Fig.2. Experimental results of the proposed scheme.

V. CONCLUSIONS

50

V5:[200V/div]

I5:[10A/div]

10→15→20
10→15→20
10→15→20
10→15→20
10→15→20
0.8→1.2→1.5
0.8→1.2→1.5

Case1:Optimal operation
8 Optimal
8
4.847
3.218
6.386
6
6 Points
3.662
2.563
5.115
4
4
3.656
2.224
2
4.781
2
2.835
1.975
0
3.718
0
10
15
20
25
5
(a-1) 5
(a-2)
Load (kW)
51
51

V6:[200V/div]

Case 3: Experiment
1
0.8
0.6
0.4
0.2
0
25
(a-3) 0
50.5

f4

f3

0.323

50
8

4
Active Power (kW)

1
0.6
2

f6
f5

49.5
0
(b-3)

0.5
Active Power (kW)

1

Load change

50.3

Load change

49.9

49.8
1

Active power

0
(c-1)
8 Operation
6 Points 3.218
2.563
4
2.224
2
1.975
0
(d-1)

0.877

0.213

1
Load (kW)

f2

51

0.587

2

Time (s)

Time (s)

2

1

7

6.386
5.115
4.781
3.718

4.847
3.662
3.656
2.835
1

49.7
3 (c-2) 0

3
1
3

(d-2)

0

2

1

Time (s)

3
6.027
5.855
4.647
3.471

5.877
4.394
3.311
1.408

4.295
3.083
2.517
0.105

5

Time (s)

2

49.5
0
(c-3)
1
0.6

20

60

40

Time (s)

0.587

0.877

0.966

0.213

0.323

0.634

0.2
3

(d-3) 0

20 Time (s)

40

Fig. 3. Simulation and experiment results of (a) OSF gi(Pi) and SOSF i(Pi), (b) droop curve, (c) frequency, (d) active power under three cases.

60

