Automatica 169 (2024) 111870

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Automatica
journal homepage: www.elsevier.com/locate/automatica

Distributed predefined-time optimal economic dispatch for
microgrids✩
Yu Zhang, Yan-Wu Wang, Jiang-Wen Xiao, Xiao-Kang Liu

∗

School of Artificial Intelligence and Automation, Huazhong University of Science & Technology, Wuhan, 430074, China
Key Laboratory of Image Processing and Intelligent Control, Ministry of Education, Huazhong University of Science and
Technology, Wuhan 430074, China

article

info

Article history:
Received 5 November 2023
Received in revised form 21 March 2024
Accepted 2 June 2024
Available online 22 August 2024
Keywords:
Distributed generator
Optimal economic dispatch
Smooth reconstruction penalty method
Distributed predefined-time optimization

a b s t r a c t
With the massive popularization of distributed generators, optimal economic dispatch has been a key
optimization problem to maintain stable and efficient work of the whole system. In this paper, a new
smooth reconstruction penalty function with continuous and piecewise linear differential is designed
to deal with generation power constraints, which promotes to obtain a better suboptimal solution
compared with the existing smooth penalty methods. A distributed predefined-time optimal economic
dispatch strategy is presented by utilizing a time-based function. By employing the proposed strategy,
the minimization of the generation cost with transmission loss under the power balance constraint
and generation minimum/maximum constraints can be realized within a predefined settling time.
The performance of the proposed optimization strategy is evaluated by simulations and hardwarein-the-loop experiments in terms of validity verification, robustness to load change and topology
reconfiguration, plug-and-play functionality, and comparison with the existing results to illustrate the
advantages of fast convergence and near optimal results.
© 2024 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and
similar technologies.

1. Introduction
As an aggregation of distributed generators (DGs), buses, and
loads, microgrid has been studied in recent years to reduce carbon
emissions and release environmental pressure (Dragičević et al.,
2016; Liu et al., 2023; Schiffer et al., 2016). In a microgrid, optimal
economic dispatch, minimizing generation power cost with transmission loss under power balance equality constraint and power
generator maximum/minimum inequality constraints, is vital for
the stable and efficient operation of the whole system (Li et al.,
2019).
To solve the optimal economic dispatch problem, many traditional optimization techniques such as the gradient method, particle swarm algorithm (Gaing, 2003), genetic approach (Chiang,
2005), and approximate dynamic programming strategy (Shuai
✩ This work is supported by the National Natural Science Foundation of China
under Grants 62233006, 62173152 and 62103156, and the Young Elite Scientists
Sponsorship Program by CAST (2023QNRC001). The material in this paper was
not presented at any conference. This paper was recommended for publication
in revised form by Associate Editor Francesco Vasca under the direction of Editor
Thomas Parisini.
∗ Corresponding author at: School of Artificial Intelligence and Automation,
Huazhong University of Science & Technology, Wuhan, 430074, China.
E-mail addresses: yuzhang_hust@hust.edu.cn (Y. Zhang),
wangyw@hust.edu.cn (Y.-W. Wang), jwxiao@hust.edu.cn (J.-W. Xiao),
xiaokangliu@hust.edu.cn (X.-K. Liu).

et al., 2019) are presented. Though with great performance,
most of the results are centralized optimization strategies. Utilizing a centralized dispatch center to collect global information
and send regulation instructions requires a communication infrastructure with high bandwidth and connectivity, which is
usually difficult in practical scenarios. Therefore, in Yang et al.
(2013), a consensus-based algorithm is designed to solve the economic dispatch problem in a distributed method. A distributed
optimization strategy is designed based on two consensus algorithms working in parallel to deal with the economic dispatch with transmission loss and generation power constraints in
Binetti et al. (2014). A fully distributed control method is presented to achieve optimal resource management through a twolevel control framework in Xu and Li (2015). In Binetti et al.
(2014), Yang et al. (2013) and Xu and Li (2015), projection
operation, an unsmooth method, is employed to deal with
generation power limits. In Zhao and Ding (2017), a distributed
initialization-free optimization approach is raised to address the
optimal charging problem of electric vehicles in a microgrid.
In Cherukuri and Cortés (2018), a distributed collaborative optimization algorithm is presented to minimize the aggregate cost
while meeting the load profile over a weight-balanced strongly
connected communication digraph. In Liu et al. (2021), a distributed discrete-time optimal dispatch method is formulated on
a weight-unbalanced digraph. In Cherukuri and Cortés (2018),
Zhao and Ding (2017) and Liu et al. (2021), exact penalty function

https://doi.org/10.1016/j.automatica.2024.111870
0005-1098/© 2024 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Y. Zhang, Y.-W. Wang, J.-W. Xiao et al.

Automatica 169 (2024) 111870

in an embedded structure is exploited to address the inequality
constraints in the large-scale optimization problem. However,
both the projection operation in Binetti et al. (2014), Yang et al.
(2013) and Xu and Li (2015) and the exact penalty function
in Cherukuri and Cortés (2018), Zhao and Ding (2017) and Liu
et al. (2021) lead to the non-differentiability of the optimization
objective, which makes it difficult to gain access to gradientbased optimization techniques and hinders the employment of
continuous-time optimization tools.
In low-inertial microgrids, rapid convergence of the power
dispatch is beneficial to keep the power balance. In Zhao and
Ding (2018), a two-layer optimization strategy is presented by
employing a smooth linear-quadratic penalty function to accomplish resource optimization with maintaining the supply–demand
balance within a finite time. To realize economic dispatch within
a finite time, a distributed consensus-based optimization algorithm is raised in Mao et al. (2021), in which the assumption of
the linear increment and boundedness on the gradient of the cost
function is relaxed. Under the finite-time optimization algorithm
in Mao et al. (2021), Zhao and Ding (2018), the convergence
time of the power dispatch is dependent on the initial condition, optimization parameters, communication network, and
the number of generation units in the microgrid. To overcome
the drawbacks, a distributed fixed-time optimization algorithm
is presented for multi-agent systems with strongly convex cost
functions in Wang et al. (2020). An initialization-free distributed
power management strategy is raised based on a projection operation or smooth ϵ -exact penalty function in Chen and Guo (2022).
A distributed fixed-time cooperative algorithm is proposed to
realize both economic dispatch and demand response for generation and load participants within a fixed time in Liu and Yang
(2023). Nevertheless, under the distributed fixed-time optimal
dispatch strategies in Chen and Guo (2022), Wang et al. (2020)
and Liu and Yang (2023), the convergence time of the reference
generation power depends on the system communication topology and many optimization parameters, which is not conducive
to system deployment. To solve the shortcoming, a predefinedtime optimal allocation approach is presented by using a smooth
linear-quadratic penalty method to give a suboptimal solution
within a predefined time in Guo and Chen (2022b). In Guo and
Chen (2022a), a distributed dynamic event-triggered optimization
strategy is presented by employing projection operator to achieve
resource allocation within a predefined time, where the initial
condition is constrained. However, the transmission loss is not
concerned in Guo and Chen (2022a, 2022b) and the suboptimal solution is far from the optimal solution in Guo and Chen
(2022b). Inspired by Shao et al. (2021), a distributed predefinedtime optimal economic dispatch strategy based on a smooth
reconstruction penalty function is proposed in this paper.

Fig. 1. An illustration of a microgrid.

dispatch strategy is proposed to minimize the generation
power cost with transmission loss under the power balance equality constraint and generation power minimum/
maximum inequality constraints exactly. The power dispatch of DGs can be given within a predefined settling
time.
(3) The performance of the proposed optimal economic dispatch strategy is evaluated in numerical simulations and
hardware-in-the-loop experiments including validity verification, robustness to load change and topology reconfiguration, and plug-and-play functionality. By comparison
with the existing results (Zhao & Ding, 2018), Liu and Yang
(2023), and Guo and Chen (2022b), the advantages of fast
convergence rate and near optimal results are illustrated.
2. Problem formulation
The microgrid under consideration consists of DGs, converters,
and loads, as shown in Fig. 1. A group of distributed various DGs
provide reliable power to distributed loads by employing converters, and the transmission lines transmit the power to release
pressure on overloaded power nodes and ensure the economic
operation of microgrids.
2.1. Optimization goals and communication graph
In a microgrid, optimal economic dispatch problem is to minimize the total generation cost with transmission loss under the
power balance equality constraints and the generation power
inequality constraints:

P

s.t.

N
∑

Ci (Pi ) ,

(1a)

Pi = PL + Ploss ,

(1b)

min C0 (P) =

i=1
N
∑
i=1

(1) Different from the projection operation in Binetti et al.
(2014), Yang et al. (2013) and Xu and Li (2015), the exact
penalty function in Cherukuri and Cortés (2018), Zhao and
Ding (2017) and Liu et al. (2021), and the smooth linearquadratic penalty function in Chen and Guo (2022), Zhao
and Ding (2018) and Guo and Chen (2022b), a novel smooth
reconstruction penalty function is designed with continuous
and piecewise linear differential, which promotes to present
a better suboptimal power dispatch compared with (Guo &
Chen, 2022b).
(2) Compared with the finite-time optimization methods in Mao
et al. (2021), Zhao and Ding (2018), the fixed-time optimal dispatch approaches in Chen and Guo (2022), Wang
et al. (2020) and Liu and Yang (2023), and the predefinedtime economic allocation strategy in Guo and Chen (2022a,
2022b), a distributed predefined-time optimal economic

Pil ≤ Pi ≤ Piu , i = 1, 2, . . . , N ,

(1c)

where Pi is the generation power of the ith DG, the generation
cost of the DG Ci (Pi ) = ai Pi2 + bi Pi + ci , and N is the number of
DGs in the microgrid. P = (P1 , . . . , Pi , . . . , PN )T ∈ RN with RN
being the set of N × 1 real column vectors. In the power
∑N balance
constraint ((1)b), the power demand of loads PL =
i=1 PL,i and
∑N
∑N
transmission loss Ploss =
ι
P
+
ι
P
with
ιi and ιL,i
i=1 i i
i=1 L,i L,i
being the transmission loss coefficients (Xu & Li, 2015). In the
generation power constraints ((1)c), Piu and Pil are the generation
power maximum and minimum bounds, respectively. Therefore,
the power balance constraint ((1)b) can be modified as follows:
N
∑
i=1

2

(1 − ιi )Pi =

N
∑
i=1

(1 + ιL,i )PL,i .

(2)

Y. Zhang, Y.-W. Wang, J.-W. Xiao et al.

Automatica 169 (2024) 111870

In a microgrid, communication graph G is used to characterize
the information interaction. Ni is the set of neighbors of node
i, i.e.,[ if ]j ∈ Ni , then node i receives information from node j.
A = aij ∈ RN ×N is the adjacency matrix with RN ×N being the set
of N × N real matrices, where aij is the communication
{ weight.
}
aij > 0 for j ∈ Ni and aij = 0 for j ∈
/
Ni . Din = diag din
is a
i
∑
diagonal matrix with din
=
a
,
and
the
Laplacian
matrix
ij
i
j∈Ni
L = Din − A.

From (6), the global optimal solution without generation power
limits is shown as follows.
Pi∗ =

λ =
∗

λ∗ (1 − ιi ) − bi
2ai
PD +

(7a)

(1−ιi )bi

∑N

i=1

∑N

,

2ai

,

(1−ιi )2

i=1

(7b)

2ai

∑N (

where PD =
i=1 1 + ιL,i PL,i .
Considering the generation power minimum and maximum
inequality constraints ((1)c), the necessary conditions for the
optimization problem can be expanded slightly as follows (Wood
et al., 2013).

2.2. Predefined-time optimization
Definition 2.1. (Becerra et al., 2018) Consider a constrained
optimization problem:
min f (x) =

N
∑

⎧ 1 ∂C
i
= λ∗ ,
⎪
⎪
⎨ 1−ιi ∂ Pi
1 ∂ Ci
≥ λ∗ ,
1−ιi ∂ Pi
⎪
⎪
⎩ 1 ∂ Ci ≤ λ∗ ,

fi (xi )

i=1
N

s.t.

∑

(3)

1−ιi ∂ Pi

xi = d0

i=1

gi (xi ) ≤ 0, i = 1, 2, . . . , N ,

Pil ≤ Pi ≤ Piu
Pi = Pil
Pi =

(8)

Piu

3.2. Suboptimality with inequality

where xi ∈ RN , fi (·), gi (·) : RN → R and d0 are the decision
variable, the local cost function, the local inequality constraint,
and the global equality constraint, respectively. The optimization
problem (3) is said to realize predefined-time convergence, if the
following conditions hold:

{

)



limt →tf xi − x∗i  = 0


 xi − x∗  = 0 ∀t > t f
i

In a microgrid, it is necessary for the stable work of DGs to
generate power within the generation maximum and minimum
limits. To deal with the generation constraints, a novel smooth
reconstruction penalty function pϵ,i (Pi ) is designed as follows

(4)

pϵ,i (Pi ) =

⎧
0,
⎪
⎪
⎪
⎪
⎨ k2ϵ hui (Pi ) ,
⎪
⎪
⎪
⎪
⎩

where tf and x∗i are
time and the optimal decision,

 the predefined
respectively. Let xi − x∗i  denote the 2-norm of (xi − x∗i ) ∈ RN .

kϵ l
h
2 i

hi (Pi ) ≤ 0
0 < hi (Pi ) ≤ ϵi , Pi > Piu

(Pi ) ,

hi (Pi ) −

0 < hi (Pi ) ≤ ϵi , Pi < Pil

ϵi −∆ϵi2
2

,

hi (Pi ) > ϵi

(

where hi (Pi ) = (Pil − Pi )(Piu − Pi ), hui (Pi ) = Piu − Pi

3. Penalty function and suboptimal solution

Ci (Pi ) + λ PL + Ploss −

N
∑

i=1

=

ai Pi2 + bi Pi + ci

Pi

)

(5)

i=1

+λ

( N
∑(

1 + ιL,i PL,i −

i=1

)

N
∑

)
(1 − ιi ) Pi ,

where λ is the Lagrangian multiplier related to the modified
power balance constraint (2). To obtain the global optimal solution, let the Lagrangian function with respect to the power and
multiplier be 0 as follows.

i=1

Piul +2∆ϵi

∆ϵi

= 1+τ2τ . Thus,
hi (Pi ) ≤ 0
0 < hi (Pi ) ≤ ϵi , Pi > Piu
0 < hi (Pi ) ≤ ϵi , Pi < Pil

(10)

hi (Pi ) > ϵi

As an extension of ϵ -feasibility in Guo and Chen (2022b), Zhao
and Ding (2018) and Pinar and Zenios (1994), ∆ϵ -feasibility is
given as follows.

i=1

⏐
⏐
∂ L ⏐⏐
∂ Ci ⏐⏐
=
− λ∗ (1 − ιi ) = 0,
∂ Pi ⏐P ∗
∂ Pi ⏐P ∗
i
i
⏐
N
N
∑
∑
(
)
∂ L ⏐⏐
=
1
+
ι
P
−
(1 − ιi ) Pi∗ = 0.
L
,
i
L
,
i
∂λ ⏐λ∗

and hli (Pi ) =

2
Piul with ∆ i

Remark 3.1. Different from the smooth linear-quadratic penalty
function in Chen and Guo (2022), Zhao and Ding (2018) and Guo
and Chen (2022b), the proposed smooth penalty function (9) is
reconstructed based on the decision function hi (Pi ) in the range of
hi (Pi ) ∈ (0, ϵi ], which turns the differential of the penalty function
(10) into a piecewise linear function. Based on the proposed
penalty function, the order of cost function gradient is reduced
compared with those in Guo and Chen (2022b), Zhao and Ding
(2018) and Pinar and Zenios (1994).

)

i=1

N
∑
(

)

⎧
0,
⎪
⎪
)
(
⎪
⎨
kϵ Pi − Piu ,
∂ pϵ,i
(
)
=
⎪
∂ Pi
kϵ Pi − Pil ,
⎪
⎪
⎩
2Pi − Piu − Pil ,

For the optimization problem (1) without the generation minimum/maximum inequality constraints ((1)c), the Lagrangian function can be presented as follows:
N
∑

( u

error, and kϵ =

3.1. Optimality without inequality

L=

)2

ϵ = τ Piul and
Pi − Pi . ϵi = hi Pi + ∆ϵi = (1 + τ )τ
Piul = Piu − Pil . τ is a positive scalar that regulates optimization

( l

In this section, a new smooth reconstruction penalty function
is designed to deal with the inequality constraints in the optimal
economic dispatch problem, and a suboptimal error and condition
are derived.

(

)2

(9)

Definition 3.1. P̃ = (P̃i , . . . , P̃i )T is defined to be ∆ϵ -feasible, if
Pil − ∆ϵi ≤ P̃i ≤ Piu + ∆ϵi , i = 1, . . . , N.

(6a)

By employing the smooth reconstruction penalty function (9),
the objective function is reconstructed as follows,

(6b)

Cϵ (P) =

i=1

N
∑
i=1

3

Cϵ,i (Pi ) =

N
∑
i=1

Ci (Pi ) + µpϵ,i (Pi ),

(11)

Y. Zhang, Y.-W. Wang, J.-W. Xiao et al.

Automatica 169 (2024) 111870

where the Lagrange multiplier µ is a positive constant to regulate
∗
∗ T
the severity of the penalty function. Let Pϵ∗ = (Pϵ,
1 , . . . , Pϵ,N )
be a ∆ϵ -feasible suboptimal solution of (11). In the following,
the optimization error between the real optimal solution and the
suboptimal solution Pϵ∗ caused by the proposed penalty function
will be discussed.

Lemma 3.2.

Lemma 3.1.
For the optimal solution Pe∗ and the ∆ϵ -feasible
suboptimal solution Pϵ∗ ,

κϵ = √

0 ≤ C0 Pe∗ − C0 Pϵ∗ ≤ µϵs ,

( )

where ϵs =

( )

∑N

i=1

N
∑

ul 2

N
∑

(13)

Ci (Pi ) + µpi (Pi ).

Cϵ (Pϵ∗ ) =

(14)

2 max maxPi ∈Pi

{

∂ Ci
∂ Pi

,

min Pi − Pi

ϵi − ∆ϵi2

2
Together with (11) and (14),

.

µ

µτ

i∈Pb

2

(19)

0 ≤ pϵ,i Pϵ,i ≤

2
From (11), (14), and (18),
0≤

( ∗)

Ci Pe,i +

i=1

−

N
∑

2
Piul

+ ⎝µ

∑

2

.

µpe,i Pe,i −

2

∑

2

,

∗
hi (Pϵ,
i)

i∈Pb

ϵi − ∆ϵi2

∗
(hi (Pϵ,
i) −

∑

Piu +Pil

2

∑

∗
∗
hi (Pϵ,
i) + µ

⎞
) − µ∗

∑

(25)

∗ ⎠
hi (Pϵ,
i)

i∈Pb

∗
hϵκ
b,i (Pϵ,i )

i∈Pb
ϵ −∆ϵ 2

− hi (Pϵ,∗ i ). From the definition of
κϵ and τ ≪ 1, we have 0 < κϵ < 12 . For hi (Pϵ,∗ i ) > ϵi ,
ϵi + ∆ϵi2
− ϵi .
2κϵ

(26)

From (25) and (26), then

µ

(20)

N
∑

∑

∗
(hi (Pϵ,
i) −

i∈Pb

ϵi − ∆ϵi2
2

) > µ∗

∑

Ci Pϵ,i

N
∑

N
∑
µτ

µpϵ,i Pϵ,i ≤

i=1

2

(21)
2
Piul

µ

.

∑ kϵ
2

i∈Pw

(
+

Substituting (19) and (20) into (21), the result is obtained. □

µ

∗
∗
hui (Pϵ,
i) = µ

∑

=µ

∑
i∈Pw

4

(1+2τ )τ
2κϵ

2

− (1 + τ )τ )P ul .

∗
hi (Pϵ,
i)

2

)
∗
hui (Pϵ,
i)

−µ

∗

∑

∗
∗
hi (Pϵ,
i) + µ

∗
hi (Pϵ,
i)

(28)

i∈Pw

(
∗

(27)

i∈Pw

∑ kϵ
i∈Pw

After obtaining an upper bound on the optimization error
incurred by pϵ,i instead of pi , the threshold of µ for ∆ϵ -feasibility
is characterized.

∗
∗ ∗
hi (Pϵ,
i ) + µ hb ,

i∈Pb

where h∗b = min{ 2κ i − ϵi } = (
ϵ
(2) For the 3rd term in (24),

( ∗)

i=1

( ∗)

2
(24)

ϵi +∆ϵ 2

( ∗)

)

∗
hui (Pϵ,
i ).

) = µ∗

i∈Pb

∗
hϵκ
b,i (Pϵ,i ) >

i=1

i=1

ϵi − ∆ϵi2

⎛

and Pϵ∗ is ∆ϵ -feasible,
(1 + 2τ )τ

ϵi − ∆ϵi2

∗ )− i
i
hi (Pϵ,
i
2
∗
where hϵκ
b,i (Pϵ,i ) =
κϵ

pe,i (Pe∗,i ) = 0,

)

∑ kϵ

(17)

(18)

(

∗
(hi (Pϵ,
i) −

i∈Pb

P ul .
2 i
Recall that Pe∗ is an optimal solution,
∗
0 ≤ Ce,i Pe∗,i − Cϵ,i Pϵ,
i ≤

∑

(16)

2

= max{Piul }.

Note that the penalty function pϵ,i (Pi ) is symmetric about
u
∗
so only the case of Piu < Pϵ,
i ≤ Pi + ∆ϵi will be analyzed.
(1) For the 2nd term in (24),

= µ∗

µτ

∗
hi (Pϵ,
i) −

i∈Pb

(15)

P ul .
2 i
Taking the infimum of (17) gives
0 ≤ Ce,i (Pi ) − Cϵ,i (Pi ) ≤

∑(

i∈Pw

}}

}
l

{ u

0 ≤ pi (Pi ) − pϵ,i (Pi ) ≤

∗
Ci (Pϵ,
i) + µ

+µ

where the generation
power feasible
∑N
∑N set for the ith DGs Pi = {Pi ∈
R| hi (Pi ) ≤ 0, i=1 (1 − ιi )Pi = i=1 (1 + ιL,i )PL,i }.
From the definition of pϵ,i (Pi ) in (9) and pi (Pi ) in (13),

N
∑

N
∑
i=1

{

( ∗)

ul

Proof. Denote Pw and Pb as the sets of indices corresponding
to constraints which are satisfied within ∆ϵi at Pϵ∗ and violated
beyond ∆ϵi at Pϵ∗ . Assume that the cardinality of Pb is at least
one. The following analysis is conducted by contradiction. The
objective function with the smooth reconstruction penalty term
is written as follows

Denote Pe∗ = (Pe∗,1 , . . . , Pe∗,N )T as an optimal solution of (14).
Pe∗ is the optimal solution if µ = µ∗ > max{µsi } with µs =
(µs1 , . . . , µsN )T being the Lagrange multiplier satisfying the KKT
condition (Bertsekas, 1975; Pinar & Zenios, 1994). The upper
bound of µs is given as follows (Kia, 2016).

)

+1

1+2τ

i=1

{ }

(23)

with PNul = (N − 1) P ul 2 , P ul = min{Piul } and P

hi (Pi ) ≤ 0
hi (Pi ) > 0

Ce,i (Pi ) =

max µsi <

,

P

i=1

(

1
ul +1
PN

(12)

The objective function for the exact penalty pi (Pi ) is formulated
as:
Ce (P) =

(22)

where the multiplier gain κϵ is chosen by

ϵi .

0,
hi (Pi ) ,

{

µ∗
.
κϵ

µ=

Proof. To obtain the optimization error, an exact penalty function
pi (Pi ) is introduced (Bertsekas, 1975):
pi (Pi ) =

Pϵ∗ satisfies ∆ϵ -feasibility if µ is given as follows

)
∑
i∈Pw

∗
hϵκ
w,i (Pϵ,i )

,

Y. Zhang, Y.-W. Wang, J.-W. Xiao et al.

Automatica 169 (2024) 111870

u
u
∗
∗
∗
l
∗
where hϵκ
w,i (Pϵ,i ) = (Pi − Pϵ,i )(kϵκ (Pi − Pϵ,i ) − (Pi − Pϵ,i )) with
kϵ
kϵκ = 2κ .

and the suboptimal condition for (35) is given as follows

( )
∇ Cϵ,i Pϵ,∗ i = λ∗ϵ ,

ϵ

dhϵκ
w,i

= (Pil − Pϵ,∗ i ) + (Piu − Pϵ,∗ i )(1 − 2kϵκ )

∗
dPϵ,
i

1 + ιL,i PL,i =

(

m,∗
. Recall
ϵκ −1)
κϵ ∆ϵi
+ 1+2τ (1−κϵ ) < Piu + ∆ϵi . At the

<κ <

<

=

i∈Pw

∗
∗
hui (Pϵ,
i) ≥ µ

2

∑

ul 2

where h∗w = − 2(1+2κτϵ(1τ −κ )) P .
ϵ
Combine (24), (27), and (31), then
N
∑

∗
∗
Ci (Pϵ,
i) + µ

∑

∗
∗ ∗
hi (Pϵ,
i ) + µ hb

i∈Pb

i=1

+µ

∗

∑

(32)

∗
∗
∗
hi (Pϵ,
i ) + µ (N − 1)hw ,

i∈Pw

By the definition of κϵ in (23),
h∗b + (N − 1)h∗w = 0.

(33)

Then,
Cϵ (Pϵ∗ ) >

≥

N
∑

∗
∗
Ci (Pϵ,
i) + µ

N
∑

i=1

i=1

N
∑

N
∑

Ci (Pe∗,i ) + µ∗

i=1

∗
pi (Pϵ,
i)

(34)

√
τ≥

pi (Pe∗,i ).

1 + 2τ

⎛√
=⎝ 1

PNul + 1

1
+ 1+τ
τ

N
∑

Piul

+ (1 + τ )τ ⎠ µ∗

N
∑

2

Pi

s.t.

2

Piul .

i=1

(1 − ιi )Pi =

(35a)
N
∑

(1 + ιL,i )PL,i ,

(37b)

1

− ,

−2
N) ∑2NP
N
ul 2

2

√

(38a)
(38b)
(38c)

where Pir is the reference generation power. The local power
mismatch Pim = (1 + ιL,i )PL,i −ῑi Pir . ξi and νi are auxiliary variables
(
)
(
)
∑
∑
ξ
ν
with ei =
j∈Ni aij ῑi ξi − ῑj ξj and ei =
j∈Ni aij ῑi νi − ῑj νj .
Motivated by Shao et al. (2021), the time-based
gain ω(t) is de{
ta
, t ∈ [0, ta )
r2
ta −t
signed as ω(t) = r1 + t Ω (t), where Ω (t) =
a
1,
t ∈ [ta , +∞)

i=1
N
∑

ul 2

( )
)
ω(t) (
−∇ Cϵ,i Pir + ῑi ξi ,
ῑi
)
ω(t) ( ξ
ξ̇i =
−ei − eνi + Pim ,
ῑi
ω(t) ξ
ν̇ i =
e ,
ῑi i

i=1

Cϵ,i (Pi ) ,

PNul + 1

Ṗ ir =

Based on Lemmas 3.1 and 3.2, the optimization problem (1)
can be further presented as
N
∑

(37a)

In the following, a distributed predefined-time strategy is
designed to achieve the optimal economic dispatch with transmission loss under power balance and generation constraints
within a predefined settling time.

Thus, the optimization error can be decreased by reducing the
adjustable parameter τ .

min

2ai

4. Distributed predefined-time optimization strategy

i=1

⎞

Pϵ,i

∑N

Remark 3.2.
From (12) and (22), the upper bound of the
optimization error can be formulated as follows.

+ 1)µ∗ (1 + τ )τ

⏐
⏐
⏐ ∗ is the modified gradient with

there exists µ i=1 ϵi ≤ (1 + N)µ∗ N ϵ which means that the
optimization error can be decreased by employing the smooth
reconstruction penalty function (9).

Note that ∆ϵi can be made arbitrarily small by choosing small
enough τ . Thus, the ∆ϵ -feasible suboptimal solution does not
hinder the stable work of the DGs.

PNul + 1

1 ∂ Cϵ,i
∂ Pi

ῑi

i=1 Pi

which contradicts (18). Hence, under (22)–(23), the set Pb must
be empty, which means Pϵ∗ is ∆ϵ -feasible. □

µϵs = (

√
(1 +

i=1

√

=

Remark 3.3. Different from the exact penalty (13), the penalty
function (9) is continuously differentiable, which makes gradientbased optimization techniques available. Compared with the
smooth linear-quadratic penalty method in Pinar and Zenios
(1994), Zhao and Ding (2018) and Guo and Chen (2022b), the
piecewise linear smooth reconstruction penalty function differential (10) can effectively reduce the order of the cost function
gradient. Moreover, the introduction of τ to ϵi in (9) generates
a ∑
variable ∆ϵi feasible set with the optimization error being
µ Ni=1 ϵi instead of the constant
ϵ feasible set with the opti√
mization error being (1 + N)µ∗ N ϵ in Pinar and Zenios (1994),
Zhao and Ding (2018) and Guo and Chen (2022b). To compare the
optimization error under the proposed penalty function and the
linear-quadratic penalty function, the relationship between ϵi in
(9) and ϵ in Pinar and Zenios (1994), Zhao and Ding (2018) and
Guo and Chen (2022b) is given as ϵ = max{ϵi }, which means the
same feasible set for the key DG. Then, when the parameter τ
satisfies the following condition:

(31)

i∈Pw

Cϵ (Pϵ∗ ) >

)

∗ 2
(Pi − Pϵ,
i) ,
ῑi
⏐
⏐
( )⏐ 2(ai + µ) + µτ ⏐
⏐∇ Cϵ,i (Pi ) − ∇ Cϵ,i P ∗ ⏐ ≤
⏐Pi − P ∗ ⏐ .
ϵ,i
ϵ,i
ῑi

(30)

∗
∗
∗
hi (Pϵ,
i ) + µ (N − 1)hw ,

(36b)

ῑi = 1 − ιi . From (10) and (11),

From (28) and (30),

∑ kϵ

(1 − ιi ) Pϵ,∗ i .

∗
∗
(∇ Cϵ,i (Pi ) − ∇ Cϵ,i (Pϵ,
i ))(Pi − Pϵ,i ) ≥

κϵ τ
2
m,∗
P ul .
hϵκ
w,i (Pϵ,i ) = −
2(1 + 2τ (1 − κϵ )) i
µ

∑
i=1

∗
where ∇ Cϵ,i Pϵ,
i

u
∗
The minimum of hϵκ
w,i (Pϵ,i ) is attained at Pϵ,i = Pi + 2(k

Piu

)

i=1
Piul

m,∗
Pϵ,i

N

∑(

(29)

= 2(kϵκ − 1)Pϵ,∗ i + 2(1 − kϵκ )Piu − Piul .
1
0
, then Piu
ϵ
2
m,∗
point of Pϵ,i ,

(36a)

N

{

(35b)

̇ (t) =
and Ω

i=1

5

1
Ω (t)2
ta

0,

, t ∈ [0, ta )
.
t ∈ [ta , +∞)

Y. Zhang, Y.-W. Wang, J.-W. Xiao et al.

Automatica 169 (2024) 111870

Theorem 4.1. For an undirected and connected communication
graph G , the reference generation power Pir in (38) converges to the
∗
∆ϵ -feasible suboptimal solution Pϵ,
i within the predefined settling
time ta .

Take the time derivative of (45) along (43)–(44), then
V̇ (t) = aPoT Ṗ o + aξoT ξ̇o + (a + b)νoT−1 ν̇ o−1

+ b(ξo−1 − νo−1 )T (ξ̇o−1 − ν̇ o−1 )
+ c(Po − ξo )T (Ṗ o − ξ̇o )
)
(
≤ aω(t) −P̃ T g − λ2 ξoT−1 ξo−1
(
)
+ bω(t) −ξoT−1 Po−1 − λ2 νoT−1 νo−1 − νoT−1 Po−1
(
(
)
+ c ω(t) −P̃ T g + PoT−1 T−T 1 LT−1 ξo−1
)
(
+ PoT−1 T−T 1 LT−1 νo−1 + PoT Po + ξoT Tg
(
)
)
−ξoT ξo − λ2 ξoT−1 ξo−1 − ξoT−1 T−T 1 LT−1 νo−1

Proof. The proposed distributed optimization strategy (38) can be
given in a compact form as follows.

(
( )
)
ῑṖ r = ω(t) −∇ Cϵ P r + ῑξ ,
)
(
ῑξ̇ = ω(t) −Lῑξ − Lῑν + P m ,

(39a)
(39b)

ῑ̇ν = ω(t)Lῑξ ,

(39c)

where ῑ = diag{ῑ1 , . . . , ῑN }, P =
,...,
,...,
ν = (ν1 , . . . , νN )T , and P m =
ν s denote the equilibrium point of (39), then
r

(P1r
m
(P1

ξ = ξ , . . . , ξN )T ,
ξ s , and

PNr )T ,
( 1
m T
PN ) . Let P rs , P ms ,

From ((37)a),

0 = ω(t) −∇ Cϵ P rs + ῑξ s ,

(40a)

(
)
0 = ω(t) −Lῑξ s − Lῑν s + P ms ,

(40b)

0 = ω(t)Lῑξ s .

(40c)

(

)

)

(

− P̃ T g ≤ −ζιm PoT Po ,
2a

ῑi

ξoT Tg ≤

= 0.

(41)

̇

)

ξ̃̇ = ω(t) −Lξ̃ − Lν̃ − P̃ ,

(42b)

ν̃̇ = ω(t)Lξ̃ .

(42c)

Jν (b) = bλ2 −
Jξ (a) = aλ2 −

(43a)

ν̇ o1 = 0,

(43c)

c
2

) }. Based on Young’s inequality,

ω(t)ξoT ξo − ω(t)Jν νoT−1 νo−1

1
2
1
2

(

ζιm − 1 −

ζιM
2

)
− λ2N ,

− c,
(
)
λ2N
1
+ c λ2 −
−
.
2

2

c
V̇ (t) ≤ − ω(t)∥ψ∥2
2
≤ −kV ω(t)V (t)
r2
≤ −kV (r1 + Ω (t))V (t),
ta

then

(43b)

ῑ2i

(50)

(51a)
(51b)
(51c)

Choose b to ensure Jν ≥ 2c , then select a to guarantee JP ≥ 2c and
Jξ ≥ 0. Then,

(
)T
(
)T
ξo1 , ξo−1 T = [T1 , T−1 ]T ξ̃ , and νo = νo1 , νo−1 T = [T1 , T−1 ]T ν̃ ,

ξ̇o1 = −ω(t)Po1 ,

(49)

µ
2(ai +µ)+ τ 2

JP (a) = aζιm − b2 + c

N

+ ξo1 ),

PoT Po ,

where

RN , and T−1 ∈ RN ×N −1 . Define the orthogonal transformation
)T
(
= [T1 , T−1 ]T P̃, ξo =
of P̃, ξ̃ and ν̃ by T as: Po = Po1 , Po−1 T

Ṗ o1 = ω(t)(−

2

− ω(t)Jξ ξo−1 ξo−1 ,

It is obvious that there exists an orthogonal matrix T = [T1 , T−1 ]T
to ensure L = T T ΛT with Λ = diag (0, λ2 , . . . , λN ), T1 = √1 1N ∈

T1T g

2

ζιM

ξoT ξo +

V̇ (t) ≤ −ω(t)JP PoT Po −

(42a)

(

1

where ζιM = max{(

Together with ((40)a) and ((40)c), we have ∇ Cϵ (Pirs ) = ∇ Cϵ (Pjrs ),
∀i, j = 1, . . . , N. To analyze the predefined-time stability of the
distributed optimization strategy (38), let P̃ = ῑP r − ῑP rs , g =
∇ Cϵ (P r ) − ∇ Cϵ (P rs ), ξ̃ = ῑξ − ῑξ s , and ν̃ = ῑν − ῑν s denote the
error variables. Then,
P̃ = ω(t)(−g + ξ̃ ),

(48)

where ζιm = min{ 2i }. From ((37)b),

Left-multiply ((40)b) by 1TN = (1, . . . , 1) ∈ R1×N , then
1TN P ms

(47)

(52)

where kV = a+3bc +2c .
(1) If t ∈ [0, ta ), from the definition of Ω (t),

and
Ṗ o−1 = ω(t)(−T−T 1 g + ξo−1 ),

dΩ (t)r2 kV

(44a)

ξ̇o−1 = −ω(t) T−1 LT−1 ξo−1 + T−T 1 LT−1 νo−1 + Po−1 ,

(44b)

ν̇ o−1 = ω(t)T−T 1 LT−1 ξo−1 .

(44c)

( T

)

dt

+
+

a
2
b
2
c
2

Let ψ =

PoT Po +

a
2

ξoT ξo +

a+b
2

Ω (t)r2 kV +1 .

(53)

r2
ta

Ω (t))V (t).

(54)

From (53) and (54),

(ξo−1 − νo−1 ) (ξo−1 − νo−1 )

d[Ω (t)r2 kV V (t)]
dt

(45)

V (t) ≤ e−r1 kV t Ω (t)−r2 kV V (0).

, ξ , ν ) , then
T T

a T
a + 3b + 2c
P Po ≤ V (t) ≤
∥ψ∥2 .
2 o
2

≤ −r1 kV Ω (t)r2 kV V (t).

(55)

Then,

(Po − ξo )T (Po − ξo ).
T

ta

Ω (t)r2 kV V̇ (t) ≤ −kV Ω (t)r2 kV (r1 +

νoT−1 νo−1

T

(PoT

r2 kV

Multiply Ω (t)r2 kV on (52),

Construct a positive definite function as follows:

V (t) =

=

(56)

Note that limt →ta Ω (t)−r2 kV = 0, then
0 ≤ lim V (t) ≤ 0.

(46)

t → ta

6

(57)

Y. Zhang, Y.-W. Wang, J.-W. Xiao et al.

Automatica 169 (2024) 111870

Fig. 3. Grid topology of IEEE 9-bus system.
Table 1
Modified IEEE 9-bus system.

Fig. 2. The hierarchical framework of a microgrid.

Therefore, limt →ta ∥P r (t) − P rs ∥ = 0, P r (t) in the distributed
predefined-time optimization strategy (38) converges to Pϵ∗ within
the predefined settling time ta .
(2) If t ∈ [ta , ∞), from (52) and the definition of Ω (t),
V̇ (t) ≤ −kV (r1 +

r2
ta

)V (t),

≤ 0,

V (ta )

b

c

Pu

Pl

ι

1

0.30

10

5

250

10

0.051

2

0.20

15

10

300

10

0.080

3

0.15

30

5

150

10

0.045

4

0.45

5

0

250

20

0.078

5

0.23

25

14

380

50

0.073

6

0.33

40

5

275

30

0.105

7

0.35

10

20

280

10

0.085

8

0.15

30

2

300

20

0.073

9

0.20

10

17

270

20

0.062

In this section, the effectiveness of the distributed predefinedtime optimal economic dispatch strategy is verified by a modified
IEEE 9-bus system, whose grid topology is shown in Fig. 3. The
generation cost and transmission loss parameters are listed in
Table 1 with τ = 0.01. PL,i = 100 and ιL,i = 0.05, i = 1, . . . , N
with the number of DGs N being 9. The optimization parameters
r1 and r2 in (38) can be chosen arbitrarily, Ω̄ need to be large to
ensure the predefined-time stability. For the predefined settling
time parameter ta = 0.1, the optimization parameters are set
as r1 = 1 and r2 = 10 with Ω̄ = 300. Based on the grid
topology in Fig. 3 and the transmission loss coefficients in Table 1
to determine aij in L, the network parameter λ2 (L) equals to 0.68.
The proposed optimal economic dispatch strategy is tested by
the following cases: effectiveness verification, load change and
plug-and-play, topology reconfiguration, and comparison with
the existing results.

r

−kV (r1 + t2a )(t −ta )

a

5. Simulation results

(58)

then
V (t) ≤ e

Node

(59)

which implies ∥P r (t) − P rs ∥ = 0, t ∈ [ta , +∞).
According to (4), the reference generation power P r (t) in the
optimization strategy (38) converges to the ∆ϵ -feasible suboptimal solution Pϵ∗ within the predefined settling time ta . □
The time-based function Ω (t) plays a decisive role in ensuring
the predefined-time stability of the system. However, when t →
ta , Ω (t) → ∞ results in numerical overflow issues in numerical
implementation. Thus, a gain limit Ω̄ is required during the implementation of the proposed strategy. Note that limt →ta ∥P r (t) −
P rs ∥ can be made arbitrarily small by choosing large enough Ω̄ .
Note that the proposed distributed predefined-time strategy (38)
will be implemented in a digital way instead of in hardware
in real applications, therefore, Ω̄ can be chosen as large as the
software can tolerate.
The hierarchical framework of the optimization strategy is
shown for practical application in Fig. 2, where the double PI
controller is employed to achieve the fast-tracking of the gridforming converter output voltage to the reference signal from
primary and secondary controllers. The distributed predefinedtime optimization strategy is deployed in the tertiary layer to
ensure the economic work of the microgrid.

5.1. Case I : Validity verification
In this case, the effectiveness of the proposed distributed
predefined-time optimization strategy is verified. Moreover, the
suboptimal generation cost from (38) is compared with the optimization result obtained by the CVX optimization tool. The
generation power, gradient, and generation cost are shown in
Fig. 4. The comparative optimization results from the proposed
strategy and CVX tool are shown in Table 2.
From Fig. 4, the gradient and generation power converge
within the predefined settling time 0.1 s, which illustrates the
validity of the proposed distributed predefined-time optimization
strategy.
From Table 2, under the conditions of power balance equality
constraints and generation inequality constraints, the generation
cost obtained from (38) is close to that from the CVX optimization
tool with an optimization error of 0.004, which verifies the feasibility of the optimization results from the proposed economic
dispatch strategy.
To demonstrate the feasibility and applicability in a more
complex and realistic case, the proposed optimization strategy is
tested in the modified IEEE 39-bus system.

Remark 4.1. Compared with the finite-time optimization strategy in Mao et al. (2021), Zhao and Ding (2018) and the fixed-time
optimal dispatch approach in Chen and Guo (2022), Wang et al.
(2020) and Liu and Yang (2023), the reference generation power
can converge within a predefined settling time, which can be regulated by one parameter ta . This makes the convergence time of
the system states very convenient to adjust. Compared with Guo
and Chen (2022a, 2022b), the optimization goal includes not only
the power balance but also the transmission loss to match the
actual application scenario.
7

Y. Zhang, Y.-W. Wang, J.-W. Xiao et al.

Automatica 169 (2024) 111870

Table 2
Comparative optimization results.
Power(Node)

P(1)

P(2)

P(3)

P(4)

P(5)

P(6)

P(7)

P(8)

P(9)

Cost

CVX

107.468

143.012

149.839

74.846

103.813

45.822

88.303

142.514

159.044

46 794.634

(38)

107.47

143.01

149.84

74.85

103.81

45.82

88.30

142.51

159.04

46 794.63

Fig. 5. The validity verification in modified 39-bus system.

Fig. 4. The validity verification.

From Fig. 5, the gradient and reference power converge within
the predefined settling time 0.1 s. Moreover, compared with the
optimization results obtained by CVX, the optimization error is
within the range of 0.005%. Therefore, the validity and feasibility
of the proposed optimization strategy in a more complex and
realistic case are illustrated.
5.2. Case II : Impact of parameter changes
In this case, the impact of parameter sensitivity is conducted
by changing one of the four optimization parameters (r1 , r2 , ta ,
and τ ) at a time. Specifically, the optimization parameters are set
as r1 = 50, r2 = 20, ta = 0.2, and τ = 0.1 under P6u = 200 and
P6l = 60 for comparison with the original values.
From Fig. 6, the convergence rates of the reference powers
and gradients increase as r1 changes from 1 to 50 and r2 changes
from 10 to 20 respectively, which verifies the theoretical result in (56). The upper bound of the convergence time of the
reference generation power changes according to the change of
the predefined-time parameter ta , which verifies Theorem 4.1.
Moreover, compared with the optimization error under τ = 0.1,
the optimization error under τ = 0.01 decreases by 0.348, which
demonstrates the instruction in Remark 3.2.

Fig. 6. The impact of parameter changes.

restarts, the predefined-time optimal economic power dispatch
converges to the steady state within the predefined settling time,
and both the gradient and generation power converge within 0.1
s to achieve the optimization goals.

5.3. Case III : Load change and plug-and-play
The effectiveness of the distributed predefined-time strategy
in load change and plug-and-play scenarios is verified. At t = 0.5
s, the load PL,5 changes from 100 W to 150 W. Moreover, the 3rd
DG quits working at t = 1 s and revises at t = 1.5 s.
From Fig. 7, after the load changes, the reference generation power increases to meet the supply–demand power balance
while minimizing the generation cost under generation power
limits in the microgrid. When the 3rd DG stops working or

5.4. Case IV : Topology reconfiguration
In this case, the effectiveness of the proposed optimization
strategy is demonstrated in the scenario of topology reconfiguration. At t = 2 s, the line between the 4th and the 5th DG
8

Y. Zhang, Y.-W. Wang, J.-W. Xiao et al.

Automatica 169 (2024) 111870

Fig. 7. Load change and plug-and-play tests.

Fig. 9. Comparative results.

resource allocation method in Guo and Chen (2022b). The finitetime optimization gain parameters in Zhao and Ding (2018) are
set as p = 1 and q = 11 to minimize the upper bound of the
ul
convergence time. According to the parameter τ and P , a large
penalty function parameter ϵ value is obtained, which does not
match the practical application. Thus, the parameter ϵ is chosen
as 3.3. The fixed-time optimal dispatch gain parameters in Liu
and Yang (2023) are set as k1 = k2 = 24 and ξ = 1 so that
the upper bound of the convergence time is equal to 0.1 s. The
parameters α and k3 are chosen as 100 and 10000 to approach
the power balance equality constraint. For the predefined-time
resource allocation parameters in Guo and Chen (2022b), the
predefined time parameter tf is set to be the same as ta . The
optimization gains are arbitrarily chosen as σ = 0.1 and ϑ = 5
with ϵ = 3.3. The load PL,5 increases by 200 W at t = 1 s. ιi and
ιL,i , i = 1, . . . , N are set to 0 for the comparison with the methods
in Guo and Chen (2022b), Zhao and Ding (2018), which means no
consideration of transmission loss. The dynamic reference power
and the generation cost are shown in Fig. 9.
From Fig. 9, the convergence rate of the system states under
the proposed distributed predefined-time optimal economic dispatch strategy is faster than that under the finite-time optimal
dispatch in Zhao and Ding (2018). Compared with Liu and Yang
(2023), an accurate power balance equality constraint is satisfied under the proposed strategy. Compared with the allocation
method in Guo and Chen (2022b), the system states under the
proposed strategy converge to a suboptimal solution closer to
the optimal one within 0.1 s. From Fig. 9, after the load changes
at t = 1 s, it obtains that the proposed strategy can ensure
faster convergence speed and get nearer optimal results in the
load-switching scenario.

Fig. 8. Topology reconfiguration.

quits running, which means the ring topology is broken in the
microgrid. At t = 2.5 s, the 1st DG is linked with the 6th
DG, which represents another ring is built. With the topology
reconfiguration, the network parameter λ2 (L) changes to 0.25 at
t = 2 s and 0.64 at t = 2.5 s.
From Fig. 8, after the topology reconfiguration, the system
states converge to the steady state within 0.1 s, which illustrates
the effectiveness of the optimal economic dispatch strategy in the
topology reconstruction scenario.

6. Experimental verification
5.5. Case V : Comparative simulation
In this section, the experiment verification is executed on a
hardware-in-the-loop platform as shown in Fig. 10. The experimental platform consists of IT6724H DC Power Supply (output voltage: 0–300 V, output current: 0–10 A, output power:
0–1500 W, resolution: 100 mV/10 mA), dSPACE 1202, boost

In this case, the proposed predefined-time optimal economic
dispatch strategy is compared with the finite-time optimization
approach in Zhao and Ding (2018), the fixed-time optimal dispatch scheme in Liu and Yang (2023) and the predefined-time
9

Y. Zhang, Y.-W. Wang, J.-W. Xiao et al.

Automatica 169 (2024) 111870

Fig. 10. Microgrid experiment platform.
Table 3
Modified IEEE 4-bus DC microgrid.
Node

a

b

c

Pu

Pl

ι(10−3 )

1

0.15

5

5

20

5

9.375

2

0.10

10

5

60

15

4.125

3

0.20

3

10

40

5

9.375

4

0.18

4

8

40

5

4.125
Fig. 11. Validity verification and load change.

converter (inductance: 2 mH, capacitance: 1 mF, switching frequency: 10 kHz), IT8615 DC Electronic Load (input voltage: 10–
600 V, input current: 0.1–20 A, input power: 0–1800 W), resistive transmission lines, MSO46 Oscilloscope, PLC, and desktop
computer.
The microgrid topology is modified from IEEE 4-bus system,
and the generation cost and transmission loss parameters are
listed in Table 3 with τ = 0.1. PL,i = 20 and ιL,i = 0, i = 1, . . . , N
with the number of DGs N being 4. For the predefined settling
time parameter ta = 1, the optimization parameters are set as
r1 = 1 and r2 = 10 with Ω̄ = 10. Based on the ring grid topology
of 1-2-3-4(-1) and the transmission loss coefficients, the network
parameter λ2 (L) equals to 23.4. The proposed optimal economic
dispatch strategy is tested by the following cases: load change,
plug-and-play, and comparative experiment with the existing
result.
6.1. Case I : Validity verification and load change
In this case, the effectiveness of the proposed distributed
predefined-time optimal economic dispatch strategy is verified.
Moreover, the load PL,i changes from 20 W to 30 W at t = 15
s. The output power, gradient, and generation cost are shown in
Fig. 11.
From Fig. 11, the output power and the gradients converge
within the predefined settling time 1 s, which illustrates the
validity of the proposed distributed predefined-time optimal economic dispatch strategy. After the load demand increases, DGs
generate more power to meet the power balance while achieving
the minimization of the generation cost with transmission loss
under generation limits.

Fig. 12. Plug-and-play.

6.3. Case III : Comparative experiment
In this case, the proposed distributed predefined-time optimal
economic dispatch strategy is compared with the predefinedtime resource allocation approach in Guo and Chen (2022b).
For the optimization parameters in Guo and Chen (2022b), the
predefined settling time parameter tf is set to be the same as ta
to guarantee the same upper bound of the convergence time. The
optimization gains are arbitrarily chosen as σ = 0.1 and ϑ = 4
with ϵ = 1. The load PL,i increases by 10 W at t = 15 s. The
output power and the generation cost are shown in Fig. 13.
From Fig. 13, compared with the allocation strategy in Guo
and Chen (2022b), the system states under the proposed strategy
converge to the steady state within 1 s. At t = 10 s, the generation
cost from (38) is 717.1. On the contrary, the generation cost
from Guo and Chen (2022b) is 718.3, which means a nearer
optimal result is obtained by the proposed strategy. Moreover,

6.2. Case II : Plug-and-play
In this case, the plug-and-play performance of the distributed
predefined-time optimal economic dispatch strategy is tested.
The 1st DG quits working at t = 25 s and revises at t = 35 s.
From Fig. 12, when the 1st DG stops working and restarts, the
output power and gradient converge to the steady state within 1
s, and the minimization of the generation cost with transmission
loss is realized under equality and inequality constraints.
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Automatica 169 (2024) 111870
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Fig. 13. Comparative results.

from Fig. 13, after the load changes at t = 15 s, it can be seen
that the proposed power dispatch strategy can obtain a faster
convergence rate and nearer optimal results in the load-switching
scenario.
7. Conclusion
In this paper, a novel smooth reconstruction penalty function
with continuous and piecewise linear differential is designed to
deal with generation constraints, which promotes to present a
better suboptimal solution. A distributed predefined-time optimal economic dispatch strategy is proposed by employing a
time-based function. The minimization of the generation cost
with transmission loss under the power balance and generation
constraints can be realized within a predefined settling time.
The proposed optimization strategy is tested by simulations and
hardware-in-the-loop experiments to illustrate the advantages
of fast convergence rate and near optimal solution. Future work
could focus on predefined-time optimization under cyber attack
in a microgrid.

Yu Zhang received the B.S. degree from Taiyuan University of Technology, Taiyuan, China, in 2020. He is
currently working toward the Ph.D. degree in control
theory and control engineering with the Huazhong
University of Science and Technology, Wuhan, China.
His research interests include multi-agent systems,
distributed control and optimization, and hybrid
microgrids.

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Yan-Wu Wang received the B.S. degree in automatic
control, the M.S. degree and the Ph.D. degree in control
theory and control engineering from Huazhong University of Science and Technology (HUST), Wuhan, China,
in 1997, 2000, and 2003, respectively. She has been
a Professor with the School of Artificial Intelligence
and Automation, HUST, since 2009. Currently, she is
also with the Key Laboratory of Image Processing and
Intelligent Control, Ministry of Education, China. Her
research interests include hybrid systems, cooperative
control, and multi-agent systems with applications in
the smart grid.
Dr. Wang was a recipient of several awards, including the first prize of Hubei
Province Natural Science in 2014, the first prize of the Ministry of Education of
China in 2005, and the Excellent Ph.D. Dissertation of Hubei Province in 2004,
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Y. Zhang, Y.-W. Wang, J.-W. Xiao et al.

Automatica 169 (2024) 111870
Xiao-Kang Liu received the B.S. degree in automatic
control and the Ph.D. degree in control science and
engineering from Huazhong University of Science and
Technology (HUST), Wuhan, China, in 2014 and 2019,
respectively. From Jul. 2017 to Aug. 2018, he was
a visiting scholar with the Department of Electrical,
Computer, and Biomedical Engineering, University of
Rhode Island, RI, USA. From Oct. 2019 to Feb. 2021,
he was a postdoctoral research fellow with the School
of Electrical & Electronic Engineering, Nanyang Technological University (NTU), Singapore. He is currently an
Associate Professor with the School of Artificial Intelligence and Automation,
HUST. His research interests include hybrid control, distributed control and
optimization, DC microgrids.

China. In 2008, she was awarded the title of ‘‘New Century Excellent Talents’’
by the Chinese Ministry of Education.
Jiang-Wen Xiao received the B.S. degree in electrical
engineering, the M.S. degree in control theory and
control engineering, and the Ph.D. degree in systems
engineering from the Huazhong University of Science
and Technology (HUST), Wuhan, China, in 1994, 1997,
and 2001, respectively. He is currently a Professor with
the School of Artificial Intelligence and Automation,
HUST, where he is also with the Key Laboratory of
Image Processing and Intelligent Control, Ministry of
Education, China. His research interests include smart
grid technologies and power markets.

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