stability of power flow analysis of different resources both on and off grid

Citation: Zheng, F.; Meng, X.; Wang,
L.; Zhang, N. Power Flow
Optimization Strategy of Distribution
Network with Source and Load
Storage Considering Period
Clustering. Sustainability 2023, 15,
4515. https://doi.org/
10.3390/su15054515
Academic Editors: Fu Gu,
Jingxiang Lv and Shun Jia
Received: 1 February 2023
Revised: 25 February 2023
Accepted: 1 March 2023
Published: 2 March 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
sustainability
Article
Power Flow Optimization Strategy of Distribution Network
with Source and Load Storage Considering Period Clustering
Fangfang Zheng , Xiaofang Meng *, Lidi Wang and Nannan Zhang
College of Information and Electric Engineering, Shenyang Agricultural University, Shenyang 110866, China
* Correspondence: xfmeng123@syau.edu.cn; Tel.: +86-158-0244-6563
Abstract: The large-scale grid connection of new energy will affect the optimization of power flow.
In order to solve this problem, this paper proposes a power flow optimization strategy model of a
distribution network with non-fixed weighting factors of source, load and storage. The objective
function is the lowest cost, the smallest voltage deviation and the smallest power loss, and many
constraints, such as power flow constraint, climbing constraint and energy storage operation con-
straint, are also considered. Firstly, the equivalent load curve is obtained by superimposing the
output of wind and solar turbines with the initial load, and the best k value is obtained by the elbow
rule. The k-means algorithm is used to cluster the equivalent load curve in different periods, and
then the fuzzy comprehensive evaluation method is used to determine the weighting factor of the
optimization model in each period. Then, the particle swarm optimization algorithm is used to solve
the multi-objective power flow optimization model, and the optimal strategy and objective function
values of each unit output in the operation period are obtained. Finally, IEEE33 is used as an example
to verify the effectiveness of the proposed model through two cases: a fixed proportion method to
determine the weighting factor, and this method to determine the weighting factor. The proposed
method can improve the economy and reliability of distribution networks.
Keywords: distribution network; power flow optimization; k-means period clustering; energy storage
system; particle swarm optimization
1. Introduction
As the power grid continues to develop in a more efficient, flexible and sustainable
direction [1,2], the large-scale grid connection of distributed new energy has become an
inevitable trend [3,4], increasing the difficulty of multi-objective power flow optimization
(DNMPFO) of distribution networks [5–7].
At present, scholars at home and abroad have carried out relevant research on DN-
MPFO. Reference [8] generated a typical wind-light-load scenario based on fuzzy C-means
clustering algorithm, and established a new energy planning model with the goal of max-
imizing the total installed capacity of distribution networks connected to scenery, but
energy storage was not involved in the model. Reference [9] clustered wind speed and
irradiance, and proposed a two-level optimal allocation model. The upper model took
the comprehensive income as the goal, and the lower model took the minimum sum of
network loss and voltage deviation as the goal, and adopted the planning-optimization
solution strategy of genetic algorithm-interior point method, but energy storage is not
considered in this reference either. Reference [10] established a mathematical model of
interval power flow optimization (PFO) of an AC/DC hybrid system with a DC power
flow controller based on interval and affine arithmetic, and used a non-dominated se-
quencing genetic algorithm to solve the model. However, the cost of unit operation is not
considered in the model. Reference [11] modeled the uncertainty of wind and solar power
generation, established a multi-objective optimization model of a battery energy storage
system coordinated with demand response, and used a non-sequential genetic algorithm
Sustainability 2023, 15, 4515. https://doi.org/10.3390/su15054515 https://www.mdpi.com/journal/sustainability

Sustainability 2023, 15, 4515 2 of 14
to solve it. However, photovoltaic units are not involved in this reference. Reference [12]
proposed a multi-objective optimization hierarchical strategy of a distribution network
considering the integration of distributed power generation and electric vehicles, which
was solved by a genetic algorithm, but the wind turbine unit was not considered in the
model. Considering the correlation, reference [13] simulated a 33-node distribution net-
work, and the results showed that although the cost and network loss decrease, the voltage
deviation increased slightly. Among the existing references on power flow optimization of
distribution networks, there are very few references considering new energy and energy
storage such as wind and solar, and none of them involve time period clustering, which
needs further study.
For the DNMPFO problem with source load storage (SLS), this paper proposes a
time-phased PFO model of a distribution network that takes into account many constraints,
and uses a particle swarm optimization algorithm to solve the model. Finally, the paper
takes a IEEE33-bus system as an example to carry out simulation analysis on the Matlab
software (Version is R2016a) to verify the effectiveness of the proposed model in solving
power flow optimization problems. The main contributions of this work are summarized
as follows:
(1) Based on the k-means clustering method, the equivalent load curve is clustered in
different periods, so as to dynamically determine the weight coefficient of the objective
function according to the periods;
(2) A solution method based on particle swarm optimization is proposed, which can
effectively solve the model.
(3) Based on the proposed power flow optimization method of distribution network
considering time period clustering, the power flow optimization of the IEEE33 system
in a certain area is analyzed, and the objective function values are all decreased, thus
improving the economy and security of the distribution network.
2. Multi-Objective Power Flow Optimization Model of Distribution Network with
Source and Load Storage
The DNMPFO model includes wind turbine (WT), photovoltaic unit (PV), micro gas
turbine (MG) and energy storage system (ESS). In order to make the optimization effect
of power flow better, in addition to the total economic cost, voltage offset and power
loss should also be considered. Since the model includes WT and PV with uncertain
characteristics, it may affect the stability of the power grid, and PFO is an important link to
ensure the stability, safety and economic operation of the power grid.
2.1. Establishment of Objective Function
(1) Objective function F1: the lowest total cost [14,15].
F1 = min

Cf + Cm + Cp,pur + Ch

(1)
where Cf is the unit fuel cost. Since PV and WT are clean energy, the fuel cost is zero.
The fuel cost of MG is calculated according to Equation (2), ten thousand yuan; Cm is the
operation and maintenance cost, calculated according to Equation (3), ten thousand yuan.
Because the structure of MG is convenient for monitoring and replacement repair, its opera-
tion and maintenance cost is not considered; Cp,pur is the cost of purchasing electricity from
the superior power grid, calculated according to Equation (4), ten thousand yuan; Ch is the
pollutant emission control cost, calculated according to Equation (5), ten thousand yuan.
Cf = 104
·
T
∑
t=1
NMG
∑
i=1
cgPunit,i
sgηMG
(2)

where cg is the natural gas price, yuan/m3; sg is the calorific value of natural gas, kW·h/m3;
NMG is the number of MG; Punit,i is the output power of the ith unit, kW; and ηMG is the
conversion efficiency of MG.
Cm = 104
·
T
∑
t=1
J
∑
i=1
comPunit,i(t) (3)
where J is the number of units; and com is the operation and maintenance cost of the unit
power, yuan/kW.
Cp,pur = 104
·
T
∑
t=1
p(t)Ppur(t) (4)
where p(t) is the electricity price at time t, yuan/kW; and Ppur(t) is the power purchased
at time t, kW.
Ch = 104
·
T
∑
t=1
NMG
∑
i=1
c1,CO2
ccetPunit,i(t)
ηMG
!
+ 104
·
T
∑
t=1
c2,CO2
ccet(1 + ε)Ppur(t)
!
(5)
where c1,CO2
and c2,CO2
are, respectively, the CO2 emission coefficients of natural gas and
coal-fired power plant combustion, kg/kW·h; ccet is the equivalent carbon tax, yuan/kg;
and is the line loss rate of the power grid, %.
(2) Objective function F2: the node voltage deviation is the smallest; that is, the voltage
distribution is the most reasonable [16].
F2 = min
T
∑
t=1
n
∑
i=1
Ui,t − UN
UN
(6)
where n is the number of independent nodes of the distribution network; t is the period
mark; T is the number of whole day periods; Ui,t is the voltage amplitude of the ith node
during t period, kV; and UN is the rated voltage of the ith node, kV.
(3) Objective function F3: minimum power loss.
F3 = min
T
∑
t=1
b
∑
k=1
Pk(t) − P0
k(t)

∆t (7)
where b is the number of branches; Pk(t) and P0
k(t) are, respectively, the head power and
end power of branch k, kW; and ∆t is a run time period, h.
2.2. Constraints
The optimization strategy needs to satisfy equality constraints such as power flow
constraint and power balance constraint, and inequality constraints such as bus voltage
constraint and climbing constraint.
2.2.1. Power Flow Constraint
For the operation of a distribution network, it is necessary to meet certain power flow
constraints [17]:







Pin,i(t) = ei,t
n
∑
j=1
(Gijej,t − Bij fj,t) + fi,t
n
∑
j=1
(Gij fj,t + Bijej,t)
Qin,i(t) = fi,t
n
∑
j=1
(Gijej,t − Bij fj,t) − ei,t
n
∑
j=1
(Gij fj,t + Bijej,t)
(8)
where Pin,i and Qin,i are, respectively, the active power (kW) and reactive power (kVar)
injected into the distribution network by node i; ei,t and ji,t are, respectively, the real part
and imaginary part of the voltage of node i at time t; ej,t and jj,t are, respectively, the real

part and imaginary part of the voltage of node j at time t; Gij and Bij are, respectively, the
real part and imaginary part of elements in node admittance matrix, Ω.
2.2.2. Power Balance Constraints
The sum of the output of all units minus the load power and power loss is equal to the
power injected into the grid [18], with the following constraints:
(
Pin,i(t) = Pg,i(t) + PWT,i(t) + PPV,i(t) + Pdch
ESS,i(t) − Pch
ESS,i(t) + PMG,i(t) − PLoad,i(t) − PLoss
Qin,i(t) = Qg,i(t) + QMG,i(t) − QLoad,i(t) − QLoss
(9)
where Pg,i and Qg,i are, respectively, the active power (kW) and reactive power (kVar)
of the power supply on node i; PMG,i and QMG,i are, respectively, the active power (kW)
and reactive power (kVar) generated by the gas turbine on node i; Pdch
ESS,i and Pch
ESS,i are,
respectively, the discharge and charging power of the energy storage device on node i;
PLoad,i and QLoad,i are, respectively, the active load (kW) and reactive load (kVar) of node
i; and PLoss and QLoss are, respectively, the active power loss (kW) and reactive power
loss (kVar).
2.2.3. Bus Voltage Constraint
The bus voltage needs to meet the following constraints:
Ui,min ≤ Ui(t) ≤ Ui,max (10)
where Ui,max and Ui,min, respectively, represent the maximum and minimum value of the
voltage amplitude of the ith node, kV.
2.2.4. Unit Output Constraint
The unit output needs to meet the following constraints:
Pi,min ≤ Pi(t) ≤ Pi,max (11)
where Pi,max and Pi,min, respectively, represent the maximum and minimum value of the
active output of the ith unit, kW.
2.2.5. Energy Storage Operation Constraints
The output of the energy storage device during operation meets the following con-
straints [19]:
Pmin
ESS ≤ PESS(t) ≤ Pmax
ESS (12)
where PESS(t), Pmax
ESS and Pmin
ESS , respectively, represent the output power, the maximum
value of the output power and the minimum value of the output power of ESS at time
t, kW.
In order to prevent overcharging and discharging from affecting the battery life, the
overall state of charge (SOC) of ESS is constrained:



SOC(t) = (1 − γ)SOC0 +
T
∑
t=1
PESS(t)∆t
SOCmin ≤ SOC(t) ≤ SOCmax
(13)
where SOC(t) is the SOC value of ESS at time t, kW·h; γ is the self-discharge rate of
electric energy storage; and SOC0, SOCmin and SOCmax are, respectively, the initial value,
minimum value and maximum value of SOC, kW·h.

2.2.6. Climbing Constraint
According to the operation characteristics of MG, its active power regulation rate
(namely climbing rate) is constrained [20]:
Rdown ≤
PMG,t ≤ PMG,t−1
∆t
≤ Rup (14)
where Rdown and Rup, respectively, represent the downward and upward climbing speed
of MG, kW/h; and PMG,t and PMG,t−1, respectively, represent the output of MG at time t
and time t−1, kW.
2.2.7. Branch Current Constraint
The branch current needs to meet the following constraints:
|Ik(t)| ≤ Ik,max (15)
where Ik,max represents the allowable maximum value of branch current, A.
2.3. Power Flow Optimization Mathematical Model
Firstly, the objective functions are normalized [21], and then the multi-objective power
flow optimization model of distribution network is established. Set F0
n as the normalized
objective function:
F0
n =
Fn
Fn,max
(16)
where Fn is the objective function, n = 1, 2, 3; and Fn,max is, respectively, the maximum
value of Fn.
The multi-objective power flow optimization model of distribution network is as follows:
minF = f1F0
1 + f2F0
2 + f3F0
3 (17)
where f1, f2, f3 are, respectively, the weighting factors of F0
1, F0
2, and F0
3, and meet f1 + f2 +
f3 = 1 [22].
3. Determination of Weighting Factors of Optimization Model Based on
k-Means Clustering
In this paper, a k-means clustering algorithm is used to cluster the periods to determine
the weighting factor of each period.
3.1. k-Means Clustering Analysis
k-means is a partition-based clustering algorithm [23], k represents clustering into k
clusters, and means represents taking the average value of data in each cluster as the center
of the cluster, also called centroid. The main steps are as follows:
Step (1) Randomly select k. sample points as the initial clustering center;
Step (2) Calculate the distance from each sample point to the “cluster center”, and
divide each sample point into the nearest cluster. The measurement strategy usually used
in this step is the Euclidean distance [24], whose calculation formula is as follows:
d(x, y) =
q
(x1 − y1)2
+ (x2 − y2)2
+ · · · + (xn − yn)2
=
n
∑
i=1
q
(xn − yn)2
(18)
Step (3) Calculate the center point of each cluster. If the center point obtained is the
same as the previous one, output it; otherwise, repeat step (2) with the newly obtained
center point as the initial point.

3.2. Elbow Rule Determines the Best Clustering Number k
The determination of clustering number k is very important to the clustering qual-
ity [25]. The larger the k value, the better the clustering effect, but the longer the calculation
time. The traditional k value is obtained through experience and lacks objectivity, so it is
necessary to choose the appropriate k value. In this paper, the elbow rule is adopted to
ensure the accuracy of k value selection. When Euclidean distance is used as the metric,
K-means takes the sum of squares for error (SSE) (also called the degree of distortion)
as the target to measure the clustering quality, and its calculation formula is shown in
Equation (19).
SSE = ∑
k
i=1 ∑p∈Ci
|p − mi|
2
(19)
where k represents the number of clusters; Ci represents the ith cluster; p represents the
sample point in Ci; and mi represents the mean value of all data in the ith cluster.
As the k value increases, the number of samples contained in each cluster decreases,
and the distance from the sample to its center will be closer, so the average distortion degree
will decrease [26]. For data with a certain degree of differentiation, when reaching a certain
critical point, the distortion degree will be greatly improved, and then slowly decreased,
and this critical point will be considered as the best k value.
3.3. Determination of Weighting Factor
In this paper, the PV output, WT output and load output of a day are superimposed
to obtain the equivalent load curve, and then the period of the equivalent load curve is
clustered, and then the fuzzy comprehensive evaluation method (FCE) is used to determine
the weighting factor of the optimization model in each period under the condition of
equivalent load [27]. Then, compare it with the weighting factor determined by the fixed
proportion method (FPM) (see Equation (22)) to establish two schemes for the weighting
factor. The calculation formula is as follows:
fn =
f 0
n
NF
∑
n=1
f 0
n
(20)
where f 0
n is the weighted value of exceeding the standard, calculated according to Equation (21);
and NF is the number of objective functions.
f 0
n =
Cn
Can,n
(21)
where Cn is actual value of the nth factor, n = 1, 2, 3; and Can,n is the maximum value of the
nth factor.
In this paper, Cn takes the average value of factors in each case, and Can,n takes the
maximum value of factors in each case.
fn =
1
NF
(22)
4. Solution of Mathematical Model
The particle swarm optimization (PSO) is used to solve the model established in this
paper. In PSO, multiple particles in the search space guide the update of speed and position
according to the global optimization and individual optimization, and co-evolve to find the
optimal solution of the problem [28]. The speed update equation is as follows [29]:
Vt+1
i,d = ωVt
i,d + r1c1(Pt
i,d − xt
i,d) + r2c2(Gt
i,d − xt
i,d) (23)

where Vt
i,d is the velocity of the ith particle in the tth iteration; r1 and r2, respectively,
represent random numbers evenly distributed in (0,1); c1 is a self-learning factor; c2 is a
global learning factor; and ω is the inertia weighting factor.
The location update formula is as follows:
Xt+1
i,d = Xt
i,d + Vt+1
i,d (24)
where Xt
i,d is the position of the ith particle in the tth iteration.
Considering the security of the power system, this paper gives the output constraints
of each unit to calculate the objective function value at each time. In order to avoid
the PSO algorithm falling into local optimization, this paper dynamically improves the
inertia weight ω and learning factors c1 and c2 [30,31], which are, respectively, shown in
Equations (25) and (26).
ω = a +
(a − b)gd
gd
max
(25)
where a and b are the maximum and minimum values of inertia weight ω; g is the current
iteration number; gmax is the maximum number of iterations; and d is a parameter.



c1 = (e1−e2)gd
gd
max
+ e2
c2 = (f1−f2)gd
gd
max
+ f2
(26)
where e1 and e2 are the maximum and minimum values of c1; f1 and f2 are the maximum
and minimum values of c2.
5. Analysis of Example
5.1. Basic Data
This paper simulates and analyzes the distribution network shown in Figure 1. ESS
is connected to 12 and 32 nodes; MG is connected to 3 and 24 nodes; PV is connected to
20 nodes; and WT is connected to 29 nodes. See Reference [6] for the network parameters.
Voltage reference value UB = 12.66 kV and power reference value SB = 1 MVA. The
specified voltage (per unit value) shall not be lower than 0.95 and not higher than 1.05; the
upper limit of the branch current is 20A; the upper limit of MG active output is 300 kW,
the lower limit is 100 kW, and the climbing speed Rdown, Rup is, respectively, −20 and
20 kW/h; and the maximum discharge power of the ESS is 250 kW, the maximum charging
power is 200 kW, and the self-discharge rate γ of the electric energy storage is 0.001. SOC0,
SOCmin, SOCmax is, respectively, 80, 40 and 160 kW · h. Select PV, WT and load data of
a certain area for analysis (see Figure 2). The time-of-use electricity price is shown in
Table 1. See Tables 2–4 for each cost coefficient in objective function F1. The DNMPFO
strategy based on the PSO algorithm proposed in this paper is used to optimize the power
flow of the distribution network for 24 h in this example to verify the effectiveness of the
proposed model.
Sustainability 2023, 15, x FOR PEER REVIEW 8 of 15







+
−
=
+
−
=
2
max
2
1
2
2
max
2
1
1
)
(
)
(
f
g
g
f
f
c
e
g
g
e
e
c
d
d
d
d
(26)
where 1
e and 2
e are the maximum and minimum values of 1
c ; 1
f and 2
f are the
maximum and minimum values of 2
c .
5. Analysis of Example
5.1. Basic Data
This paper simulates and analyzes the distribution network shown in Figure 1. ESS
is connected to 12 and 32 nodes; MG is connected to 3 and 24 nodes; PV is connected to
20 nodes; and WT is connected to 29 nodes. See Reference [6] for the network parame-
ters. Voltage reference value kV
66
12
= .
UB and power reference value MVA
1
=
B
S .
The specified voltage (per unit value) shall not be lower than 0.95 and not higher than
1.05; the upper limit of the branch current is 20A; the upper limit of MG active output is
300kW, the lower limit is 100kW, and the climbing speed down
R
, up
R
is, respectively,
−20 and 20 kW/h ; and the maximum discharge power of the ESS is 250 kW, the maxi-
mum charging power is 200 kW, and the self-discharge rate γ
of the electric energy
storage is 0.001. 0
SOC
, min
SOC
, max
SOC
is, respectively, 80, 40 and 160 h
kW ⋅ . Se-
lect PV, WT and load data of a certain area for analysis (see Figure 2). The time-of-use
electricity price is shown in Table 1. See Tables 2–4 for each cost coefficient in objective
function 1
F
. The DNMPFO strategy based on the PSO algorithm proposed in this paper
is used to optimize the power flow of the distribution network for 24 h in this example
to verify the effectiveness of the proposed model.
Figure 1. IEEE33 node radial distribution network system diagram. “1, 2, 3” stands for nodes, and
“(1), (2), (3) stands for branches.
500
1000
1500
Initial load PV WT
Figure 1. IEEE33 node radial distribution network system diagram. “1, 2, 3” stands for nodes, and

Figure 1. IEEE33 node radial distribution network system diagram. “1, 2, 3” stands for nod
Figure 2. Wind–solar output and initial load curve.
0 5 10 15 20 25
t/h
0
500
1000
1500
Initial load PV WT
Figure 2. Wind–solar output and initial load curve.
Table 1. Time-of-use electricity price.
Period Electricity Price (Yuan/kW·h)
0:00–4:00 0.2294
4:00–8:00 0.0287
8:00–12:00 0.4932
12:00–16:00 0.9462
16:00–20:00 0.3823
20:00–24:00 0.4251
Table 2. Coefficient setting in power generation cost.
Parameter Value
cg 0.9462
sg 0.3823
ηMG 0.4251
Table 3. Setting of coefficient in operation cost.
Unit Type Value of com
PV 0.3456
WT 0.5386
ESS 0.2610
Table 4. Coefficient setting in environmental protection cost.
Parameter Value
c1,CO2
0.19
c2,CO2
0.80
ccet 0.30
ε 0.001
5.2. Analysis of Power Flow Optimization Results Based on PSO
Firstly, K-means is used to cluster the equivalent load curve in different periods, so as
to dynamically determine the weighting factor. Then, the optimization model established
in Section 2 is solved based on PSO, and the power flow optimization results include unit
output results, voltage results and network loss results.
5.2.1. Time-Interval Clustering Results of Equivalent Load Curve
In this paper, the PSO algorithm is used to solve the DNMPFO model. The maximum
number of iterations is set to 200, the search space dimension is 96, the number of particles
is set to 50, and the inertia weight ω and learning factors c1, c2 are shown in Table 5.

Table 5. Parameter settings.
Parameter Value
a 0.9
b 0.4
d 2.0
e1, f2 0.5
e2, f1 2.5
Because the outputs of PV and WT can be regarded as negative loads, the equivalent
load of the system can be obtained by subtracting the output of PV and WT from the initial
load of the system (Figure 3).
Sustainability 2023, 15, x FOR PEER REVIEW
Because the outputs of PV and WT can be regarded as negative loads,
load of the system can be obtained by subtracting the output of PV and W
tial load of the system (Figure 3).
Figure 3. Equivalent load curve.
Cluster analysis of the equivalent load curve: firstly, the SSE curve
the elbow rule (see Figure 4). It can be found that after k = 4, the improv
tends to be slow obviously, that is, k = 4 is the critical point, so 4 is taken
value.
Figure 4. SEE curve.
Then, the equivalent load curve is clustered by K-means, and the foll
ing results can be obtained (see Figure 5).
0 5 10 15 20 25
t/h
400
600
800
1000
1200
0 2 4 6 8 10
k
0
1
2
3
4
5
6
105
0 5 10 15 20 25
400
600
800
1000
1200
Cluster analysis of the equivalent load curve: firstly, the SSE curve is obtained by the
elbow rule (see Figure 4). It can be found that after k = 4, the improvement of SSE tends to
be slow obviously, that is, k = 4 is the critical point, so 4 is taken as the best k value.
Sustainability 2023, 15, x FOR PEER REVIEW
Because the outputs of PV and WT can be regarded as negative loads, t
load of the system can be obtained by subtracting the output of PV and WT
Cluster analysis of the equivalent load curve: firstly, the SSE curve i
the elbow rule (see Figure 4). It can be found that after k = 4, the improv
tends to be slow obviously, that is, k = 4 is the critical point, so 4 is taken
value.
Then, the equivalent load curve is clustered by K-means, and the follo
Figure 5. Period clustering diagram. “*” stands for clustering center.
0 5 10 15 20 25
t/h
400
600
800
1000
1200
0 2 4 6 8 10
k
0
1
2
3
4
5
6
105
0 5 10 15 20 25
t/h
400
600
800
1000
1200
Then, the equivalent load curve is clustered by K-means, and the following clustering
results can be obtained (see Figure 5).
Because the outputs of PV and WT can be regarded as negative loads, the equivalent
load of the system can be obtained by subtracting the output of PV and WT from the ini-
Cluster analysis of the equivalent load curve: firstly, the SSE curve is obtained by
the elbow rule (see Figure 4). It can be found that after k = 4, the improvement of SSE
tends to be slow obviously, that is, k = 4 is the critical point, so 4 is taken as the best k
value.
Then, the equivalent load curve is clustered by K-means, and the following cluster-
As can be seen from the above figure, the period can be clustered into the following
four segments (Table 6).
Table 6. Period clustering result of equivalent load.
0 5 10 15 20 25
t/h
400
600
800
1000
1200
0 2 4 6 8 10
k
0
1
2
3
4
5
6
105
0 5 10 15 20 25
t/h
400
600
800
1000
1200

As can be seen from the above figure, the period can be clustered into the following
four segments (Table 6).
Table 6. Period clustering result of equivalent load.
Period Sequence Time Period
Period 1 12:00~15:00
Period 2 2:00~7:00, 10:00~11:00, 16:00~17:00
Period 3 24:00~1:00, 8:00~9:00, 18:00
Period 4 19:00~23:00
5.2.2. Determination of Power Flow Optimization Scheme
Cn and Can,n are shown in Table 7.
Table 7. Actual value and maximum allowable value of each period.
Parameter Period 1 Period 2 Period 3 Period 4
C1 (ten thousand yuan) 4.5833 2.9135 3.2874 4.1105
C2 (kV) 0.9729 1.2616 1.3901 1.7288
C3 (kW·h) 0.0691 0.1134 0.1393 0.2092
Can,1 (ten thousand
yuan)
5.0894 4.5857 3.6964 4.4428
Can,2 (kV) 1.1313 1.5232 1.5409 1.8572
Can,3 (kW·h) 0.0921 0.1559 0.1739 0.2421
According to Equation (21), the overweight weight of each objective function in each
period can be calculated (Table 8).
Table 8. Weighted value of each time period exceeding the standard.
f 0
1 0.8383 0.9043 0.5760 0.5809
f 0
2 0.7310 0.7969 0.7608 0.9567
f 0
3 0.7545 0.8189 0.7977 0.9615
According to the normalization of Equation (20), the weighting factor of each objective
function in each time period can be obtained, which is shown in Table 9.
Table 9. Weighting factor of each period.
f1 0.3607 0.3588 0.2699 0.2324
f2 0.3146 0.3162 0.3564 0.3828
f3 0.3247 0.3249 0.3737 0.3847
In this paper, the following optimization schemes are set up: the weighting factor
determined by FPM is adopted in the whole period, which is set as scheme 1; The weighting
coefficient determined by FCE is adopted in each period, which is set as scheme 2, and then
the two schemes are compared and analyzed with those before optimization.
5.2.3. DNMPFO Results under Different Schemes
(1) Units Output results under each scheme
The output results of MG and ESS are shown in Figures 6 and 7, respectively.

(a) (b)
Figure 6. MG output: (a) scheme 1; and (b) scheme 2.
(a) (b)
Figure 7. ESS output: (a) scheme 1; and (b) scheme 2.
(2) Voltage results under each scheme
The voltage results are shown in Figure 8.
(a) (b)
Figure 8. Voltage results: (a) scheme 1; and (b) scheme 2.
(3) Network loss results under each scheme
The network loss results are shown in Figure 9.
(a) (b)
Figure 9. Network loss results: (a) scheme 1; and (b) scheme 2.
(4) Comparison results of different schemes
The comparison results of different schemes are shown in Figure 10.
0 5 10 15 20 25
t/h
0
100
200
300
400
500
600
MG1 MG2
0 5 10 15 20 25
t/h
0
100
200
300
400
500
MG1 MG2
Node number
t/h
25
20
15
0.02
0.04
30 10
0.06
20
0.08
5
10
0.1
0
t/h
Node number
30
20
0
0.02
40 10
0.04
30
0.06
20
0.08
10
0.1
0
0
t/h
Branch number
40
0
10
20
40
30
40
50
20
60
20
0
0
Branch number
t/h
40
0
10
20
20
30
40
40
50
60
30 20 10 0
0
(a) (b)
(a) (b)
(a) (b)
(a) (b)
0 5 10 15 20 25
t/h
0
100
200
300
400
500
600
MG1 MG2
0 5 10 15 20 25
t/h
0
100
200
300
400
500
MG1 MG2
Node number
t/h
25
20
15
0.02
0.04
30 10
0.06
20
0.08
5
10
0.1
0
t/h
Node number
30
20
0
0.02
40 10
0.04
30
0.06
20
0.08
10
0.1
0
0
t/h
Branch number
40
0
10
20
40
30
40
50
20
60
20
0
0
Branch number
t/h
40
0
10
20
20
30
40
40
50
60
30 20 10 0
0
(a) (b)
(a) (b)
(a) (b)
(a) (b)
0 5 10 15 20 25
t/h
0
100
200
300
400
500
600
MG1 MG2
0 5 10 15 20 25
t/h
0
100
200
300
400
500
MG1 MG2
Node number
t/h
25
20
15
0.02
0.04
30 10
0.06
20
0.08
5
10
0.1
0
t/h
Node number
30
20
0
0.02
40 10
0.04
30
0.06
20
0.08
10
0.1
0
0
t/h
Branch number
40
0
10
20
40
30
40
50
20
60
20
0
0
Branch number
t/h
40
0
10
20
20
30
40
40
50
60
30 20 10 0
0
(a) (b)
(a) (b)
(a) (b)
(a) (b)
0 5 10 15 20 25
t/h
0
100
200
300
400
500
600
MG1 MG2
0 5 10 15 20 25
t/h
0
100
200
300
400
500
MG1 MG2
Node number
t/h
25
20
15
0.02
0.04
30 10
0.06
20
0.08
5
10
0.1
0
t/h
Node number
30
20
0
0.02
40 10
0.04
30
0.06
20
0.08
10
0.1
0
0
t/h
Branch number
40
0
10
20
40
30
40
50
20
60
20
0
0
Branch number
t/h
40
0
10
20
20
30
40
40
50
60
30 20 10 0
0

tainability 2023, 15, x FOR PEER REVIEW 13 of 15
(a) (b)
(c)
Figure 10. The comparative results of each objective function (a) objective function ; (b) objective
function ; and (c) objective function .
The results of each objective function before optimization, after optimization
through scheme 1 and after optimization through scheme 2 are shown in Table 10.
Table 10. Objective function values under different schemes.
Type
Before Optimiza-
tion
scheme 1 scheme 2
Total cost (ten thousand yuan) 82333.3 81673.9 80176.9
Total VD (kV) 34.9305 32.3813 31.4875
Total power loss (kW ∙ h) 3716.2 3190.2 3041.6
The results are analyzed in combination with Figures 6–10 and Table 10.
scheme 1: The total output of MG is higher at 7:00, 12:00, 16:00 and 21:00; ESS1 is in a
state of discharge most of the time, with more discharge at 7:00~8:00 and 16:00~17:00; and
ESS2 is more charged at 7~9:00, 17:00 and 23:00. The three indexes have been effectively
improved compared with those before optimization, which verifies the effectiveness of
the model established in this paper. Among them, the total cost decreased by 0.80%, the
voltage offset decreased by 7.30% and the power loss decreased by 14.15%.
scheme 2: The total output of MG is higher at 9:00, 15:00 and 19:00; the charging
power of ESS1 reaches the limit at 8:00; ESS2 is in a state of discharge most of the time;
and the discharging power reaches the maximum at 4:00. The improvement effect of the
three indexes is better than that of scheme 1, which verifies the effectiveness of the
strategy of determining the weight coefficient by time clustering and FCE adopted in this
paper. Among them, the total cost decreased by 2.62%, the voltage offset decreased by
9.86% and the power loss decreased by 18.15%.
The MG output in scheme 2 is more than that in scheme 1, and the energy storage
output is more concentrated. Both schemes conform to the law that the energy storage
battery is charged at low load and discharged at peak load, thus realizing the balance
between power production and consumption.
6. Conclusions
0 5 10 15 20 25
t/h
0.8
1
1.2
1.4
1.6
1.8
2
Before optimization
Scheme 1
Scheme 2
0 5 10 15 20 25
t/h
50
100
150
200
250
300
Before optimization
Scheme 1
Scheme 2
Figure 10. The comparative results of each objective function (a) objective function F1; (b) objective
function F2; and (c) objective function F3.
The results of each objective function before optimization, after optimization through
scheme 1 and after optimization through scheme 2 are shown in Table 10.
Table 10. Objective function values under different schemes.
Type Before Optimization Scheme 1 Scheme 2
Total cost (ten
thousand yuan)
82,333.3 81,673.9 80,176.9
Total VD (kV) 34.9305 32.3813 31.4875
Total power loss
(kW·h)
3716.2 3190.2 3041.6
The results are analyzed in combination with Figures 6–10 and Table 10.
Scheme 1: The total output of MG is higher at 7:00, 12:00, 16:00 and 21:00; ESS1 is in a
state of discharge most of the time, with more discharge at 7:00~8:00 and 16:00~17:00; and
ESS2 is more charged at 7~9:00, 17:00 and 23:00. The three indexes have been effectively
improved compared with those before optimization, which verifies the effectiveness of
the model established in this paper. Among them, the total cost decreased by 0.80%, the
voltage offset decreased by 7.30% and the power loss decreased by 14.15%.
Scheme 2: The total output of MG is higher at 9:00, 15:00 and 19:00; the charging
power of ESS1 reaches the limit at 8:00; ESS2 is in a state of discharge most of the time;
and the discharging power reaches the maximum at 4:00. The improvement effect of the
three indexes is better than that of scheme 1, which verifies the effectiveness of the strategy
of determining the weight coefficient by time clustering and FCE adopted in this paper.
Among them, the total cost decreased by 2.62%, the voltage offset decreased by 9.86% and
the power loss decreased by 18.15%.
The MG output in scheme 2 is more than that in scheme 1, and the energy storage
output is more concentrated. Both schemes conform to the law that the energy storage
battery is charged at low load and discharged at peak load, thus realizing the balance
between power production and consumption.

6. Conclusions
In this paper, the DNMPFO model is established, and the model is solved by PSO
algorithm. The effectiveness of this method is verified based on an IEEE33-bus system. The
conclusions are as follows.
(1) When establishing the DNMPFO model, taking the lowest total cost, the lowest voltage
deviation and the lowest power loss including fuel cost, operation and maintenance
cost, power purchase cost and pollutant emission control cost as objective functions,
and comprehensively considering multiple constraints such as power flow constraint,
climbing constraint and energy storage operation constraint, it can be more in line
with the actual operation situation of distribution network, and the comprehensive
optimization effect is better;
(2) K-means clustering is used to divide the equivalent load after the superposition of
PV, WT and load output for 24 h a day into different periods, and FCE is used to
dynamically determine the weighting factor of each period, so that the determination
of the weighting factor is more reasonable and the subsequent optimization effect
is better;
(3) The results of PSO algorithm show that the calculation results of this strategy can
effectively reduce the economic cost, improve the voltage deviation and reduce the
power loss, thus improving the economy and reliability of power grid operation.
Author Contributions: Conceptualization, F.Z. and X.M.; methodology, X.M.; software, F.Z.; val-
idation, F.Z., X.M. and L.W.; formal analysis, X.M.; investigation, L.W. and N.Z.; resources, L.W.;
data curation, N.Z.; writing—original draft preparation, F.Z.; writing—review and editing, X.M.;
visualization, N.Z.; supervision, X.M.; project administration, N.Z.; funding acquisition, N.Z. All
authors have read and agreed to the published version of the manuscript.
Funding: This research was funded by Youth Program of National Natural Science Foundation of
China (Grant number 61903264).
Informed Consent Statement: Informed consent was obtained from all subjects involved in
the study.
Data Availability Statement: Not applicable.
Conflicts of Interest: The authors declare no conflict of interest.
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