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2016 China International Conference on Electricity Distribution (CICED 2016)
Xi’an, 10-13 Aug, 2016
The Bi-level Programming Model of Load Transferring
Strategy Based on Topological Units of High-voltage
Distribution Network
First A. JIN Yong1,3 ,Second B. LIU Junyong1, Third C. LI Hongwei2, Fourth D. ZHANG Xi1
1. College of Electrical Engineering and Information Technology，Sichuan University,
2. College of Electrical Engineering and Information Technology，Southwest Petroleum University,
3. State Gird of China in Chengdu.
Abstract—A
load
transferring
strategy
for
High-voltage distribution network (HVDN) is
presented in this paper to mitigate operation
congestion in urban power system. A novel
bi-level programming model based on topological
units (group) is established in this paper instead of
traditional intelligent algorithm with 0-1 integer
variables. In the upper level, the balance factor is
defined to guide the reasonable load allocation of the
system, while the load shedding factor is to coordinate
the unbalance between transformer capacity and load
in emergency; In the lower level, the supply channel
matrix is used to execute orders from upper and
feedback the results. The simulation using proposed
models is verified in simplifying decision space which
is very suitable for online analysis.
Index Terms—HVDN,Load transfer,PSO, Topological
units (group).
Fig.1 the topological units
Typically, there is at least 1 line reserved for an 110kV
substation, as well as large amount of bus bar breakers
inside the substation which makes the operation mode
of HVDN very complex. A local urban power system
can be converted into the schematic diagram with
symbol of topological units in Fig 2(a), (b).
Fig.2 (a) the topological diagram of HVDN
I. I NTRODUCTION
WITH the rapid development of urban load in china,
the congestion of transmission section and short-term
overload risk of transformer is appearing significantly.
Load transfer is a nonlinear optimization problem of
multi-objective with constraints to get the best optimal
performance or index which is always modeled as an
0-1 mixed integer programming solved with intelligent
algorithm.
Fig.2 (b) the HVDN with topological unit
II. THE TOPOLOGICAL ANALYSIS OF HVDN
A. The topological unit
The definition of topological unit is the cluster of
those equipments which is responsible for the load
transferring from high voltage to the lower voltage,
including the main bus bar, and the transformer as well
as the switches or breakers besides. The letter U is
used in Fig 1 and the transformer in operation is called
available topological unit.
B. The topological unit group
From Fig2 (b) above, topological units of the 110kV
HVDN connecting with each other constitute a group
which is supplied by 220kV substations. The 220kV
line which is used to supply power to the group is
called power supply channel in this paper, such as
11 12 33 ….in Fig3 (b).Accessible matrix A(n×n) is
defined to express such connected relationship of n
topological units across the network. The meaning of
each elements of A is shown as below:
1, at least one pathbetweenU i and U j
Aij  
(1)
0, else
Accessible matrix A can be transformed to a new
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matrix like (2) where o means all the element is 0 and
the matrix Aα, Aγ means the square matrix of   
and
   to express connected units of
 or  ,respectively.
 A

A'  
o

o
(2)


A 
Acquiring the number of the column in Aα and making
a new vector rα of 1  . Each element in rα
corresponds to a certain unit. Then the topological unit
group can be descibed as,
G = Ui | i  r 
(3)
An example like Fig3 is demonstrated as follow:
Fig.3 the schematic diagram based on topological unit
The accessible matrix about Ui and Uj with
deliberately disordering the sequence is expressed as,
1 2 3 4 5 6
1

0
1
A
1
1
 0

0
1
0
0
0
1
1
0
1
1
1
0
1
0
1
1
1
0
1
0
1
1
1
0
(4)
0

1
0

0
0

1 
1

1
1
A'  
1
0
 0

1
1
1
1
0
0
1
1
1
1
0
0
1
1
1
1
0
0
0
0
0
0
1
1
(5)
0

0
0

0
1

1 
1 1 1

1 1 1
1 1 1

1 1 1
So it can be concluded that there are 2 topological unit
group of G and G which is correspond to r and
r in (6).
(6)
The topological units in G and G is also shown
in (7).
G  {U1 ,U 3 ,U 4 ,U 5 }

G  {U 2 ,U 6 }
(7)
Now, it can be concluded that topological unit is the
minimum operable unit to adjust the structure of
HVDN, and could be used to make appropriate
allocation of load among 220kV substations in order to
CICED2016 Session x
 11

Bus (ij )  

 n1
1N 


nN 
(8)
 ij Means the possibility weather unit could supply
by some sources.  ij =1 means unit i could supply by
source j, or else  ij =0. And Bus.initial can be got with
depth searching as a constant matrix as an initial
status.
The vector Lp is used to present the load of each unit in
a group where P(Ui ) is the load of unit Ui.
Lp  {P(U1 ), P(U 2 ),..., P(U n )}
(9)
The group load allocation matrix D can be deduced
D = L p Rus
(10)
j 
D( j )
D
100%
(11)
III. THE PSO BASED BI-LEVEL MODEL
1 1
A  

1 1
r  (1,3, 4,5)

r  (2, 6)
C. The supply channel matrix
Supply channel matrix is presented in this paper to
express the connection relationship of topological unit
and power sources (mainly the 220kV substations in
HVDN) in case that there is no 220kV/110kV
electromagnetic looped network and units without
power supply. So the supply channel matrix of
n  N with n units and N power sources in a certain
topological group can be illustrated as (8).
Where  j means the actual load proportion of power
source j, D(j) means the load of supply channel j in the
group.
Where A , A is respectively shown as,
1

1
A  
1

1
achieve the equilibrium of system.
D is a vector of 1 N and each element of D means
the load supplied by certain power channel in the
group. Thus, the load proportion of a group can be
calculated as,
The transformer of matrix A to A' is shown in (5),
1 5 3 4 2 6
Xi’an, 10-13 Aug, 2016
Paper No xxx
A novel bi-level programming model based on PSO is
presented in this paper to mitigate transmission
congestion and the disequilibrium of system. In the
upper level, the balance factor and load shedding
factor is used to generate the pre-allocation of the
system load; in the lower level, the actual load
allocation and the interruptible load is getting closer to
those of the upper according to the iteration of the
algorithm.  jm is the balance factor; hm is the load
shedding factor for the topological unit group of
Gm ;the dotted circle indicate the total load of
topological units before and after shedding
respectively;  jm
is the load proportion of the
group Gm (see (11)). The  jm is used to guide  jm
getting closer to  jm to balance the load ratio of
220kV substations.
A. Programming Strategies in Upper Level
The main purpose of upper level is to realize the load
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2016 China International Conference on Electricity Distribution (CICED 2016)
Xi’an, 10-13 Aug, 2016
balance of the system and avoid transmission line
congestion.
Where Pm  is the load of Gm before shedding.
A.1. the Balance Factor and The Balance Degree
To ensure the reliability of power supply, the total
Taking Fig.5 as an example, there are 5 power sources
shedding load could not be too much. Thus, the
in Fig.5, S1 , ... , S5 , 8 power supply channels: 11 ,
shedding degree H could be defined as the sum of all
hm for the whole system:
12 , 21 , 32 , 33 , 42 , 43 , 53 (  jm indicates
H   hm
the power supply channel between power source S j
(15)
and topological unit group Gm ) and 3 topological units
The shedding degree H should be as small as
group:
,
possible. If H is too large, there will be a serious
,
unbalance between load and power sources, so it is
G1  U1 ,U2 ,U3 ,U4 ,U5 ,U6 
G2  U8 ,U9 ,U10 ,U11 ,U12 ,U13 
G3  U14 ,U15 ,U16 ,U17 ,U18 ,U19 . The balance factor
 jm can be defined as the expected load proportion
from power source S j to the topological units group
of Gm through power supply channel  jm . And
 jm
should be subjected to,
(12)
 m  1
Where  m means the sum of all the balance factors
related to the topological units group of Gm . For
example, for G1 in Fig.5, 11   21  1 .
Then, the load rate of each 220kV substation could be
calculated by,
 ( S j )  P 
'

j
j
/ Pm ja x
'
j 
factor related to S j ; P
of power source S j .
To evaluate the load balance results, the balance
degree is defined based on the form of 2-norm as,
0 
, here
1
  ( S j ) is the
N
mean load rate for all power sources.
A.2. The Load Shedding Factor and The Shedding
Degree
Sometimes
(especially
for
emergency),
m
(16)
1
0  hm  hm max
Sij  Sij .max
Where Sij refers to the apparent power transforming
from S i to S j ;  j refers to the voltage at 220kV
node j ; 1 and 2 are the weights of K and H .
B. Programming Strategies in Lower Level
By adjusting the structures of topological units inside
and outside the substation, the strategies in lower level
aim to make load proportion of each topological unit
closer to the balance factor.
B.1 the close factor and the close degree
In order to describe the close degree between the actual
distributional percentage of load and the balance factor,
the close factor c m can be defined as,
cm 
suitable
solutions could not be obtained because some
constraints cannot be satisfied at all. Thus, the load
shedding factor

 j min   j   j max
shedding; Pj max refers to the maximum active power
2
F  Min(1 K  2 H )
s.t.
1  2  1
indicates the total active power supplied by S j after
K   ( S j )  0
A.3. The Upper level Programming Model
The goals in upper level are to balance the load of all
the substations and to minimize the shedding degree,
so the optimized objective function is defined as (16)
and the constrains include transmission line capacity,
source apparent powers, voltage bounds, shedding
degree and shedding factor, the equation (12) and so
on.
(13)
where  ( S j ) indicates the load rate of power source
S j ;  j  is the balance
necessary to take more investment of equipment.
hm is introduced and defined. Load
 (
m
  m ) 2
(17)
Where cm is the close factor of the topological unit
group of Gm ,  m is the balance factor that related to
Gm and
 m is the actual distributional percentage
shedding factor hm (hm  hm max ) reflects the shedding
for all the power supply channels m that connected
load percentage of Gm .
to Gm . Those factors can be calculated by (8) ~ (11)
After shedding, the load
Pm'  of Gm is,
respectively.
Pm'   Pm  (1  hm )
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The close factor c m reflects the closeness of the
actual load distribution in each power supply channel
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2016 China International Conference on Electricity Distribution (CICED 2016)
and the expected load distribution with balance factor
from the upper level. It can be inferred that the bigger c m ,
the more different between the actual load distribution
and the expected load distribution, and vise versa.
Considering the topology constraints of each unit, it is
difficult that the distributional percentage of load in each
unit to match the balance factor very well. Here, in order
to describe the adaptability of balancing plan to the
actual structure, the close degree can be defined by
summing up all of the close factor,
   cm
(18)
It is obvious that the smaller  , the stronger the
adaptability.
Xi’an, 10-13 Aug, 2016
f  min(3  4 O )
s .t .
N
Pi  ui  u j ( gij cos ij  bij sin ij )
(22)
j 1
N
Qi  ui  u j ( gij sin ij  bij cos ij )
j 1
ui min  ui  ui max
S pq min  S pq  S pq max
3  4  1
where ui is the node voltage; S pq is the apparent
power transforming from S p to S q ; The constraints
consists of power flows constraints, apparent power
constraints and voltage bound constraints; 3 and 4
indicate the weights of  and O .
B.2 The shedding vector
In upper level, the shedding ratios have been obtained
for all topological units. Here, considering difference
for any unit in each topological unit, for example,
some units have much more important load and less
load can be curtailed, the shedding vector is defined
as,
(19)
  [ x1 ,..., xn ]
C. PSO based solving strategy
C.1 the encoding
The balance factors and the load shedding factors are
adopted to be the variables of PSO, that is
Where xi is the shedding load percentage of
indicates the number of topological unit groups.
Gi ,
0  x1  1 ; n is the total number of units in Gi .
Then, the actual load of each unit after shedding can be
calculated by,
Pj'  Pj  Pi hi 
Pj x j
(20)
ni
P x
j 1
j
where ni is the total number of units in Gi ; Pj x j
ni
P x
j 1
j
j

where
j
 1, j  1, 2,..., N '
(23)
xi indicates the position of particle i ; N '
C.2 The fitness function
After determining the particle positions, the fitness
function could be built by considering the upper-level’s
target F and the lower-level’s target f . F and f
j
means the shedding load of unit j;
xi  ( 11 , 12 ,..., 21 , 22 ,..., h1,...hN ')
means the
can be calculated by formulas (16) and (22). Then, the
fitness is defined as,
Fobj  Min( F  f )
(24)
The fitness function aims at reflecting the close degree of
the upper-level and the lower-level.
total shedding load in Gi ; Pi hi means the expected
shedding load from shedding factor hi . Pj and Pj' are
the active powers of unit i before and after shedding
respectively.
B.3 Number of switch operations
The supply channel matrix Bus can be obtained by
enumeration. After iteration, comparing with initial
supply channel matrix Bus .initial , the number of switch
operations
O can be calculated by,
O
| B
us
(i, j ) - Bus.initial (i, j ) |
2
(21)
B.4 The lower level Programming Model
The lower-level programming model should be close to
the upper-level’s results with the minimum number of
switch operations. The lower-level programming model
could be built as,
C.3 the steps
Firstly, the balance factors and the load shedding factors
are solved obtained based on PSO and then the expected
load assignments are calculated with (16). Secondly, the
feasible supply channel matrixes Bvalid are obtained by
enumeration. Then, the close degrees are calculated with
(17) and the feasible supply channel matrix with
minimum close degree is selected. The F and f are
decreased gradually by iterations, and finally the load
assignments among every topological unit tend to be a
feasible balancing state.
The position and velocity of the particles are updated
with,
vidk 1  vidk  c1r1k ( pbestidk  xidk )
(25)
c2 r2k ( gbestidk  xidk )
xidk 1  xidk  vidk 1
The process can be divided into 7 steps as follow:
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2016 China International Conference on Electricity Distribution (CICED 2016)
Step1: Initialize the positions of m particles randomly
(the balance factors and the load shedding factors).
Step2: Calculate the load pre-distribution in 220kV
power grid according to (16).
Step3: Enumerate the supply channel matrix of each
topological unit, then calculate the close degrees with (18)
and choose the topology with the minimum close degree.
Step4: Check the results whether it meet the flow
constraints.
Step5: Calculate the fitness with (24).
Step6: Judge whether it meets the quit criteria: if “yes”,
quit; if “no”, go to Step7.
Step7: Update the particle speed and particle position
with (25) and (26), then go to Step2.
Fig.4 is the flow chart of proposed method. It is
necessary to point out that the Step 3 among all
topological units are independent, so the speed can be
improved by using distributed parallel computation.
Xi’an, 10-13 Aug, 2016
(b) The initial state for 110kVnetwork
Fig.5 the structure of a local urban HVDN based on
functional unit
B. The results
The load ratios of the 220kV substation and transmission
lines are shown in Tabs. 2 and 3. It can be seen that the
load ratio is reasonably unbalanced. Some substations
(like S1 , S 2 , S6 ) are heavily over-loaded and some (like
S3 , S5 ) are lightly over-loaded. And the over-loaded
line S3-S5 extremely exceeds their limits.
Using the method discussed above, the results under
different load levels are shown without the number limits
of switch operations in Fig.6.
Tab.2 The initial load ratio of 220kV substations
Substations
Fig.4 the flow chart of BPSO
S1
S2
S3
S4
S5
S6
Load ratio 94.9% 87.5% 30% 47.2% 16.9%
95.2%
IV. Numerical test
A. the description of test
Taking a part of a high voltage distribution network as
an example, the topological structure is shown in Fig.5
based on function units. There are more than 44
transfer paths and coupling switches in the system. Six
topological unit groups can be formed and the total
load of peak period is 830MW.
Tab.3 The initial load ratio of 220kV lines
Line
BS-S1
Load ratio 85%
BS-S2 BS-S3 BS-S4
78%
83%
79%
S1-S2
64%
Line
S1-S5
Load ratio 43%
S1-S6
87%
S4-S6
32%
S3-S5
129%
S4-S5
23%
S6
x61
x64
S5
x15
S1
x45
x35
x12
S2
S4
S3
x01
x04
x02 x03
(a) Below 90% load ratio and without the limit of switch
operations
BS
(a) 220kV network
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2016 China International Conference on Electricity Distribution (CICED 2016)
(b) Below 80% load ratio and without the limit of switch
operations
Xi’an, 10-13 Aug, 2016
algorithms.
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[1]
(c) Below 70% load ratio and without the limit of switch
operations
Fig.6 the results for different load level control
Based on the results from Fig.6, when the load ratio
control decreases from 90% to 70%, the balance
degree decreases obviously, but the close degree
increases somewhat. This reflects that although some
of the balancing effects can be obtained, it is much
difficult for the lower-level getting much closer to the
upper-level in order to balance the whole system.
Moreover, when the control is changed from 90% to
80%, the load need not to be curtailed, but the
numbers of switch operations increases from 3 to 8.
When the load control decreases to below 70%, there
are no feasible solutions for no load shedding. It is
necessary to curtail load (here 8% curtailed below
70% load ratio) to get a feasible balancing load
distribution. From Fig. 6 (a),(b) and (c), it can be seen
that the (c) has a better balancing result, but the cost is
8% load shedding.
Fig.7 the balancing effects of 220kV lines under different
load control
The power flows for transmission 220kV lines are shown
in Fig.7 where the line S3  S5 is heavily over-loaded in
the initial structure, but it will not be over-loaded after
the optimal reconfiguration based on the proposed
method. Considering the number of switch operations is
not too large in actual project, this number limit is added
to the fitness function.
V. CONCLUSION
In this paper, the models of topological units and
power supply channels have been introduced based on
the topological structures of 220kV and 110kV high
voltage distribution networks. On the basis of those
models, some strategies for load balance are inferred.
And then a 2-level programming optimization method
is used to get the feasible and optimal solution. The
method can reduce the dimension of the load
transferring problem and improve the calculating
speed. The strategies and method can apply to the
online analysis of the load transferring problem for
high voltage distribution networks. The test system
proved the validity and the feasibility of the proposed
CICED2016 Session x
Paper No xxx
First A. JI Yong He received B.S. degree in electrical engineering
from Sichuan university in China in 2000. His research interests are
Distribution system analysis and control, electrical marteking.
Second B. LIU Junyong He received his Ph.D. from Brunel
University, UK, in 1998. He is a professor in the School of Electrical
Engineering and Information, Sichuan University, China. His main
research areas of interest are power market, FACTS, power system
planning, operation, stability, and computer applications.
Third C.LI Hongwei He received M.S. degree in electrical engin
eering from Southwest Jiaotong university in China in 2005. His
research interests are Distribution system analysis and control, Dis
tributed Generation modeling and Control.
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