1
An Optimal Power Flow plus Transmission
Costs Solution
Pablo Oñate Y., Juan M. Ramirez, Carlos A. Coello Coello
Cinvestav – Unidad Guadalajara. Av. Cientifica 1145. Col. El Bajio. Zapopan, Jal., 45015. Fax (+52)(33)
3777 3600, email : ponate[jramirez]@gdl.cinvestav.mx, ccoello@cs.cinvestav.mx.
Abstract: This paper is aimed at the solution of the optimal power flow (OPF) problem
with embedded security constraints (OPF-SC) by the particle swarm optimizer. The
major objective is to minimize the overall operating cost while satisfying the power
flow equations, system security, and equipment operating limits. The overall operating
cost is composed by the generation cost, transmission cost, and the consumer benefit. A
modification of the conventional particle swarm optimizer (PSO) has been used as the
optimization tool, which uses reconstruction operators and dynamic penalization for
handling constraints. The reconstruction operators allow the increase of the number of
particles within the feasible region. The power equations mismatch, loss active power
transmission, and voltages are calculated by the Newton-Raphson method. To
demonstrate its robustness, the proposed algorithm was tested on systems from the open
literature. Several cases have been studied to test and validate the proposed approach.
Keywords: Optimal power flow, Particle swarm optimizer, Security constraints,
Evolutionary computation.
1. Introduction
The OPF objective is to meet the required demand at minimum production cost,
satisfying units’ and system's operating constraints, by adjusting the power system
control variables [1 - 3]. The power system must be capable of withstanding the loss of
some or several transmission lines, transformers or generators, guaranteeing its security;
such events are often termed probable or credible contingencies. Different security
criteria have been used to ensure sufficient security margins. One of them is the N-1
criterion, widely used nowadays. To determine the credible contingencies, in this paper
a contingency ranking is used [4].
The problem of the two precedent paragraphs, when they are put together, is termed
the optimal power flow with security constraints (OPF-SC). Through an optimal power
flow formulation, a method for computing the optimal pre- and post-contingency
2
operating points is presented. Likewise, constraints in generating units' limits, minimum
and maximum up- and down-time, slope-down and slope-up, a voltage profile improved
and coupling constraints between the pre- and the post-contingency states have been
taken into account.
Nowadays, with the electrical power industry deregulation, the transmission system
can be considered as an independent transmission company that provides open access to
all participants. Any pricing scheme should compensate transmission companies fairly
for providing transmission services and allocate entire transmissions’ costs among all
users. Different methods to allocate transmission costs among network users have been
applied worldwide [5, 6]. This paper employs a transmission pricing scheme using a
power flow tracing method [7, 8, 9] to estimate the actual contribution of generators to
each flow in the links.
Some conventional optimization techniques have been utilized for solving the OPFSC, for instance: linear programming, sequential quadratic programming, generalized
reduced gradient method, and Newton methods [1-3]. Although some of these
techniques have good convergences’ characteristics, some of their major drawbacks are
related to their convergence to local solution instead of global ones, if the initial guess is
located within a local solution neighborhood. The theoretical assumptions behind such
algorithms may be not suitable for the OPF-SC formulation. Optimization methods such
as simulated annealing (SA), evolutionary programming (EP), genetic algorithms (GA)
and particle swarm optimizer (PSO) have been employed to overcome such drawbacks.
Important contributions on power systems optimization applications have been
presented in [10 - 19].
In [20], PSO has been used to solve the OPF. Additionally, in [21 - 22] a modified
PSO, with improved convergence characteristics, is applied to the same aim. Three
types of PSO algorithms are proposed in [23 - 24], all of which have been applied to
solve optimization problems related to reactive power and voltage control.
In this paper, for solving the OPF-SC, a particle swarm optimizer with reconstruction
operators (PSO-RO) is proposed. For handling constraints, such reconstruction
operators plus an external penalization are adopted. The reconstruction operators allow
that all particles represent a possible solution, satisfying the units’ operating constraints,
and increasing the number of particles able to search for optimal solution, thus
3
improving the achieved solution quality.
2. Particle Swarm Optimizer summary
The PSO earliest version was intended to handle only nonlinear continuous optimization
problems [26]. However, its further development elevated its capabilities for handling a
wide class of complex optimization problems [25 - 28]. A PSO algorithm consists of a
population continuously updating the searching space knowledge. This population is
formed by individuals, where each one represents a possible solution and can be
modelled as a particle that moves through the hyperspace. The position of each particle
is determined by the vector xi ∈ Rn and its movement by the particle’s velocity vi ∈ Rn,
G
G
G
(1)
xiiter +1 = xiiter + viiter +1
The information available for each individual is based on both its own experience and
the knowledge of the performance of other individuals in its neighborhood. Since the
relative importance of these two factors can vary, it is reasonable to apply random
weights to each part, and therefore the velocity is determined by:
JJJJJJG G
G
G
viiter +1 = wviiter + C1 × rand() × ( pbest i − xi )...
JJJJJG G
+ C2 * Rand()* ( gbest i − xi )
(2)
where iter is the current iteration; C1 and C2 are two positive learning factors; rand()
and Rand() are two randomly generated values within [0, 1]; w is known as the inertia
weight, and it plays the role of balancing the global and local search performed by the
algorithm [29, 30].
3. Objective function
The OPF-SC goal is to optimize an objective function subject to equality and
inequality constraints. The optimization problem is formulated as follows:
Min F obj ( u, y )
(3)
g( u, y ) = 0
(4)
hmin ≤ h( u, y ) ≤ hmax
(5)
subject to
where g(·,·) represents the equality constraints set; h(·,·) is the inequality constraints set;
u are the state variables; y represent both integer and continuous control variables.
3.1 Objective function
In this paper, the minimization of the total cost includes the pre-contingency cost
4
(superscript 0) besides each credible contingency cost (superscript k). Thus, the
objective function is constituted by three terms: (i) the generating costs, (ii) the
transmission costs, and (iii) the consumer benefit [31],
F
obj
⎧ o K k⎫
= min ⎨C + ∑ C ⎬
k =1
⎩
⎭
(
(6)
)
C o = ∑ U i0 * f ( PGio ) + SUCi + TC o ( FLWmo− n ) −
NG
i =1
(
)
C k = ∑ U ik * f ( PGik ) + SUCi + TC k ( FLWmk− n ) −
NG
i =1
∑ B(P )
NLoad
∑ (B(P ) − D(P
NLoad
j =1
k
Loadj
(7)
Loadj
j =1
k
Loadj
o
,PLoadj
)
)
(8)
where:
K
is the total number of credible contingencies
Co
represents the base case operating cost
U io
i-th unit’s pre-contingency state , 1-ON, 0-OFF
NG
is the total number of available generators
f ( PGio )
i-th generator’s function cost at time t
PGio
active power supplied by the i-th generator at the pre-contingency state
SUCi
i-th generator’s start- up cost
TC o (.)
pre-contingency transmission cost, (9)
NLoad
total number of load buses.
B( PLoadj ) is the consumer benefit curve for j-th load at time t
PLoadj
pre-contingency active power consumption at the j-th load
Ck
credible contingencies’ cost
D(.)
represents the load interruption cost
k
PLoadj
post-contingency active power consumption at the j-th load
TC k (.)
post-contingency transmission cost, defined as (9):
M
TC = ∑ cm ( MWgi ,m )Lm MWgi ,m
m =1
where
M
lines set
cm (.)
cost per MW per unit length of line m
(9)
5
Lm
length of line m in miles
MWgi ,m
flow in line m, due to the i-th generator.
3.2 Constraints
In the following, different constraints included in the OPF-SC formulation are
detailed.
3.2.1 Equality constraints at the pre- and post-contingency states
This is the set of nonlinear power balance equations that govern the steady-state
power flow formulation, both- at pre and post-contingency state, defined as follows:
NG
0 = ∑P −
i =1
o
Gi
∑ (P ) − P
NLoad
j =1
NG
0 = ∑ QGio −
i =1
NG
NLoad
i =1
j =1
0 = ∑ PGik −
NG
0 = ∑ QGik ,t −
i =1
∑
NLoad
∑Q
j =1
o
Loadj
k
PLoadj
− PLoss k
NLoad
∑Q
j =1
o
Loadj
k
Loadj ,t
− QLosstk
o
Loss
(10)
− QLoss o
(11)
for k = 1, 2,....,K
(12)
for k = 1, 2,....,K
(13)
where PLoss represents the total active power losses, QLoss are the total reactive power
ones, and K is the total number of credible contingencies.
3.2.2 Inequality constraints at the pre- and post-contingency states
This is the set of continuous and discrete constraints representing the system’s
operational and security limits as bounds.
(i) The active power generated by each unit must satisfy the maximum and minimum
operating limits, both for pre- and post-contingency states.
PGi _ MIN ≤ PGio ≤ PGi _ MAX
(14)
PGi _ MIN ≤ PGik ≤ PGi _ MAX
(15)
(ii) Voltage magnitudes at each load bus must be close enough to the reference voltage
min Vref − V j
(16)
where Vref is the reference voltage magnitude, Vj is the voltage magnitude at the j-th
load bus.
(iii) The active power flow through each branch of the network must satisfy the security
limits
6
FLOWijo ≤ FLOWijo_ MAX
∀ i, j, i ≠ j
(17)
FLOWijk ≤ FLOWijk_ MAX
∀ i, j, i ≠ j
(18)
where FLOWij0_ MAX , FLOWijk_ MAX , represent the maximum active power that should flow
through the branch connecting the buses i-j, during the pre-contingency and each postcontingency state, respectively.
4. Transmission cost allocation
To calculate the transmission cost allocation, the average participation factor (APF)
method [5, 8, 9] has been used. It is based on a general transportation problem of how
the flows are distributed on a meshed network. The only requirement is related to the
Kirchhoff’s first law fulfillment.
Once the power flow scenario is determined, the main idea of this methodology is to
determine the agents’ (generator and loads) participation factor in the flow in links. In
this sense, it will be possible to trace the flow of electricity from a generator to the load
buses. The principle adopted to trace these flows is the proportional sharing principle,
schematized in Fig 1. The assumption made is that the bus is a perfect mixer of all
incoming flows, so that it is impossible to determine which particular inflowing electron
goes into which particular outgoing line. It may be assumed that each element flow
leaving the i-th bus can be decomposed into M shares, where M is the number of
incoming flows (generator and elements injecting power into bus i). The amount of such
shares has the same proportion that the incoming flows in the total power injected into
bus i [5, 8].
5. Proposed solution
The proposed methodology runs as follows:
STEP 1:
randomly generated initial population
STEP 2:
for each particle, the reconstruction operators are applied
STEP 3:
the Newton-Raphson routine is applied to each particle
STEP 4:
fitness function evaluation
STEP 5:
compare particles’ fitness function and determine pbest and gbest
STEP 6:
change of particles’ velocity and position according to (1) - (2)
STEP 7:
go to step 2 until a criterion is met (maximum number of generations)
5.1. Initial population
7
In this paper, a particle is composed by continuous and discrete control variables. The
continuous ones include the generators’ active power output and the load’s active
power; the discrete variables are associated with the transformers-tap setting and varinjection values of the switched shunt capacitors/reactors, Fig 2.
The population is constituted by K+1 matrices, one for the base case and one for each
of the K credible contingencies (subpopulations) of dimension (NG+ND) x NIND;
where K represents the total number of accounted contingencies; NG is the number of
continuous variables; ND is the number of discrete variables; NIND is the number of
particles.
For each scenario (pre- and post-contingency states), the initial active power is a
percentile and is randomly allocated among the NG available thermal units, such that
the load is satisfied (19),
PGin =
rand( NG )
∑ rand( NG )
Nload
*
∑P
j =1
LOADj
for n = 0 ,1,....,K
(19)
where, rand(NG) represents a vector of size NG randomly generated within the interval
[0, 1].
The transformer-tap setting and var-injection values of the switched shunt
capacitor/reactor (discrete variable) are randomly generated between upper and lower
limits. An operator is included to ensure that each discrete variable is rounded to its
nearest decimal integer value that represents the physical operating constraint of a given
variable.
(
round random ⎡⎣ STi min ,STi max ⎤⎦ , η
)
(20)
where η represents the most significant decimal.
5.2. Reconstruction operators
In this paper, the reconstruction operators have been used to satisfy the units'
constraints. Taking into account the inequalities within a conventional formulation, by
using penalty functions, for instance, results in an execution where it is very likely to
make wrong decisions, due to the use of excessively high penalty factors. Altogether,
the PSO and the operators, accomplish such handling in a more efficient way, limiting
the use of penalization. That is, such mechanisms control that each continuous variable
fulfills the generation limits, load and discrete variable limits for the pre- and post-
8
contingencies states. For each state the following reconstruction operators have been
applied.
5.2.1. Control variables operating limits
This operator is defined as
⎧ PGi _ MAX
⎪
⎪P
PGi' = ⎨ Gi
⎪ PGi _ MIN
⎪⎩0
if
PG ,i > PGi _ MAX
if
PG ,i _ MIN < PGi < PG ,i _ MAX
if
λ * PGi _ MIN < PGi ≤ PGi _ MIN
(21)
otherwise
The operator, besides satisfying the active power generation limits, is useful for
avoiding the use of additional variables in order to determine the units’ state; thus, for
those units in which the active power is below a predefined percentage, the off state is
chosen (PGi=0). Likewise,
'k
PLoad
,j
o
⎧ PLoad
,j
⎪ o
−
= ⎨ PLoad
, j − Δ Load , j
⎪Pk
⎩ Load , j
if
k
o
PLoad
, j > PLoad , j
−
k
o
if
PLoad
, j < PLoad , j − Δ Load , j
otherwise
(22)
k = 1, 2,.....,K and j = 1,2,.....,NLoad
ST j' k
⎧u max
j
⎪ min
= ⎨u j
⎪ ST
⎩ j
if
ST j > u max
j
min
if
ST j < u j
otherwise
(23)
k = 0 ,1,.....,K and j = 1,2,.....,ND
where Δ −Load , j is the maximum allowed load interruption at bus-j.
5.2.2. Handling constraints for load balance
After the execution of the precedent operation, the total active power assigned to
the thermal units is not necessarily equal to the demanded load. Thus, a re-dispatch is
required for taking into account the available units. Such re-dispatch must deal with
constraints [31].
5.3. Newton-Raphson load flow
Once the reconstruction operators have been applied and the control variables’
values are determined, for each particle a load flow run is performed. Such flow run
allows evaluating the branches’ active power flow and the total losses, which are
assigned to the slack bus.
5.4. Fitness function
9
To handle the active power flow limits and voltage constraints, the objective
function penalization is used. By this technique, the fitness function is composed by the
objective function (6) plus penalty terms for particles violating some power flow and/or
voltage constraint. Such fitness function can be expressed as follows:
Fi fit = Fi obj + Cte( iter )
N _ OFLW
∑
j =1
OverFlow j + Fi obj
NLoad
∑
j =1
Vref − V j , i = 1,..,NIND (24)
i
where, N_OFLW represents the total number of lines with overflow; NIND is the
number of individuals; OverFlowj,t is defined as in (25).
abs( FLOWi , j _ MAX − FLOWi , j )
(25)
FLOWi,j_MAX, is the maximum allowable flow across the line connecting buses i-j;
Cte(iter) is a dynamically modified penalty value: Cte(iter ) = 300 iter , this one, allows an
increasing penalization throughout generations.
5.5. Updating
The new position and velocities can be evaluated by (1)-(2).
6. Test results
The proposed algorithm was implemented in Matlab, and the IEEE New England test
system is employed to validate its potential. The test system consists of ten thermal
units, 39 buses, 46 transmission lines and tap-changing transformers, feeding a total
load of 6097.5 MW and 1409 MVAR. A detailed description of the system’s data is
available in [32]. The operating range of all transformers is set between [0.9, 1.1] with a
discrete step size equal to 0.01. A quadratic fuel cost function is used, and its
coefficients are displayed in Table 1. In this paper, the transmission costs are counted
as a portion of the generating ones,
¡Error! No se encuentra el origen de la referencia.; their proportions are displayed in
Table 2.
TCgi ,m = ( agi MWgi2,m + bgi MWgi ,m + cgi )* Pm
(26)
Three cases are analyzed.
Case 1: An OPF-SC neglecting transmission costs while improving the voltage profile.
Case 2: Both the transmission costs and voltage profile improvement are considered
within the objective function and fitness function, respectively.
Case 3: In the previous cases, the tap-changing transformers are not included as control
variables; in this case, these ones are included as such ones. In all cases, the tripping of
10
line 26-27 is considered as the contingency.
To demonstrate the consistency and robustness of the developed algorithm, 20
independent runs are conducted for each case to count for the number of times in which
the optimal or near optimal solution is reached; different stopping criteria are considered
(maximum iterations). Results and computational times are exhibited in Table 3.
Table 3 shows that with 30 iterations, the best solution is as good as with 50 iterations,
although the standard deviation in the latter case is better. The best results obtained
using the PSO-RO are shown in Table 4. The inclusion of a penalization term into the
objective function has allowed the voltage profile improvement. The total deviation
voltage is reduced from 1.1170 p.u. (case 1) to 0.9246 p.u. (case 2), and to 0.5905 p.u
(case 3) for the pre-contingency state, and from 0.9738 p.u. (case 1) to 0.9058 p.u.
(case 2), and to 0.5885 p.u (case 3) for the post-contingency state.
The routines utilized in these studies are implemented in Matlab 7.0 on a personal
computer with a Pentium 4 processor, running at 2.99Ghz and with 1GB of RAM.
7. Conclusions
This paper investigates the applicability of the PSO-RO in solving the OPF-SC
problem including transmission costs. Moreover, inequality constraints have been added
in the formulation in order to improve the voltage profile. Traditionally, PSO handles
constraints by the objective function penalization. The reconstruction operators are an
alternative mechanism for managing the units' operative constraints, while particles
violating some power flow or bus voltage are penalized. The use of reconstruction
operators allows increasing the number of suitable particles in the searching space.
Thus, the dependency of the heuristic algorithms on the appropriate definition of the
penalization terms is reduced.
The proposed methodology is able to take into account feasible and satisfactory
solutions for both the base case and for a set of credible contingencies. In case that the
power system is capable of getting ahead from the pre-contingency state to a postcontingency state, generation and branches' operative constraints are satisfied.
The PSO-RO is able to deal with mixed type control variables, so that the OPF-SC
solution assures an improved voltage profile.
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11
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15
An Optimal Power Flow plus Transmission Costs Solution
Pablo Oñate Y., Juan M. Ramirez, Carlos A. Coello Coello
j
l
40 MW
12 MW
18 MW
60 MW
i
28 MW
42 MW
k
m
Fig. 1. Proportional sharing concept
16
An Optimal Power Flow plus Transmission Costs Solution
Pablo Oñate Y., Juan M. Ramirez, Carlos A. Coello Coello
i-th PARTICLE
Fig. 2. Particle’s representation.
….
STND
STi means discrete variables
….
STi
….
ST2
ST1
PLoad,NL
….
PLoad,i
….
PLoad,2
PLoad,1
PG,NG
PG,i
PG,2
PG,1
….
17
An Optimal Power Flow plus Transmission Costs Solution
Pablo Oñate Y., Juan M. Ramirez, Carlos A. Coello Coello
Table 1: Fuel cost coefficients
2
Bus a i [$/MW ] b i
30
0.0193
31
0.0111
32
0.0104
33
0.0088
34
0.0128
35
0.0094
36
0.0099
37
0.0113
38
0.0071
39
0.0064
[$/MW] c i [$]
6.9
3.7
2.8
4.7
2.8
3.7
4.8
3.6
3.7
3.9
0
0
0
0
0
0
0
0
0
0
18
An Optimal Power Flow plus Transmission Costs Solution
Pablo Oñate Y., Juan M. Ramirez, Carlos A. Coello Coello
Table 2: Coefficients of proportion (New England power system)
1
L1-2
2
L1-18
3
L2-3
4
L2-7
5
L2-8
Transmission cost coefficients (Pm)
6
7
8
9
10
11
L2-13 L2-26 L3-13 L4-8 L4-12 L5-6
12
L6-7
13
L6-11
14
L6-18
15
L6-19
16
L6-21
0.033 0.040 0.040 0.056 0.042 0.058 0.037 0.042 0.046 0.053 0.047 0.053 0.048 0.047 0.056 0.061
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
L7-9 L8-12 L9-10 L10-12 L10-19 L10-20 L10-22 L11-25 L11-26 L12-14 L12-15 L13-14 L13-15 L13-16 L14-152
L7-8
0.043
33
L15-16
0.036 0.054 0.060 0.058 0.049 0.060 0.042 0.046 0.064 0.049 0.051 0.037 0.054 0.055 0.058
34
35
36
37
38
39
40
41
42
43
44
45
46
L16-17 L16-20 L17-18 L17-21 L19-23 L19-24 L19-25 L20-21 L20-22 L21-24 L22-23 L22-24 L23-25
0.055
0.045 0.065 0.050 0.060 0.047 0.049 0.057 0.052 0.042 0.042 0.046 0.054 0.054
19
An Optimal Power Flow plus Transmission Costs Solution
Pablo Oñate Y., Juan M. Ramirez, Carlos A. Coello Coello
Table 3: Statistical data for cases 1-3
CASE 1
CASE 2
CASE 3
Gener
NIND
20
30
50
20
30
50
30
50
36
36
36
36
36
36
36
36
Objective function
Mean
Best
Worst
123970.0
123420
124580
123518.7
122990
124410
123223.9
122918
123810
217804.7
216130
219400
216474.7
215860
218120
216140.7
215660
216620
190090.7
176620
207440
184352.4
174940
188810
Standart
Deviation
373.67
391.44
216.33
865.11
616.93
323.05
9552.04
3847.3
Average
time (seg)
290
436
610
406
707
960
554
875
20
An Optimal Power Flow plus Transmission Costs Solution
Pablo Oñate Y., Juan M. Ramirez, Carlos A. Coello Coello
Table 4: Optimization results for the New England power system
CASE 1
CASE 2
CASE 3
PG [MW]
pre-cont 26-27 out pre-cont 26-27 out pre-cont 26-27 out
239.83
250.98
243.69
229.53
393.03
393.03
611.28
587.86
468.68
509.26
643.10
643.10
698.20
629.00
641.35
530.67
739.50
739.50
750.00
683.98
750.00
703.48
708.32
708.32
439.58
416.80
650.00
551.24
175.51
175.51
653.45
750.00
750.00
750.00
713.32
713.32
531.51
549.67
750.00
750.00
624.03
624.03
435.64
452.13
447.26
513.79
593.18
593.18
783.20
900.00
889.50
900.00
641.95
641.95
994.87
939.37
580.52
739.23
908.59
908.59
Position taps
1.025
1.025
1.025
1.025
0.91
1.00
1.070
1.070
1.070
1.070
1.10
1.10
1.070
1.070
1.070
1.070
1.10
1.10
1.006
1.006
1.006
1.006
1.04
0.95
1.006
1.006
1.006
1.006
0.99
1.09
1.060
1.060
1.060
1.060
1.09
1.10
1.070
1.070
1.070
1.070
1.09
1.04
1.009
1.009
1.009
1.009
1.00
1.10
1.025
1.025
1.025
1.025
0.94
0.96
1.025
1.025
1.025
1.025
1.02
0.90
1.025
1.025
1.025
1.025
0.97
0.97
Loss active power
40.06
62.29
73.51
79.70
43.03
48.59
Total deviation voltage [p. u.]
1.1170
0.9738
0.9246
0.9058
0.5905
0.5885
Total transmition cost [$]
6374.50 5895.90 5473.60 5044.90
Objective function [$]
61339.0 61579.0 63209.0 62398.0
63564
63552
Fitness function
122918.0
215660
174940
21
An Optimal Power Flow plus Transmission Costs Solution
Pablo Oñate Y., Juan M. Ramirez, Carlos A. Coello Coello
Figures caption
Fig. 1. Proportional sharing concept
Fig. 2. Particle representation.
22
An Optimal Power Flow plus Transmission Costs Solution
Pablo Oñate Y., Juan M. Ramirez, Carlos A. Coello Coello
Figures caption
Table 1: Fuel cost coefficients
Table 2: Proportions coefficients (New England power system)
Table 3: Statistical data for cases 1-3
Table 4: Optimization results for the New England power system