1
Abstract — In conventional optimal power flow (OPF), the
parameters of electrol components (e.g. resistance and thermal
ratings of the overhead lines) are assumed to be constant despite
the fact that they are strongly sensitive to the weather effect (e.g.
temperature or wind speed) which influences the accuracy of
optimal power flow results. This paper introduces a weather-
based optimal power flow (WB-OPF) algorithm with wind farm
integration by considering the temperature related resistance and
the dynamic line rating (DLR) of overhead transmission lines. A
method of calculating the current-temperature relationship of
bare overhead lines, given the weather conditions, is presented as
a set of coupled temperature and power flow equations. A
simplified general model is proposed to calculate the dynamic
line rating (DLR) for maximizing the utilization of wind power. A
Primal-dual Interior Point (PDIP) method is developed to solve
the WB-OPF problem and the effectiveness of the proposed
method is evaluated and demonstrated in the paper by two
example power systems.
Index Terms— Electro-thermal Coupling, Dynamic Line
Rating (DLR), Wind generation, weather effects, Weather-Based
Optimal Power Flow (WB-OPF).
NOMENCLATURE
Indices and Sets
i, j Indices.
k State index (k=0 denotes pre-contingency,
k>0 denotes post-contingency).
kc Index of the most severe contingency.
1, ,
dc
n
dc
N
The node set of the DC network.
C
1, ,c
Set of N-1 outage contingencies including
both the traditional AC contingencies and
outage of DC lines.
MAX MAX
{ , }
Gj pcci
U P P
Set of control variables for Differential
Evolution (DE) method.
Dr. Jun Cao, Dr. W. Du (corresponding author) and Prof. H. F. Wang are
with the State Key Laboratory of Alternate Electrical Power System with
Renewable Energy Sources, North China Electric Power University,
Changping, Beijing, China.
This work was supported in part by the State Key Laboratory of Alternate
Electrical Power System with Renewable Energy Sources, the National Basic
Research Program of China (973Program) (2012CB215204), the National
Natural Science Foundation of China (No. 51407070, 51311122), China, and
in part by the Fundamental Research Funds for the Central Universities (No.
2014QN01), the project of State Grid Corporation of China (Grant:
SGHB0000KXJS1400040)
P
C
S
Set of contingencies that are preventively
controlled.
C
C
S
Set of contingencies that are correctively
controlled.
UC
S
Set of contingencies that cause infeasible
solution.
kc
S
Set of the most severe contingency.
*
c
P
Set of the SCOPF optimal solution.
Functions
f, f
0
Objective functions of the CSCOPF
model with and without penalty term
added.
0
0
,,
kk
h h h
Inequality constraints of the CSCOPF
model for the base case and post-
contingency short-term, long-term period.
0
0
,,
kk
g g g
Equality constraints of the CSCOPF
model for the base case and post-
contingency short-term, long-term period.
Parameters
s
W
Incident solar energy.
c
d
The conductor diameter.
dci
U
,
P
The voltage and power references of the
droop control strategy.
MAX
Gj
P
Upper limits of active power generation
of the jth unit (MW).
MAX
pcci
P
Upper limits of active power generation
of the ith converter (MW).
MAX
h
Vector of long-term flow limits.
MAX
convi
S
Maximal apparent power of the ith
converter.
MAX
sk
u
Maximal allowed adjustment variables of
long-term control actions.
MAX
k
LineFlow
Maximal Line flow during the kth
contingency.
Parameter defining how much the short-
term post-contingency security
constraints can be temporarily relaxed
from the permanent limits.
kc
r
Occurring probability of the kcth
contingency.
Weather-Based Optimal Power Flow
with Wind Farms Integration
Jun Cao, Member, IEEE, W. Du, H. F. Wang, Senior Member, IEEE
2
Penalty coefficient.
dcij
Y
The element of bus admittance matrix Y
dc
of the DC network.
NP The number of the DE population number.
D The dimension of the DE parameter
vectors.
Variables
j
Q
Heat gain due to resistive line losses (
loss
P
[W/m])
s
Q
Heat absorbed by solar radiation
c
Q
Forced convection heat loss
r
Q
Radiated heat loss
period.
I. INTRODUCTION
ncread wind power generation has been connected to the
power systems all over the world. These lead to significant
challenges for the economical operation of power systems
with large-scale wind power integration due to the stochastic
characteristic of the wind speed. The variation nature of wind
speed will introduce the changing of not only the wind power
generation, but also the parameters of eletronic components
(e.g. the resistance and thermal ratings of the overhead lines,
as these parameters are strongly related to the weather effects,
such as temperature, wind speed, et al. [1, 2]), which would
affect allocating the system load power between conventional
generators and wind-powered ones. Thus, there is a great need
to incorporate these effects, bringing with the wind power
integration, into the traditional OPF problem.
The variations of actual weather conditions will influence
the resistances and thermal ratings of the system overhead
lines and then affects the results of power flow and optimal
power flow (OPF) [1, 2]. For example, the resistance of the
power system equipment is a strong function of temperature
and the line thermal rating varies with the weather conditions,
such as the wind speed and direction, ambient temperature and
solar radiation. However, traditional optimal power flow
algorithms neglect the weather effects and take the resistance
and thermal ratings of transmission lines as constant. These
negligence will bring with two limitations: (1) some weather-
related error is inherent in the OPF results by using the
inaccurate resistance value [2]; (2) normally, the Dynamic (or
real-time) thermal Line Rating (DLR) is higher than the static
rating most of the time. Experience shows that an average of
50% extra capacity of overhear lines in favorable locations can
be safely exploited by using the DLR technique [3]. Thus
traditional OPF does not exploit the full capabilities of
existing lines which results in higher cost of the total system.
To accurately analysis the weather (mainly temperature, wind
and solar radiation) related effect, this paper presents a
weather-dependent optimal power flow technique to take
account of the estimate of branch element weather factors in
the wind power integrated OPF formulation.
In common practice, the steady state analysis such as power
flow, optimal power flow and state estimation assume that the
resistance of the overhead lines are constant and the
calculation of system admittance was based on constant
temperature (normally, the worst-case situation [1, 4]).
However, the temperature, resistance and losses are
interrelated and vary significantly in the real operation.
Without considering the temperature-related variational part of
resistance will lead to significant errors in loss estimations,
especially under heavily loaded conditions [2]. To reduce the
temperature-related error in the power flows analysis, Dr.
Stephen Frank firstly introduces the temperature dependent
power flow algorithm which integrates an estimate of branch
temperatures and resistances with the conventional power flow
equations [2, 5]. The work of [6] studies the influence of
changes of the transmission lines resistance due to temperature
on state estimation performance. The impact of transmission
line temperature variations, resulting from loading and
weather conditions changes, on system dynamic performance
is analyzed in [7]. Although many research have focused on
the environmental effect on system steady state and dynamic
response, considering the weather related condition in the
traditional optimal power flow algorithm with wind farms
integration is still a blank field till now.
One of the key challenges faced during integration of wind
powers with the grid is the spillage of wind energy due to the
transmission constraints [8]. Many techniques have been
introduced to minimize the spillage of wind power by using
FACTs [9] and energy storage devices [10], which, however,
is mostly cost expensive. The technology of Dynamic Line
Rating (DLR) has attracted many attentions from the academic
and industry, especially for maximizing the utilization of wind
power [11]. Conventional static line rating (fixed
summer/winter thermal rating) used in OPF is determined
based on worst case weather assumption for operation. The
analysis results are generally conservative and expensive. But
in reality, the real capacity is not static and is a complex
function of air temperature, solar radiation, local wind and
actual current et al. [12]. One of the application example using
the DLR technology is a leading UK distribution network
which was concerned about network capacity problem when
offshore and onshore wind farms were connected. Alstom
Grid successfully alleviates the wind power integration
problem by using the DLR technology. The trial showed that
50% or more wind generation could be connected to the grid
compared to using the fixed summer/winter thermal ratings
[11]. The work of [13] proposes a new general DLR
calculation model and based on this, it develops an economic
optimization simulation model regarding wind power
integration by using a general DLR calculation model on
overhead lines. The use of DLR technology will provide the
true transfer capacity of the grid in real time accounting for
actual weather conditions and improve the system reliability
and utilization of the existing system [11].
This paper presents a novel treatment of the weather-
dependent optimal power flow algorithm with wind farms
integration which considers the temperature related resistance
and the dynamic line rating (DLR) of overhead lines. A
resistance-weather relationship and calculation modeling of
I
3
DLR for overhead lines are present in Section II. Section III
proposes the weather-based optimal power flow model with
wind farms integration. A Primal-Dual Interior Point (PDIP)
method is developed to solve the WB-OPF problem.
Numerical solutions of two case studies using the MATLAB
are discussed in Section IV. Finally, conclusions are drawn in
Section V.
II. WEATHER-DEPENDENT MODELING OF OVERHEAD LINES
A. Resistance-weather relationship of bare overhead
conductors
(1) Thermal balance of overhead lines
Fig. 1 illustrates the thermal balance model of overhead
lines. The resistance of bare overhead conductors is a function
of the ambient weather conditions according to the following
steady state thermal balance equation [12]:
j s c r
Q Q Q Q
(1)
Fig. 1 Thermal balance model of overhead lines
The solar heat gain can be formulated as
s s c
Q W d
(2)
The radiated heat loss
r
Q
is nonlinear function of line
temperature. However, it can be approximated as linear
function of the conductor temperature rise over ambient [1]:
44
( 273) ( 273)
()
r r c a
r c a
Q A T T
K T T
(3)
Forced convection heat loss can be written as
( T )
c c c a
Q K T
(4)
where
r
K
and
c
K
are the radiation and convective radiation
heat transfer coefficients, respectively (equations are given in
Appendix A).
Using (3) and (4), (1) can be rewritten as
( ) ( )
loss s c r c c a
P W d K K T T
(5)
Thus, by rearranging (5)
1
()
c a loss s c
rc
T T P W d
K
(6)
Where
rc r c
K K K
is the coefficient of heat loss.
(2) Resistance equation
The resistance of metallic conductors varies with the
conductor temperature according to
[1 ( )]
c a c a
R R T T
(7)
where
,
ca
RR
conductor resistance at temperature
and
ca
TT
,
ca
TT
conductor and ambient temperature (
o
C
)
temperature coefficient
Fig. 2 represents the conductor temperature variation
against the ambient temperature and wind speed. It shows that
the wind speed, by changing the coefficient of
c
K
, is almost
linear relationship with the conductor temperature.
20 22 24 26 28 30 32 34 36 38 40
30
35
40
45
50
55
Ambient Temperature (solid)
Conductor Temperature
1 2 3 4 5 6 7 8 9 10
Wind speed (dotted)
31.05
31.10
31.15
31.20
31.25
31.30
Fig. 2. Influence of the single parameter variation on the conductor
temperature T
c
B. Calculation modeling of Dynamic Line Rating (DLR)
In reality, the transmission line rating are varied with: 1)
current flowing in the conductor; 2) conductor size and
resistance; 3) ambient weather conditions (temperature, wind
speed and direction, solar radiation). A simple way to
calculate the static line rating is based on the worst scenario.
Engineering recommendation (ER) P27, which can be
formulated as
0.52
max
SLR
[1.01 0.0371 ( ) ] [ ( ) ]
SLR
cf
SLR SLR
f angle c a
f
SLR
dv
k K T T
I
R
(8)
where
SLR
R
line resistance used in the scenario of SLR
calculation
SLR
v
wind speed in the SLR calculation
,,
f f f
k
the density, dynamic viscosity and thermal
conductivity of air
SLR
angle
K
the angle between wind speed and the conductor
axis in the scenario of SLR calculation
The Dynamic line rating (DLR) can be computed by using
a variety of methods: conductor sag and tension monitoring,
physical modeling and prediction techniques [13]. The
simplified calculating model of DLR is based on the capacity
ratio between DLR and SLR, which are given as below [13]:
max max
DLR SLR
vT
II
(9)
where
v
and
T
are the ratios related to the wind speed and
temperature, respectively.
Fig. 3 shows the dynamic line rating varies with the wind
speed and ambient temperature. The current rating reaches
maximum when wind speed is 10m/s and the temperature is
20
o
C
. Normally, when the wind speed is higher, the ambient
temperature is lower, thus more wind powers can be
transferred by using the modeling of dynamic line rating.
4
Fig. 3. Dynamic current rating versus wind speed and ambient temperature
III. FORMULATION OF THE WEATHER-BASED OPF PROBLEM
WITH WIND FARMS INTEGRATION
A. Weather-Based OPF Formulation
To incorporate the weather condition into the traditional
OPF problem with wind farm integration, the following
modifications should be made:
1) the addition of branch temperatures to the vectors of
system state variables;
2) the construction of a set of mismatch equations for
thermal balance equations of overhead lines;
3) the incorporation of dynamic line rating model in the
transmission line constraints;
4) the addition of wind generation cost to the objective
functions;
5) the modeling of interdependence of temperature and
the system via an augmented Jacobian and Hession
matrix.
Fig. 4 shows a representation model of the weather-based
OPF incorporating the wind farms which explains the state
vectors and interactions of each module.
Weather-dependent OPF
Electro-thermal model of
over-head lines (6), (7)
Traditional OPF (16) (17)
State vector: T
State vectors:
θ,V
loss
P
State vector: T
Dynamic Line
Rating Model
(9)
Weather Datas:Wind
speed; Ambient
temperature;Solar
Measurement
Wind
Generation Cost
Model (12)
Fig. 4. Illustration of Weather-Based OPF model incorporation of Wind
energy
Mathematically, the OPF can be stated as the following
constrained nonlinear optimization problem [17]:
Minimize
min ( )f y
(10)
Subject to
( ) 0; ( ) 0; y y y y yhg
(11)
where
min ()f
is the objective function; the vector of y is
the optimization variables which include the state variables x
and control variables u. The lower and upper limits of y are
represented by
y
and
y
, respectively.
()yh
represents a set of
equality constraints which includes both the nonlinear power
flow mismatch equations and control equations.
()yg
is a set
of inequality constraint functions to avoid the violation of the
system limits.
B. Wind Generation Cost Model
The optimal schedule of wind farms output are highly
dependent on the accuracy of wind power forecast
technologies. However, the current forecast error from day-
ahead point prediction technique can be as high as 25%~40%
[14]. Therefore, it is necessary to include the forecast
uncertainty into the objective cost function of OPF model.
The actual available wind power generation
Avai
Wind
P
is a
random variable which can hardly be predicted accurately.
The probability density function (PDF) of
Avai
Wind
P
(or forecast
error, forecast error =
Avai
Wind
P
-forecast value) is a conditional
probability function with respect to the forecast value [15]. Fig.
5 shows the distribution of wind power output
Avai
Wind
P
for a
forecast value range [0.20, 0.24] by using the versatile
probability distribution model in [15].
The integration of wind farms will introduce two types of
generation cost: 1) wind spillage opportunity cost (green area
in Fig. 5). Although there is no fuel cost for wind energy, the
construction and operation of wind farms will introduce the
initial investment and the maintenance costs, no matter
whether the wind power is scheduled or not [16]. Thus wind
power spillage, which refers to the amount of the unused wind
power production (
Avai Sche
Wind Wind
PP
), will add extra opportunity
cost to the objective function; 2) Reserve cost (red area in Fig.
5). When
Avai
Wind
P
is larger than
Sche
Wind
P
, it will bring with extra
reserve cost. Thus the expected wind generation cost can be
defined as
( ) ( )z(x)dx ( ) (x)dx
r
C x x z
Rated Rated
Wind Wind
Sche Sche
Wind Wind
PP
Sche Sche Sche
Wind Wind Wind
PP
P P P
w
cc
(12)
Where
1 (1 ) 1
1
8760 D(1 )
n
I
n
C
D
m
nD
design rated
Wind wind
CF P
w
c
(13)
I
C
initial investment
design
Wind
CF
capacity factor
rated
wind
P
rated wind power
D
discount rate
m
annual maintenance cost function
n
life time of plant
20
25
30
35
40
0
5
10
1.5
2
2.5
3
3.5
4
4.5
Ambient temperature (degree)
Wind speed (m/s)
Current rating (KA)
5
0 0.1 0.2 0.3 0.4 0.5
0
2
4
6
8
10
12
PDF
Probabilistic wind power output(p.u.)
Sche
Wind
P
Avai
Wind
P
()
Sche Avai
Wind Wind
PP
Forecast Value
Expected reserve
cost area
Expected opportunity cost
of wind power spillage
area
()
Sche Avai
Wind Wind
PP
Fig. 5 Probabilistic distribution of wind power output for a given forecast
value
Sche
Wind
P
the set of scheduled active power output of wind
farms;
Rated
Wind
P
Rated active power output of wind farms;
w
c
opportunity cost coefficient of wind power
spillage, relating to under-forecasting of wind
generated electricity [16];
r
c
reserve cost coefficient, relating to over-
forecasting of wind generated electricity;
z the probability distribution function (PDF) of
wind power output
Avai
Wind
P
;
C. Objective Function with Wind Generation
The main goal of objective function of the WB-OPF model
with wind farms integration is to minimize the generation cost
of the whole grid, meanwhile, reduce the opportunity cost and
reserve cost of wind power. Hence the objective function can
be written as
min ( ) ( ) ( )
g
f C C
Sche
G G Wind
P P P
(14)
Where
2 1 0
()
g g g g
C c c c
2
G G G
P P P
(15)
g
C
cost function for the conventional generators
C
penalty cost function for wind power forecasting
error.
G
P
the set of scheduled active power output of
generators;
2 1 0
,,
g g g
c c c
are the cost coefficients of generators.
It should be noted that the quantification of the wind
generation cost is a complex process. Wind energy as a clean
and environment-friendly technology will bring with social
benefit, which are not include in the analysis.
D. Equality and Inequality Constraints
The equality constraints of WB-OPF include the
conventional power flow mismatch equations and electro-
thermal coupling constrains which can be formulated as
follows:
1
( , , )
( )cos( ) ( )sin( )
i i i
a
i Gi Wi Li
i GN i GW i LN
N
i j ij ij i j ij ij i j
j
P P P P
VV G T B T i
ac
δ VT
N
(16)
1
( , , )
( )sin( ) ( )cos( )
i i i
a
i Gi Ci Li
j GN j CN j LN
N
i j ij ij i j ij ij i j
j
Q Q Q Q
VV G T B T i
ac
δ VT
N
(17)
22
,
( , , )
1
( ) ( ) 2 ( ) cos( )
,
ij ij
a ij ij i j ij ij i j i j
rc ij
HT
T G T V V G T VV
K
ij
ac
δ VT
N
(18)
,min ,maxGi Gi Gi
P P P
(19)
,min
Sche Rated
Wi Wind Wind
P P P
(20)
,maxWi
II
(21)
ij,maxij
TT
(22)
,min ,maxGi Gi Gi
Q Q Q
(23)
where
ij
G
and
ij
B
are the mutual (or self when i=j)
conductance and mutual (or self when i=j) susceptance,
respectively.
ij
T
and
ij,max
T
are the conductor temperature and
temperature limit of line i-j, respectively;
,rc ij
K
is the heat loss
coefficient of line i-j.
,maxWi
I
is the branch thermal rating (static
or dynamic).
Note that the optimization variables of the WB-OPF are
, , , ,
T
GG
x U T P Q
. Where,
U
is the bus voltage
magnitude state variable vector,
bus voltage angle state
variable vector,
conductor temperature state variable vector,
G
P
active power generation control vector,
G
Q
reactive
power generation control vector, respectively.
To include the temperature related components in the WB-
OPF algorithm, the Jacobian and Hessian matrices need to be
modified accordingly. Modification of the Jacobian and
Hessian matrix is to add one state variable (temperature T) per
overhear line, one sets of equality constraints (Equation 18)
and one sets of inequality constraints (Equation 22).
IV. CASE STUDIES
In this section, the WB-OPF algorithm is verified by use of
a modified IEEE 9 node system and the New England
transmission grid which are available in MATPOWER [17].
