UNSW - SCHOOL OF ELECTRICAL ENGINEERING AND TELECOMMUNICATIONS
TELECOMMUNICATIONS
ELEC4612 POWER SYSTEM ANALYSIS LABORATORY
EXPERIMENT 4
ECONOMIC DISPATCH AND OPTIMAL POWER FLOW
1. AIMS:
With the aid of the PowerWorld Simulator program, the objectives are:


To analyse the problem of economic dispatch.
To analyse the problem optimal power flow.
2. BACKGROUND:
2.1 Economic Dispatch:
Economic dispatch is a method that determines the allocation of the loads to the electricity
generation resources in the power system, and minimizes the generation cost. Consider an
ideal system with n generating units, neglecting unit constraints and system losses. The
operating cost of generator i is C , which varies with the real power output P of the
i
i
generator, and
(1)
Ci   a  bPi  cPi 2  dPi 3    fu
fuel
el cost
ost 
$/hr
hr 
 $/
where a, b, c, d are the coefficients of the cost curve.
Generally, “a” represents the fixed costs that do not vary with the generator output, such as
the cost of installing the generators; “b”, “c”, “d” are coefficients related to generator power
outputs. The fuel input to the generator is in Btu/hr and represented by bPi  cPi 2  dPi 3 . The
fuel cost is in $/Btu. The operating cost can be reduced by optimizing the operation strategies.
Practically, the curve of operating cost C i versus generator output is a piecewise-continuous
and monotonically increasing curve. The discontinuities may be caused by the incremental
firing of the equipment, like additional boilers or condensers. The incremental cost curve,
dCi dPi versus Pi , which indicates the cost of producing one more MW power by the
electricity generating unit, can be obtained by taking the slope or derivative of the operating
cost curve. When C i consists of only fuel costs, dCi dPi specifies the heat rate of the unit.
Heat rate is the ratio of the amount of heat energy required in Btu to generate one more MW
of power, which indicates the fuel efficiency of the generator over its operating range. The
lower this number is, the higher the efficiency of the unit is. Generally, generators reach their
highest efficiency somewhere in the middle of their operating range, and become least
efficient at the minimum and maximum MW output.
For an area consists of n units which operate on economic dispatch, the total operating cost of
the area is
CT 
n
C
i
 C1 P1   C 2 P2  

 C n Pn 
(2)
$/hr
i 1
Neglecting system
syste m losses, the total load demand
d emand in the area is
Elec4612 - Experiment 4 – Economic dispatch and optimal power flow
Page 1
PT 
n
P  P  P
1
i
2


 Pn
(3)
i 1
Assuming the load demand PT is constant, the economic dispatch problem becomes the
determination of the values P1 , P2 , Pn that minimize the total area operating cost CT , and
the sum of unit outputs should be equal to the total load demand.

The minimum total operating cost occurs when
CT
CT
CT
T
1
2
dC  P dP  P dP   P dPn  0
1
2
n
From Equation 2, the above can be written as:
dC n
dC1
dC 2
dCT 
dP1 
dP2  
dPn  0
dP1
dP2
dPn

(4)
(5)

Since the load demand PT is assumed constant,
dP1  dP2 

 dPn  0
(6)
Multiplying Equation 6 by  and subtracting the resulting equation from Equation 5, the
following equation can be obtained,

 dC

 dC1

  dP1   2   dP2 

 dP2

 dP1


 dC
  n   dPn  0

 dPn
(7)
According to Equation 6, Equation 7 is satisfied when
 dC1      dC 2    

  dP
 dP

  2
 1

dC
  n     0

 dPn
(8)
Thus,
dC1
dP1

dC 2



dP2
dC n

(9)
dPn
Therefore, all units in the system should operate at the same incremental operating cost under
economic dispatch, which is a criterion for the solution to economic dispatch problem.
dC1 dC 2
dC n

 
(10)
dP1
dP2
dPn

The criterion given in the Equation 10 can be explained as follows: neglecting the system
loss, if a unit operates at a higher incremental cost than others, the output power shift from
this unit to the unit with lower incremental cost will certainly reduce the total operating cost
in the system. Thus all the electricity generating units must operate at the same incremental
cost to achieve the minimum operating cost.
If the unit’s output constraint, Pi min  Pi  Pi max , is considered, the units that have reached
their limit values are held at their limits, and the others that are not at their limits will
distribute remaining loads equally. In this case, the incremental operating cost of the area is
determined by the units that are not at their limits.
When the transmission losses are included, the problem becomes more complicated. The total
cost of transmitting 1MW power includes the incremental cost and the cost due to the
transmission losses. The unit with lower incremental operating cost may be so far away from
the load centre that the total cost is higher than those with higher incremental cost. The total
load demand is as follows: n
PT 
Pi  PL  P1  P2 


 Pn  PL
(11)
i 1
Elec4612 - Experiment 4 – Economic dispatch and optimal power flow
Page 2
where PL is the total transmission loss in the system. Generally, PL depends on the unit
outputs P1 , P2 ,
, Pn . Equation 6 can be rewritten as

dP1  dP2 

 P
P
 dPn    L dP1  L dP2 
P2
 P1



PL
dPn   0
Pn

(12)
Thus, Equation 7 can be rewritten as
 dC1

 dC2

PL
PL





d
P





 1 
 dP2 


d
P
P
d
P
P
 1

 2

1
2

According to Equation 12, Equation 13 is satisfied when
PL
PL
dC1
dC
dC2
dCn

dP1
P1
 

dP2
 
P2


 dC

P
  n   L    dPn  0
Pn
 dPn


dPn
PL
  0
Pn
(13)
(14)
or
dC1
dP1

dC 2



dP2
 P 
  1  L 
dPn
 Pn 
dC n
(15)
Equation 15 is the new criterion when the transmission losses are considered. When the
transmission losses are negligible, which means PL P1  0 , this equation reduces to
Equation 9.
2.2 Optimal Power Flow:
Optimal Power Flow (OPF) is a generalized non-linear programming formulation of the
economic dispatch problem with system constraints. Under economic dispatch, only output
constraints of electricity generating units are considered. However, each transmission line or
transformer has a limit as specified by its power rating. The solution to the Optimal Power
Flow problem not only solves the dispatch problem economically, but also follows the
transmission network constraints. An OPF problem is typically a non-linear programming
problem with multi-objective, which aims to optimize the steady-state performance of a
power system network and satisfy the equality and inequality constraints at the same time.
t ime. It
can be formulated as:
f  x, u 
Minimize
g  x, u   0
(16)
h  x, u   0
subject to.
where x is the vector of state variables of a power system network which include slack bus
real power output PG1 , load bus voltage VL , generator reactive power outputs QG , and
transmission line loadings S l . u is the vector of control variables, which are self-constrained
and include generator voltages VG , generator real power outputs except at the slack bus,
transformer tap settings T , and reactive powers QC injected from shunt VARs. Therefore, x
and u can be expressed as:
x  PG1 , VL1 VLN , QG1 QGN , S l1 S ln
(17)
u  VG1 VGN , PG2 PGN , T1 Tn , QC1 QC N 







OPF is the optimized power flow, and the OPF equality constraints g x  can be expressed as:
Elec4612 - Experiment 4 – Economic dispatch and optimal power flow
Page 3
N
PP  PGi  PDi  Vi  YijV j cos i   j   ij 
i  N 
j1
(18)
N
QP  QGi  QDi  Vi  YijV j sin  i   j   ij 
i  N 
j1
where
PGi , QGi — real and reactive power generation at bus i;
PDi , Q Di — real and reactive power demand at bus i;
Vi ,
i
Yij ,
 ij
— voltage magnitude and angle at bus i;
— magnitude and angle of bus admittance.
The system operational and security limits which are represented by the OPF inequality
constraints hx  can be shown as follows:
PGi min  PGi  PGi max
(i  1,
QGi min  QGi  QGi max
Vi min  Vi  Vi max
S li  S li max
(i  1,

(i  1,
(i  1,

, N)

, N)
(19)
, N)
, n)

There are many methods for solving OPF problem. The most popular traditional optimization
techniques are linear programming, Newton-based techniques, sequential quadratic
programming, and generalized reduced gradient method. However these techniques are
naturally local optimizers and each technique only suits a specific OPF problem. Nowadays,
various modern algorithms, including Genetic Algorithm (GA), Evolutionary Programming
(EP), Artificial Neural Network (ANN), Simulated Annealing (SA), Ant Colony Optimization
(ACO), and Particle Swarm Optimization (PSO), have been proposed to avoid local minima.
They allow engineers to solve the OPF problem more efficiently and effectively for large
power systems.
Elec4612 - Experiment 4 – Economic dispatch and optimal power flow
Page 4
3. SIMULATIONS:
Procedure:
This experiment involves the use of PowerWorld Simulator. A 10-bus power system is shown
in Figure 1 below.
Figure 1: A 10-bus power system.
This system consists of 10 buses, and three generators (G1 at bus 1, G2 at bus 10, G3 at bus
3). The corresponding data of the system is listed in Table 1 and Table 2. AGC are available
for all the generators and enforced MW limits are applied.
Table 1: Bus data
Bus
No.
P
P
P
Q
Generated
max.(MW)
Generated
min.(MW)
Load
(MW)
1
300.0
80.0
2
0.0
3
Q
Q
Voltage
Load
(MVar)
Bus
Type*
Generated
max.(MVar)
Generated
min.(MVar)
Level
(kV)
0.0
0.0
2
50.0
-6.0
13.8
0.0
21.7
12.7
3
0.0
0.0
13.8
200.0
60.0
54.8
19.0
2
60.0
-6.0
13.8
4
0.0
0.0
77.3
16.6
3
0.0
0.0
13.8
5
0.0
0.0
0
0
3
0.0
0.0
13.8
6
0.0
0.0
0
0
3
0.0
0.0
13.8
7
0.0
0.0
9
5.8
3
0.0
0.0
13.8
8
0.0
0.0
6.1
1.6
3
0.0
0.0
13.8
9
0.0
0.0
57
5.8
3
0.0
0.0
13.8
10
150.0
30.0
14.9
5.0
1
80.0
-6.0
13.8
*Bus Type: (1) swing bus, (2) generator bus, (3) load bus.
Elec4612 - Experiment 4 – Economic dispatch and optimal power flow
Page 5
Table 2: Line data
From Bus
To Bus
Resistance (p.u.)
Reactance (p.u.)
Line Charging (p.u.)
1
2
0.01938
0.05917
0.0528
1
5
0.05403
0.22304
0.0492
2
3
0.04699
0.19797
0.0438
2
4
0.05811
0.17632
0.0374
2
5
0.05695
0.17388
0.034
3
4
0.06701
0.17103
0.0346
4
5
0.01335
0.04211
0.0128
4
7
0.03181
0.08450
0.0
4
10
0.12711
0.27038
0.0
5
6
0.09498
0.1989
0.0
5
8
0.12291
0.25581
0.0
5
9
0.06615
0.13027
0.0
6
7
0.08205
0.19207
0.0
8
9
0.22092
0.19988
0.0
9
10
0.17093
0.34802
0.0
*Rating for transmission lines are 150MVA.
The fuel cost is 1.00 $/Mbtu, and the operating costs of the generators, using quadratic cost
model, can be specified as follows:
C1   2 P1  0.375 P12    fuel cost 
 $ / hr 
C2  1.75 P2  1.75 P2    fuel cost 
 $ / hr 
C3   3.25 P3  0.834 P3    fuel cost 
 $ / hr 
2
2
where P1 , P2 , P3 are in MW units.
1.
Obtain
the Incremental
the of
Fuel
curve ofunder
the generators
the
same scale,
and forecastCost
the curve
outputand
trends
theCost
generators
Economic using
Dispatch
state.
2.
Run the simulation and write down the total MW losses, the total MVar losses, and the
hourly cost of the system, under Economic Dispatch state. Record the marginal cost as
well. To change the AGC Status, select Case Information
Aggregation Areas, and
then double-click on the AGC Status field.
3.
Repeat (2) by using LP OPF algorithm. To solve the OPF, first toggle the AGC Status
field to OPF. Then select Add Ons
Primal LP to solve the power flow using the LP
OPF algorithm. Comment on the results.
4.
Add a capacitor bank to bus 9. Gradually increase the voltage of bus 9 above 0.9800, and
resolve the OPF problem. Comment on the results.
Remove the capacitor bank before proceeding to the next part.
Elec4612 - Experiment 4 – Economic dispatch and optimal power flow
Page 6
5.
If the maximum MW output of G1 is 150MW, what is the marginal cost of the area in
OPF? What if the maximum MW output of G1 is 100MW?
Set the maximum MW output of G1 back to 300MW before proceeding to the next
question.
6.
Increase the fuel cost of G1 to 2.0 $/MBtu. Resolve the OPF problem. Write down the
total MW losses, the total MVar losses, the hourly cost and marginal cost of the system.
Comment on the results.
4. DISCUSSION:
1. Explain the difference between economic dispatch and optimal power flow.
2. Comment on the relation between the output of the generators and the marginal cost of
the area.
_____________
____________
_
Elec4612 - Experiment 4 – Economic dispatch and optimal power flow
Page 7