Power System Planning And
Operational Problems
Definition
Power system planning is the recurring process of
studying and determining what facilities and
procedures should be provided to satisfy and promote
appropriate future demands for electricity.
Power system planning
• Electric energy has to be delivered
instantaneously, no bulk storage is available
• Difference between demand and supply both in
shortage or excess has big economic impact
Planning considers:
• Options - the choices available to the planner,
• Uncertainties - parameters whose values are not
known precisely and
• Attributes - measures of ‘‘goodness.’’
Power system planning (cont’d)
The planning problem is therefore to
identify and choose among options, in
the presence of uncertainties, so as to
maximize the attributes
Planning and operational problems
Economic dispatch
Determine the power that must be generated by given n
generators so that the total cost of generation is minimum
subject to the constraints of power demand
FT F1 F2 F3 ... FN i 1 Fi ( Pi )
N
F
PLoad i 1 Pi 0
N
Objective function, FT
-is the total cost for
supplying the given load.
Here, the problem is to minimize FT subject to the
constraint that the sum of the powers generated
must equal the received load.
Input-output characteristics of
generating units
• The fundamental of the economic dispatch
problem is the set of input - output
characteristic of a power - generating unit.
• To determine the total cost of generating a
given power, the input output characteristics
has to be known
I/O characteristic - Thermal
Volume of
steam
Electric
power
Fuel
Input-Output Characteristics of the generating
units
B T G Input MBTU/h or $/h
Input
Fuel Electric Power
(Input) (Output)
MW
Pmin Pmax
a, b, c – coefficients of Output
the I/O Ch.
1/6 barrel of oil = 1 million BTU
1 BTU = 0.293071 watt hours
1 kWh = 3413 BTU 7
I-O characteristic – Hydroelectric unit
Electric
power
Volume of
water
I-O characteristic … cont’d
• Input output is almost linear
F=aP+b
Calculation of Input - Output Characteristic
Parameters
• Analyzing the fuel and the output power data set
(Fk, Pk), we can determine the shape of the input -
output characteristic and the corresponding
parameters.
• If the quadratic curve is the best match according
to the statistical data, we can use the least
squares method to compute the parameters.
Let ( F k , P k ) be obtained from the statistical
data, where k = 1, 2, … n , and the fuel curve will
be a quadratic function. To determine the
coefficients a , b , and c , compute the following
error for each data pair ( F k , P k ):
Calculation of Input - Output Characteristic
Parameters using least square method
According to the principle of least squares, we
form the following objective function
Taking the first derivative of the above function J
with respect to each of the independent variables
a, b, and c and setting the derivatives equal to
zero will give the necessary conditions for an
extreme value of the objective function
Calculation of Input - Output Characteristic
Parameters (Cont’d)
Solving the following equations coefficients a,
b, and c can be obtained.
Calculation of Input - Output Characteristic Parameters
Home assignment: Determine equation of input-
output parameters using least square method and
simulate the input output curve.
• Determine the coefficients a, b, c for the thermal
generating unit and simulate the input- output curve.
Sample data K=1 K=2 K=3 K=4 K=5
Unit consume fuel (Btu/MWh) 0.40 0.38 0.365 0.395 0.4
Power output (MW) 150 170 185 200 220
Consume fuel (Btu/h) 60 64.6 67.5 79 88
Other Solution Methods of Economic
Dispatch
1. Equal incremental cost method
– The incremental cost of all generators should be
equal
2. Lagrangian method
– Use Lagrangian multiplier to reformulate the
equation
3. Lambda Iteration Method
Equal incremental cost method
• Principle of Equal Incremental Rate
– Considering the inequality constraints
Principle of Equal Incremental Rate
• Two generators connected to a single bus serving an
electrical load PD .
• I/O characteristics of two generating units
- F1(PG1) and F2(PG2 )
- F - sum of the fuel consumptions of
the two generating units.
The economic power dispatch problem of the system is
to minimize F
minF = F1 (PG1) + F2 (PG2)
PG1 + PG2 = PD
Principle of Equal Incremental Rate
• According to the principle of equal incremental rate ,
the total fuel consumption F will be minimal if the
incremental fuel rates of two generators are equal,
that is
Is the incremental fuel rate of generating unit i
Equal incremental cost method
Considering the inequality constraints
• Problem statement
Minimize
Subject to
Equal incremental …
• Solution: Compute
dF1 PG1 dF2 PG 2 dF3 PG 3
dPG1 dPG 2 dPG 3
Algorithm of equal incremental cost
1. Neglect the inequality equation and distribute the
power among the units according to the equal
incremental principle.
2. Check the power output limit for each unit
– If any limit is violated, use
3. Handle the violated units as a negative load
P′Dk = −PGk
4. Re-compute the load dispatch with
OR
5. Go back to step (1) until all inequalities of units are
met.
Example: The I/O characteristics of two generating units is
given by the following equation. Total power demand is
500MW. The constraints are also given. Determine the
economic operation for these two units.
• Equations
• Constraints
• Solution: the equal incremental cost method gives us
• Checking the inequality constraints of generators, we can
see that the power output of unit 2 is violated
(As per algorithm 2),
The final solution therefore is:
Pg2 = 300 MW and Pg1 = 200MW
Home assignment 2 on constrained equal incremental cost
rate
For the following three generators, determine
the equal incremental cost operating point when
delivering a) 500MW b) 850MW
𝐹1 = 0.0006𝑃𝑔1 2 + 0.05𝑃𝑔1 + 6 𝐵𝑡𝑢/ℎ
𝐹2 = 0.0005𝑃𝑔2 2 + 0.06𝑃𝑔2 + 5 𝐵𝑡𝑢/ℎ
𝐹3 = 0.0007𝑃𝑔3 2 + 0.04𝑃𝑔3 + 3 𝐵𝑡𝑢/ℎ
Constraints are:
2. Lagrangian Method
In order to establish the
necessary conditions for an
extreme value of the
objective function, add the
constraint function to the
objective function after
the constraint function has
been multiplied by an
undetermined multiplier.
Incremental cost rate of all units = λ
(Necessary condition for the existence of
Minimum cost operation)
Necessary condition
• Incremental cost has to be equal to the
Lagrangian variable
1 Constraint
Participation Factor
• This assumes that the ED is solved by moving
from starting point for small change of power
Relation between change in incremental cost and change in
power
Participation factor ….
• The change in generation of each unit for a
change in demand
The new value of generation is calculated using
Example
The economic operating point for three thermal units when delivering a total of
850 MW is given as:
$
𝐹1 = 561 + 7.92𝑃1 + 0.001562𝑃1 2
ℎ
𝐹2 = 310 + 7.85𝑃2 + 0.00194𝑃2 2 $/ℎ
𝐹3 = 78 + 7.97𝑃3 + 0.00482𝑃3 2 $/ℎ
Inequality costraints are:
150 ≤ 𝑃1 ≤ 600 𝑀𝑊
100 ≤ 𝑃2 ≤ 400 𝑀𝑊
50 ≤ 𝑃3 ≤ 200 𝑀𝑊
The optimal economic solution is found to be:
P1 = 393.2 MW
P2 = 334.6 MW
P3 = 122.2 MW
Example (Cont’d)
Using the participation factor method calculate the dispatch for a total load
of 900 MW.
Solution
Since:
The new value of generation is calculated using
Pnew_1 = 393.2+0.47*50=416.7 MW
Pnew_2 = 334.6+0.38*50=353.6 MW
Pnew_3 = 122.2 +0.15*50=129.7 MW
Inequality constraint limit satisfied
Lambda iteration method for Economic
Dispatch
(Page 39-41 and Example 3D)[1]
It is a method used to compute the economic
power generation level of generators using
iteration
• Start with an initial guess of
• Compute the power generation of each unit
• Compute the error (difference between demand
and the total generation)
• Repeat until the error is less than tolerance
value
Lambda iteration contd…
HA
Home Assignment 3
• Assume the cost function of thermal
generators is given by
• Where the parameters A, B, C and D are
• Fuel cost is 1.0$/MBtu for all
Home assignment contd…
• If the power limits are given by
• The load demand is 2500MW
• Solve the problem using the algorithm given
above and with starting of =6.0 & =9
Based on the flow chart two sample calculations are shown. In this
calculation, the value for on the second iteration is always set at 10%
above or below the starting value depending on the sign of the error.
For the first example
For the second example
Calculation using B matrix formula
([1] pp 111-123)
Assignment
Problems 4.3 & 4.5
