This chapter will focus on the optimization and security of a power system. basically it will focus on economic dispatch analysis without considering transmission line losses.

1. ASTU
SCHOOL OF ELECTRICAL ENGINEERING AND
COMPUTING
DEPT. OF POWER AND CONTROL ENGINEERING
COMPUTER APPLICATION IN POWER SYSTEM
(PCE5307)
CHAPTER SIX
OPTIMIZATION AND SECURITY
BY: MESFIN M.

2. INTRODUCTION
 To provide a secure energy supply at a minimum operating
cost is a very complex process that relies heavily upon on-
line computer control.
Optimization and security are often conflicting
requirements and should be considered together.
In an interconnected power system, the objective is to find
the real and reactive power scheduling of each power plant
in such a way as to minimize the operating cost.
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3. CONT.…
The aim of optimal power system operation is to try and
make the best use of resources subject to a number of
requirements over any specified time period.
Here are some examples of power system optimization
studies, with time scales given in brackets.
Long-term scheduling for plant maintenance and availability
of resources (months/years)
Short-term scheduling for unit commitment (days).
Economic allocation of generation base points (minutes)
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4. CONT.…
The economic criterion which appears to have universal
acceptance is that of minimizing production costs of which
only those of fuel and maintenance vary significantly with
generation output.
The security objective determines local plant loading limits.
It also imposes limitations on network structures and
loading patterns on a system scale which often conflicts with
the economic objective.
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5. FORMULATION OF THE OPTIMIZATION
PROBLEM
 With reference to power system operation the optimization
problem consists of minimizing a scalar objective function
(normally a cost criterion) through the optimal control of a
vector [u] of control parameters, i.e.
𝑚𝑖𝑛𝑓( 𝑥 , [𝑢])
Subject to
Equality constraints of the power flow equation
𝑔 𝑥 , 𝑢 = 0
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6. CONT.…
Inequality constraints on the control parameters
(parameter constraints).
𝑢𝑖,𝑚𝑖𝑛 ≤ 𝑢𝑖 ≤ 𝑢𝑖,𝑚𝑎𝑥
Dependent variables and dependent functions (functional
constraints)
𝑥𝑖,𝑚𝑖𝑛 ≤ 𝑥𝑖 ≤ 𝑥𝑖,𝑚𝑎𝑥
ℎ𝑖 𝑥 , 𝑢 ≤ 0
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7. CONT.….
The optimal dispatch of real and reactive powers can be
assessed simultaneously using the following control
parameters:
 voltage magnitude at slack node
 voltage magnitudes at controllable P, V nodes
 taps at controllable transformers
 controllable power PGi
 phase shift at controllable phase-shifting transformers
 other control parameters.
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8. CONT.…
Let us assume that only part (PGi) of the total net power
(PNi) is controllable for the purpose of optimization.
The objective function can then be defined as the sum of
instantaneous operating costs over all controllable power
generation:
𝑓 𝑥 , 𝑢 =
𝑖
𝑐𝑖 (𝑃𝐺𝑖)
Where 𝑐𝑖 is the cost of producing 𝑃𝐺𝑖.
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9. UNCONSTRAINED PARAMETER
OPTIMIZATION
The mathematical tools that are used to solve
unconstrained parameter optimization problems come
directly from multivariable calculus.
The necessary condition to minimize the cost function
𝑓 𝑥1, 𝑥2, … , 𝑥 𝑛 is obtained by setting the derivative of 𝑓 with
respect to variables equal to zero:
𝜕𝑓
𝑥𝑖
= 0 𝑖 = 1, … , 𝑛
Or
∇𝑓 = 0 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑣𝑒𝑐𝑡𝑜𝑟
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10. EXAMPLE
Find the minimum of
𝑓 𝑥1, 𝑥2, 𝑥3 = 𝑥1
2
+ 2𝑥2
2
+ 3𝑥3
2
+ 𝑥1 𝑥2 + 𝑥2 𝑥3 − 8𝑥1 − 16𝑥2 − 32𝑥3 + 110
Soln.
Find the gradient
Check the values using the hessian matrix
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11. CONSTRAINED PARAMETER
OPTIMIZATION; EQUALITY CONSTRAINTS
This type of problem arises when there are functional
dependencies among the parameters to be chosen. The
problem is to minimize the cost function.
𝑓 𝑥1, 𝑥2, … , 𝑥 𝑛
Subjected to the equality constraints
𝑔𝑖 𝑥1, 𝑥2, … , 𝑥 𝑛 = 0
Such problems may be solved by the Lagrange multiplier
method.
This provides an augmented cost function by introducing k-
vector l of undetermined quantities.
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12. CONT.….
The constrained cost function becomes.
ℒ = 𝑓 +
𝑖=1
𝑘
𝜆𝑖 𝑔𝑖
The resulting necessary conditions for constrained local
minima of ℒ are the following.
𝜕ℒ
𝜕𝑥𝑖
=
𝜕𝑓
𝜕𝑥𝑖
+
𝑖=1
𝑘
𝜆𝑖
𝜕𝑔𝑖
𝜕𝑥𝑖
= 0
𝜕ℒ
𝜕𝜆𝑖
= 𝑔𝑖 = 0
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13. OPERATING COST OF THERMAL
PLANT The factor influencing power generation at minimum cost are
operating efficiencies of generator, fuel cost and transmission
losses.
The most efficient generator the system doesn’t guarantee
minimum cost as it may be located in an area where fuel cost is
too high
Also, if the plant is located far from load center, transmission
losses may be considerably higher hence the plant may be
overly uneconomical.
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14. CONT.….
 Hence, the problem is to determine the generation of different
plants such that the total operating cost is minimum.
The input to the thermal plant is generally measured in Btu/hr,
and the output is measured in MW.
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15. CONT.….
 In all practical cases, the fuel cost of generator i can be
represented as a quadratic function of real power generation.
𝐶𝑖 = 𝛼𝑖 + 𝛽𝑖 𝑃𝑖 + 𝛾𝑖 𝑃𝑖
2
An important characteristics is obtained by plotting the
derivative of the fuel cost curve versus the real power. This is
known as incremental fuel cost curve
𝑑𝐶𝑖
𝑑𝑃𝑖
= 2𝛾𝑖 𝑃𝑖 + 𝛽𝑖
The incremental fuel cost curve is a measure of how costly it
will be to produce the next increment of power.
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16. ECONOMIC DISPATCH NEGLECTING LINE
LOSS
 The simplest economic dispatch problem is the case when
transmission loss is neglected.
This problem model doesn’t consider system configuration and
line impedance.
The model assumes the system is only one bus with all
generators and loads connected to it.
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17. CONT.….
 since transmission loss is neglected, the total demand is the
sum of the generation.
A cost function Ci is assumed to be known for each plant.
The problem is to find the real power generation for each plant
such that the objective function (i.e., total production cost) as
defined by the equation is min. subjected to the constraint.
𝐶𝑡 =
𝑖=1
𝑛 𝑔
𝐶𝑖
=
𝑖=1
𝑛 𝑔
𝛼𝑖 + 𝛽𝑖 𝑃𝑖 + 𝛾𝑖 𝑃𝑖
2
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18. CONT.….
Is to minimize subjected to the constraint.
𝑖=1
𝑛 𝑔
𝑃𝑖 = 𝑃 𝐷
A typical approach is to augment the constraints into objective
function by using the Lagrange multipliers.
ℒ = 𝐶𝑡 + 𝜆 𝑃 𝐷 −
𝑖=1
𝑛 𝑔
𝑃𝑖
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19. CONT.….
The minimum of the constrained function is found at the point
where the partials of the functions to its variable are zero.
𝜕ℒ
𝜕𝑃𝑖
= 0
𝜕ℒ
𝜕𝜆
= 0
First condition,
𝜕 𝐶𝑡
𝜕𝑃𝑖
+ 𝜆 0 − 1 = 0
Since
𝐶𝑡 = 𝐶1 + 𝐶2 + ⋯ + 𝐶 𝑛
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20. CONT.….
Then
𝜕𝐶𝑡
𝜕𝑃𝑖
=
𝑑𝐶𝑖
𝑑𝑃𝑖
= 𝜆
Therefore, the condition for optimum dispatch is
𝑑𝐶𝑖
𝑑𝑃𝑖
= 𝜆
Or
2𝛾𝑖 𝑃𝑖 + 𝛽𝑖 = 𝜆
𝑃𝑖 =
𝜆 − 𝛽𝑖
2𝛾𝑖
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21. CONT.….
Second condition
𝑖=1
𝑛 𝑔
𝑃𝑖 = 𝑃 𝐷
Solving for 𝜆 we have
𝑖=1
𝑛 𝑔
𝜆 − 𝛽𝑖
2𝛾𝑖
= 𝑃 𝐷
𝜆 =
𝑃 𝐷 + 𝑖=1
𝑛 𝑔 𝛽𝑖
2𝛾𝑖
𝑖=1
𝑛 𝑔 1
2𝛾𝑖
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22. CONT.….
Using iterative method
Δ𝜆(𝑘)
=
Δ𝑃(𝑘)
1
2𝛾𝑖
Where Δ𝑃(𝑘)
= 𝑃 𝐷 − 𝑃𝑖
And therefore
𝜆(𝑘+1)
= 𝜆(𝑘)
+ Δ𝜆(𝑘)
The process continued until Δ𝑃(𝑘)
is less than specified
accuracy
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23. EXAMPLE
The fuel cost function for three thermal plants in $/hr are given
by
𝐶1 = 500 + 5.3𝑃1 + 0.004𝑃1
2
𝐶2 = 400 + 5.5𝑃2 + 0.006𝑃2
2
𝐶3 = 200 + 5.8𝑃3 + 0.009𝑃3
2
Where 𝑃1, 𝑃2, and 𝑃3 are in MW. The total load 𝑃 𝐷 is 800MW.
Neglecting line losses and generator limits, find the optimal
dispatch and the total cost in $/hr, by
A. analytical method.
B. By iterative Method
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24. CONT.….
The economic dispatch with generator limit, the same
procedure is followed except checking the limit for the
generators.
If it exceeds the upper limit fix the generator value at the
maximum value and solve for the remaining generators power.
Example: do the above example with Pd is at 975 MW and
200 ≤ 𝑃1 ≤ 450
150 ≤ 𝑃2 ≤ 350
100 ≤ 𝑃3 ≤225
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25. SECURITY ASSESSMENT
The aim of security assessment is:
the detection of operating limit violations through the
continuous monitoring of power flows, voltages, etc.
 contingency analysis, a far more demanding task, which
first considers all the possible outages in order of severity
and uses that information to alter the pre-contingency
operating state to try and reduce the effect of the
disturbance.
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26. END OF CHAPTER SIX
Thank You
NEXT CHAPTER FIVE
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