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Chapter two

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Mathematical simulation of transmission system costs and constrained
equations

2.1 Introduction

The first step in the planning
process of any system is the representation of the different costs associated
with this system in a correct or accurate mathematical form. Also, the
different constrained equations governing this system should be developed in
mathematical form, if possible. The form used to simulate these two items, cost
and constraints, should be compatible with the solution technique in hand.

In this chapter, the different
mathematical forms utilized to simulate both costs and constraints are
analyzed.

2.2
mathematical optimization planning problem

The general form of the mathematical
optimization planning problem in its static or dynamic form, can be stated as
follows;

Optimize a certain objective function;

Subject to a certain set of
constrained equations;

Where:

F      objective function of any form

X
set of variables of n dimension

gi         constrained equations on the x
variables with i=1,2,……….m

In the formulation of the above forms
for any optimization problem, it is noticed that:

a-
F
and g can take any form

b-
X
may be real variable or integer variable

c-
n
&m has no relation

d-
g
equation may have equality, inequality or both constrained equation.

In the transmission system planning
problem, the following points are noticed:

1-
the
optimization process is a minimization one

2-
the
cost function to be minimized is the total cost related to the transmission
network, the costs can be given as follows:

a-
capital
cost of new or added lines

b-
ohmic
loss cost associated with old and new lines

c-
demand loss cost associated with old and new
lines

d-
Operation
and maintenance cost of transmission network.

e-
Outage
(reliability) cost

3-
The
constrained equations are mainly:

a-
flow equations, they are equality constrained equation.

b-
Quality
constrained equations, they are limit constraints on:

1-
Line
flow on both existing and new lines

2-
Bus
voltage magnitude

3-
Line
angle for stability consideration

c-
Reliability
level constrained equations.

In developing the set of constrained
equations, we should notice the following important points;

1-
They
should be in a form that permits the existing of large number of solutions. The
goal of the solution optimization technique is to determine that solution which
has the minimum cost function.

2-
They
should be developed in the same variables utilized in the cost function. If
not, they should be related to the variables utilized in the cost function.

2.3
mathematical simulation of the system’s cost items

2.3.1 Capital
cost of new lines

This cost is the cost of purchasing
and installing of a new line, including the right of way cost, if any. This can
be given in an exact form if an integer variable is used as follows:

a-
of a new line with known size and circuits

Where:

Ci
= capital cost of line i

Zi
= zero-one integer variable related to line i

b-
If
number of circuits are variable

Where:

nc(i)    maximum number of circuits associated with
line i

Cij       capital
cost of step j associated with line I

Zij       zero – one integer variable associated with
step j of line I

If this form is used, a logic
equation should be added to the set of constraints to guarantee that one stop
only can be added if required as follows:

This process is shown in figure
(2.1).

Figure (2.1)

In addition to the above exact form
which utilizes the zero – one integer variable, the capital cost can be
simulated accurately in a nonlinear form as follows:

Where:

Ki = is a constant

Xi = is current or power
flow on line i

If Xi is zero, no current
or power will flow, then

And if xi has some value,
then,

This form is shown in figure (2.2).

Figure (2.2)

2.3.2 Capital
cost of reconductoring an existing line

Reconductoring process means the
addition of one or two sub conductors to an existing line to increase its capacity.
This process can be successfully utilized on some important lines if the tower
and insulators carrying this line can withstand the new force acting upon them.

This process can be mathematically
simulated as follows:

Where:

This process is shown in figure
(2.3). The last constrained equation guarantees that if no reconductoring
process is made; the original line will be treated as an existing line.

Capital cost

Ci3

Ci2

Figure (2.3)

2.3.3
Cupper (energy) loss cost

This
is a running cost paid annually due to the flow of electric currents on both
the existing and new lines.

This
cost is given mathematically on line i as follows:

Where:

Annual energy loss cost associated with line i

Resistance of line I in ohms

Magnitude of current flow on line i

Loss factor

Time in hours (8760 hr)

Cost of unit energy in LE/KWh

In
transmission planning process, it is found that this cost is very small with
respect to capital cost due to:

a-
Most
transmission lines have large cross sectional area ( small resistance )

b-
Capital
cost of transmission lines is normally high

c-
Maximum
current flow on transmission line is normally less than thermal limit due to
voltage and stability limitations.

The
above cost, on the other side, is largely used in distribution systems
planning.

The
above equation can be written as follows:

Where
is constant equal to  and when  = 8760, then

As

So

Multiplying
by V2 and dividing on V2

So,

Considering
is equal to unity or near to it, and considers
is a fixed percentage of,
then

Where

The real part of current flow

The imaginary part of current flow

Real power flow on line i

Imaginary power flow on line i

Annual loss cost for unit power transmitted on
line i

2.3.4 Demand
loss cost

This cost, for a transmission
system, is the cost of the added generation capacity required to feed the ohmic
power loss of a given transmission network.

It can be given as:

Where

Annual
cost of demand loss associated with line i

Cost of peaking
generation system in LE/KVA

Reserve factor
of generation system

Annual charge rate of generation system

This cost,
although paid one time only as capital cost of lines, is converted to annual
cost using annual charge rate of generation system. The reserve factor which is
the ratio of total generation capacity to total demand is normally in the range
of 1.1 to 1.2.

Considering
the same procedure used in the section 2.3.3, this equation can be given as
follows:

Where

Is annual
demand cost for unit power transmitted on line i

Actually,
both annual loss cost and demand loss cost are small values relative to the
line capital cost. So, they are neglected in most researches.

2.3.5 Operation and maintenance cost

This cost,
which is paid annually represents:

·
Cost of wages for operation and maintenance staff

·
Cost of maintenance and repair materials.

·
insurance cost

·
other costs

·
Good maintenance process is made to:

a-      Reduce line faults

b-      extend line life

c-      Improve system reliability

In planning
phase of transmission networks, this cost is normally considered as a constant
percentage of line capital cost as there is no acceptable accurate relative
between the line flow and this cost.

2.3.6 Lines outage cost

The forced
outage of a transmission line due to faults can lead to load curtailment. So,
this cost is the cost of loss revenue occurring throughout the life of a line.
If this cost, although a normal one, exceeds some value on limit, the
transmission system is considered to be of poor level of reliability.

To obtain a
high reliability level and reduce the cost of line outages, more lines are to
be added to the system in an optimum manner. This set of new lines is added up
to the point Where

·
Required reliability level is reached, or

·
The cost of adding a new line is greater than the cost
saved due to reducing outage cost.

It is well
known that the line outage cost is very difficult to obtain in an accurate
manner as it depends on:

·
Network configuration

·
Time of outage

·
Duration of outage

·

·
Other factors

SO,
reliability level constraint is dealt with in a separate phase where for transmission
networks most planners prefer to plan a network that can withstand one line
outage or double line outages.

2.3.7 Summation of lines cost

Due to the
fact that the cost associated with a transmission line has both capital cost,
which is paid one time, and a running cost which is paid annually, these costs
cannot be directly added. So, we have to convert one type to another as
follows:

a) Static planning

As the
annual cost in the static planning is fixed; it is preferred to convert the
fixed capital cost to a recurrent annual payment considering the interest rate
and the line life. Then, this cost is added to the annual running cost to get
the total annual cost, If fi is the capital cost added at present,
the recurrent annual payment Ai is get as:

Where

Line life in
years

Interest rate

b) Dynamic planning

In this
case, it is noticed that:

i.
Capital cost of lines is paid at different years

ii.
Annual running cost is not fixed at the whole planning
period

So, it is
preferred to convert all costs (capital and running) into their present worth
value at the start of the first planning years.

The summation
the capital cost and the running cost, after converting one cost to another,
gives the following total cost associated with line i :

For existing
line, only line running cost is existing.

With respect
to a new line i of multi steps, the total cost equation become:

2.3.8 Approximate forms of line cost

The correct
line cost equation given in 2.25 can only be handled if the mixed integer
non-linear is the solution tool. If other tools of the classical optimization
techniques are to be used, this exact form should be approximated as follows:

a-       Mixed integer linear programming technique

In this
case, the non-linear part of the cost function should be approximated with one
or more lines segments. So, the approximation cost of step j of new line i is:

Where

Number of
linear segments used for step j

Cost of unit
power transmitted on segment l of step j of new line i

Power flow on segment l of step j of new line i

Similarly,
the non-linear running cost of existing lines is approximated in a similar
manner. Note that each segment l has a certain power limit.

In this
case, one second order function in the power flow on new line i is used to
approximate the whole cost of the new line I as follows:

Where

and  are constant
cost coefficients for new line i

c-       linear programming technique

In this
case, the whole cost of a new line (i) is approximate by one linear segment as
follows:

The same
process will be made for existing lines

2.4 Mathematical simulation of transmission system constraints

The mathematical programming technique, optimization
tool, should search the optimum set of lines to be added while realizing the
set of constraints imposed on the transmission network.

These
constraints consist of:

a-       Equality constraints

There are the load flow equations governing the power
flow on the different lines and relate them to the bus injection.

b-      In equality constraints

There are normally limit constraints imposed on the
different system variables to force them to be within acceptable limits. They
include:

I.
constraints on line flow

II.
Constraints on bus voltage magnitude

III.
constraints on line angle ( stability constraint )

IV.
Right-of-way constraints, if any

V.
Logical constraints associated with integer variables,
if any

There are
the equality constraints relating the bus power injections with bus voltages on
line flow or line voltage angle.

equations

These are
non-linear equations relating the bus power injection (active and reactive) to
the bus voltages (magnitude and angles).

Considering
the bus frame of reference, these equations are given as follow:

For bus (i),

Where

Number of buses (where bus N is the
reference bus)

Active power injected at bus i

Reactive power injected at bus i

The magnitude of the element ijth of

Angle of the element ijth of the bus

Angle of
voltage of bus i

The
utilization of the above AC load flow equation in the mathematical planning
model gives many difficulties as follows:

1-      They are highly non-linear where
convergence problems exist for many real size transmission networks

2-      The reactive power injection of the
different generators are not known where it changes within a pre specified
positive and negative values. This complicates the planning problem and leads
normally to divergence.

3-      The elements of bus admittance matrix
should include both existing and proposed (new) lines. If one or more new lines
are deleted, the bus admittance matrix is in error.

4-      If the range of reactive power injections
of all generators are not enough to regulate the bus voltages, new shunt
compensation elements should be included in the planning process, the fact
which may increase the complexity of the TEPP, which is already complex.

In general,
if the AC load flow equations are to be utilized in the planning process, they
should be firstly simplified and modified such that the deletion or addition of
any new line will not affect the correctness of the load flow equations.

These
equations are developed for transmission systems assuming that line resistances
are neglected. This is correct in most transmission system where line reactance
is larger than line resistance.

These
equations are developed as follows:

1-      First Kirchhoff’s law ( power reserve ) at
bus i

2-      Voltage law for basic loop i

Where

Number of buses

Number of basic
loops

Power flow on
line j

Set of lines (existing
and new) connected to bus i

Set of lines (existing and new) found in basic loop i

Reactance of
line j

Real power
injected at bus i

This form of

a-      They are linear

b-      The power flow on the lines explicitly
exists. So, no linking relations are required as the cost function is normally
giving in the power flow on the lines

The basic
disadvantage found in this linear form is that the deletion of any new line
from a basic loop will represent an error in the loop equation as it is not an
actual loop.

The rest of
the lines existing in this loop does not form a loop while the constrained
equation represent it is still found as a closed loop.

If this form
is to be utilized, some modification in the loop equations containing new lines
should be made in order to make the equation correct whatever a new line is

2.4.2 Inequality constrained equations

These
constraints are mainly used to realize the quality degree imposed on the
transmission system. They are given as follows:

a-      Thermal limit constraint

They are
mainly used to prevent overloading of both existing and new lines as follows:

For line i,

The equation
used depends on the type of the line flow variable utilized in the developed
planning model. If line i have many steps, the equation is written for each
step.

Where

Current
flow on line i

Apparent power flow on line i

Real
power flow on line i

Maximum current magnitude permitted on line i

Maximum apparent power permitted on line i

Maximum real power permitted on line i

The maximum
flow limit on line i is a fixed value that depends on line type and area
irrespective of line length

b-      Bus voltage magnitude constraint

If the AC
load flow is the main constrained equation utilized, this constraint is used as
follows:

For bus i,

Voltage magnitude of bus i

Maximum permissible voltage magnitude.

Minimum permissible voltage magnitude.

In case DC
load flow is utilized instead of AC one, this constraint can be considered
partially through limiting the maximum real power flow on the line depending on
line length.

This limit
is necessary in order to get rid of the voltage problems associated with too
high or too low voltage magnitude.

c-      line angle constraint

This
constraint can be considered for line K of the two end buses I and j as
follows:

Where

Angle of voltage of bus i

Angle of voltage of bus j

Maximum line angle permitted

This
constraint can be used only when the bus voltage angle is considered as a
variable in the planning process.

It is used
to guarantee a good degree of transient stability of the transmission system.

Again, if
angle ? is not considered in the planning process, a maximum limit can be put
on real power flow from the stability point of view depending on line length.

d-     Right-of-way constraint

This
constraint is used to guarantee the feasibility of the planned network.
Depending on the right-of-way constraint, there is a maximum number of circuits
that can be installed on each new line.

This
constraint can be taken into consideration as follows:

1-      If a number of steps is considered on a new
line, then the last or largest step is pre-known depending on the right-of-way
of the route

2-      If the new line is simulated by the power
flow on the line, then the maximum power permitted to flow is considered
corresponding to the maximum number of circuits allowed.

2.5 Conclusion

This chapter
presents, in a detailed manner, the different mathematical forms in which both
cost function and constrained equations can be represented. The classical optimization
mathematical tool to be used to solve the TEPP depends mainly on the form of
both cost function and constrained equations. In other words, the form of these
equations (cost and constraints) determines the optimization tool to be used.