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Chapter two

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-

Load

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-

Addition

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

·

Nature of loads

·

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.

b- Quadrate programming technique

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

2.4.1 Load flow constrained equations

There are

the equality constraints relating the bus power injections with bus voltages on

line flow or line voltage angle.

a- AC load flow

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

bus admittance matrix

Angle of the element ijth of the bus

admittance angle

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.

b- DC 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

load flow equations offers the following advantages:

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

added or deleted.

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.