thyristor controlled series capacitor fair copy 1234
TRANSCRIPT
CHAPTER-1
INTRODUCTION
1.1 PROBLEM DEFINITION AND FORMULATION
The electric power flow problem is the most studied and
documented problem in power system engineering. In essence, this problem
is the calculation of line loading given the generation and demand levels.
Load flow calculation provide power flows and voltages for a specified
power system subject to the regulating capability of generators, condensers,
and tap changing under load transformers as well as specified net
interchange between individual operating systems. This information is
essential for the continuous evaluation of the current performance of a power
system and for analyzing the effectiveness of alternative plans for system
expansion to meet increased load demand. These analysis require the
calculation of numerous load flows for both normal and emergency
operating conditions.
The power flow problem is formulated as a set of nonlinear
equations. Many calculation methods have been proposed to solve this
problem. Among them, N-R method and fast-decoupled load flow method
are two very successful methods. In general, the decoupled power flow
methods are only valid for weakly loaded network with large X/R ratio
network. For system conditions with large angles across lines (heavily
loaded network) and with special control that strongly influence active and
reactive power flows, Newton-Raphson method may be required.
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Therefore, when the AC power flow calculation is needed in
systems with FACTS devices, Newton-Raphson method is a suitable power
flow calculation method in the system with TCSC when high accuracy is
required. The basic requirement of power system is to meet the demand that
varies continuously. That is, the amount of power delivered by the power
companies must be equal to that of consumer’s need. Unfortunately, nobody
guarantees that unexpected things such as generator fault or line fault and
line tripping would not happen.
Due to its fast control characteristics and continuous
compensation capability, FACTS devices have been researched and adapted
in power system engineering area. There are so many advantages in FACTS
device; it can increase dynamic stability, loading capability of lines and
system security. It can also increase utilization of lowest cost generation.
The key role of FACTS device is to control the power flow actively and
effectively. In other words, it can transfer power flow from one line to
another within its capability. This project work focuses on the operation of
the TCSC under line overloaded that may cause any other transmission line
to be overflowed, the operation at emergency state should be different from
the operation at normal state.
Thyristor controlled series capacitor (TCSC) is one of
generation FACTS controller which control the effective line reactance by
connecting a variable capacitive reactance in series with line. The variable
capacitive reactance is obtained using FC-TCR combination with
mechanically switched capacitor sections in series. In load flow, studies the
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TCSC can represented in many forms. For example, the model presented is
based on the concept of a variable series compensator whose changing
capacitive reactance adjusts itself in order to constrain the power flow across
the branch to a specified value. The power transmitted over an ac
transmission line is a function of the line impedance, the magnitude of
sending end and receiving end voltages, and the phase angle between these
voltages.
Traditional techniques of reactive line compensation and step
like voltage adjustment are generally used to alter these parameters to active
power transmission control. Fixed and mechanically switched shunt and
series reactive compensation are employed to modify the natural impedance
characteristics of transmission line in order to establish the desired effective
impedance between the sending and receiving ends to meet power
transmission requirements. Voltage regulating and phase shifting
transformers with mechanical tap-changing gears are also used to minimize
voltage variation and to control the power flow.
These conventional methods provide adequate control under
steady state and slowly changing conditions, but are largely ineffective in
handling dynamic disturbances. The traditional approach to contain dynamic
problems is to establish generous stability margins in enabling the system to
recover from faults, line and generator outages, and equipment failures. This
approach, although reliable, results in a significant under utilization of the
transmission system. As a result of recent environmental restrictions, right of
way issues, and construction cost increases, and deregulation policies, there
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is an increasing recognition of the necessity to utilize existing transmission
system assets to the maximum extend possible.
The Thyristor Controlled Series Capacitor (TCSC) is a
transmission system which uses reliable high-speed Thyristor based high-
speed controllable elements. Issues include increased utilization of existing
facilities such as secure system operation at higher power transfers across
existing transmission lines which are limited by stability constraints, the
development of control designs for FACTS devices, and determination of
functional performance requirements for FACTS components.
Voltage control and stability problem are not new to the
electrical utility industry but are now receiving special attention in many
system. Voltage stability problems normally occur in heavily stressed
system while the disturbance leading to voltage collapse may be initiates by
a variety of causes; the underlying problem is an inherent weakness in the
power system. The principle factor contributing to voltage collapse are the
generator reactive power / voltage control limits, load characteristics,
characteristics of voltage control devices such as transformer under load tap
changer (ULTC) and FACTS (Flexible AC Transmission System) devices.
Voltage Stability improvement is one of the main objective for
power systems to provide quality and reliable supply. This makes
interesting and investigations in the area of voltage stability improvement,
FACTS device can quench the burning need of voltage stability.
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Though many FACTS device available like SVC, TCSC, UPFC
etc.., TCSC (Thyristor Controlled Series Capacitor) inspires power systems
engineers to make use of it for the improvement of voltage stability.
Obviously this project use TCSC for voltage stability improvement.
To know the voltage level of all the buses in the power systems,
Load Flow among all the buses should be known. This can be found by
using standard Load Flow programs. This project uses NR- method to know
load flow among all the buses in the power system. Voltage levels of the
buses are compared before and after connection of TCSC in the power
system which demonstrates voltage stability improvements.
1.2 VOLTAGE STABILITY
According to the IEEE definition, “the voltage stability refers to
the ability of a power system to maintain steady voltages at all busses in the
system after being subjected to a disturbance from a given initial operating
condition”.
Voltage stability like any other stability is a dynamic problem.
However, from the static analysis certain useful information such as
loadability limit and proximity of the operating point to this limit can be
obtained. The pattern of the voltage decay in the vicinity of the voltage
collapse point will not be known from the static analysis.
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1.3 VOLTAGE COLLAPSE
A power system at a given operating state and subjected to a
given disturbance undergoes voltage collapse if post-disturbance equilibrium
voltages are below acceptable limits. Voltage collapse may come total
blackout or partial in the power system.
The progressive decline the voltage leads to voltage collapse. A
system enters to a state of voltage instability when a disturbance, increase in
load demand, or change in system conditions causes a progressive and
uncontrolled decline in bus voltage. The main factor causing the instability
is the inability of the power system to meet the reactive power demand.
1.3.1 Causes of Voltage Collapse
1. Load on the transmission line is too high
2. Reactive power sources are too far from the load centers
3. Insufficient load reactive compensation
1.3.2 Voltage Stability analysis methods
Various methods that are available in the literature for the
analysis of the voltage stability are,
1. P-V Curve
2. Q-V Curve
3. Modal Analysis
4. Minimum Singular Value Decomposition
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1.3.3 P-V Curve
The P-V Curves (Thierry VAN CUTSEM, 1998) have been
used to analyze voltage stability of a power system. P–V curves are useful to
calculate the amount of corrective measure needed to achieve a desired MW
margin. For constant current and constant admittance loads the system is
voltage stable throughout the P-V curve. For constant power load, voltage
stability limit and maximum loadability limit are the same.
The point at which the loadability limit is attained is called as
the nose point or collapse point. The loadability limit at the bus depends on
the reactive power support that this bus can receive from the neighboring
buses. Corrective measures are required when the current operating point is
either very close to the loadability limit or following a contingency the
loadability limit itself becomes lower than the bus load. Reactive power
compensation can be used to increase the loadability limit.
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PmaxP0
Nose Point
P(MW)System Load
V
VSM
Fig1.1 Typical P–V Curve
Curve
0
From Figure 1.1 Voltage Stability Margin (VSM) is given by,
VSM = Pmax – P0(1.1)
where,
P0 : Active Load on the system (operating point)
Pmax : Maximum loading of the system
1.3.4 Voltage Stability Enhancement
Voltage Stability Enhancement is important in planning of
power system. If the state of the power system is found to possess a voltage
stability margin less than the desired level, then it is subjected to voltage
stability enhancement process. Voltage stability enhancement process
enforces voltage stability margin to the desired level.
1.4 REVIEW OF LITERATURE
Literature survey was carried out under three category to cater
the objective of the project. As voltage stability problem is one of the
important problem in the power system and FACTS device attracts power
system engineers to solve these kind of problem the project aims to improve
voltage stability margin by incorporating TCSC into the power system.
Three category of the literature survey are,
1) Voltage Stability
2) FACTS device
3) Power Flow techniques for FACTS device connected power
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system.
In category two, some of the important literatures are: Rusejla
Ponjavic Sadikoic, Mevludin Glavic research work paper in the title “Effect
of FACTS device on steady State Voltage Stability” which demonstrates the
effect of SVC and TCSC on IEEE 39 bus system. A.J.F. Keri, X.Lombard,
and A.A. Edris provide tool for power utility engineers to evaluate the
application of TCSC, its impacts on Power System.
In category three, some of the important literatures are: A.
Nabavi-Niaki and M. R. Iravani (1996) use sequential method for solving
these kind of problem i.e. line flow in the line to which TCSC is to be
connect is found by NR-method and then four equation each corresponding
one TCSC control variable is solved iteratively. This method has some
drawback to overcome this drawback Fuerte-Esquivel C. R., and Acha E
use simultaneous method for solving load flow problem with TCSC. In
simultaneous method four equations each corresponding to one TCSC
control variable is included in jacobian matrix itself, so one can get system
state variables and control variables simultaneously. In this project the
concept of simultaneous method is used and, follow the newly derived
equation which makes different and ease to use for the TCSC connected
power system
The project mainly deals with the enhancement of static voltage
stability of power system. This project proposes a decoupled model of
TCSC, for maintaining desired voltage profile and voltage stability with the
prescribed stability margin.
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1.5 OUTLINE OF THE THESIS
It includes derivation power flow injections of the TCSC for
two voltage sources, and incorporating the newly derived equation into the
power flow problem. Here Newton Raphson (NR) method is used to solve
the power flow and also TCSC inclusion, the Jacobian matrix and the
number equation to be solved are altered according to the need. MATLAB
program were developed to demonstrate the proposed work. It is generalized
program and may use for the multiple TCSC connection in the power
systems. The results for various conditions are obtained and performance of
the power system is compared before and after TCSC connection. Six bus
systems is considered in this project.
A brief outline of the various chapters of the thesis is as
follows. In Chapter 1, a brief introduction of the project and basic definition
of the frequently used technical terms are given. Literature survey for the
project work, proposal for the project work, introduction of the Flexible AC
Transmission (FACTS) device and their needs, different types of FACTS
device and their benefits and finally advantage of TCSC which makes
suitable for the proposed work are explained.
In Chapter 2, Operating principle and working of the system
without TCSC is given in the power flow model is deeply explained and the
needed equations are derived.
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In Chapter 3, Operating principle and working of the TCSC is
given and then how the TCSC in included into the power flow model is
deeply explained and the needed equations are derived.
In Chapter 4, Ward and Hale Six bus system is considered with
TCSC connection and also final conclusion of the project work are carried
out with future work of TCSC are presented.
1.6 INTRODUCTION TO FACTS DEVICES
Flexible alternating current transmission systems (FACTS)
devices are used for the dynamic control of voltage, impedance and phase
angle of high voltage AC lines. FACTS devices provide strategic benefits
for improved transmission system management through: better utilization of
existing transmission assets; increased transmission system reliability and
availability; increased dynamic and transient grid stability; increased quality
of supply for sensitive industries (e.g. computer chip manufacture); and
enabling environmental benefits. Typically the construction period for a
facts device is 12 to 18 months from contract signing through
commissioning.
1.7 NECESSITY OF FACTS DEVICES
The need for more efficient electricity systems management has
given rise to innovative technologies in power generation and transmission.
Worldwide transmission systems are undergoing continuous changes and
restructuring. They are becoming more heavily loaded and are being
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operated in ways not originally envisioned. Transmission systems must be
flexible to react to more diverse generation and load patterns. In developing
countries, the optimized use of transmission systems investments is also
important to support industry, create employment and utilize efficiently
scarce economic resources.
Flexible AC Transmission Systems (FACTS) is a technology
that responds to these needs. It significantly alters the way transmission
systems are developed and controlled together with improvements in asset
utilization, system flexibility and system performance.
1.8 TYPES OF FACTS DEVICES
1) Static VAR compensator (SVC)
2) Thyristor controlled series compensator (TCSC)
3) Static Synchronous compensator (STATCOM)
4) Unified power flow controller (UPFC)
1.8.1 Static VAR compensator (SVC)
Static VAR Compensators (SVC’s), the most important FACTS
devices, have been used for a number of years to improve transmission line
performance and improves the voltage profile of the bus by injecting VAR
power in to the bus where it needs by resolving dynamic voltage problems.
The accuracy, availability and fast response enable SVC’s to provide high
performance steady state and transient voltage control compared with
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classical shunt compensation. SVC’s are also used to damp out the power
swings, improve transient stability, and reduce system losses by optimized
reactive power control.
1.8.2 Thyristor controlled series compensator (TCSC)
Thyristor controlled series compensators (TCSCs) are an
extension of conventional series capacitors through adding a Thyristor
controlled reactor. Placing a controlled reactor in parallel with a series
capacitor enables a continuous and rapidly variable series compensation
system. The main benefits of TCSCs are increased energy transfer, damping
out of power oscillations, damping out of sub synchronous resonances, and
control of line power flow.
1.8.3 Static Synchronous compensator (STATCOM)
STATCOM are GTO (gate turn-off type thyristor) based
SVC’s. Compared with conventional SVC’s they don’t require large
inductive and capacitive components to provide inductive or capacitive
reactive power to high voltage transmission systems. It is used only to
compensate real power calculation. This results in smaller land
requirements. An additional advantage is the higher reactive output at low
system voltages where a STATCOM can be considered as a current source
independent from the system voltage. STATCOM have been in operation for
approximately 5 years.
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1.8.4 Unified power flow controller (UPFC)
Connecting a STATCOM, which is a shunt connected
device, with TCSC which is a series branch in the transmission line via
its DC circuit results in a UPFC. This device is comparable to a phase
shifting transformer but can apply a series voltage of the required phase
angle instead of a voltage with a fixed phase angle. The UPFC combines
the benefits of a STATCOM and a TCSC. This can be overloaded
shortly. Also it is used for compensating both real and reactive power.
1.9 BENEFITS OF FACTS DEVICES
(A) Better utilization of existing transmission system assets
In many countries, increasing the energy transfer capacity and
controlling the load flow of transmission lines are of vital importance,
especially in de-regulated markets, where the locations of generation and the
bulk load centers can change rapidly. Frequently, adding new transmission
lines to meet increasing electricity demand is limited by economical and
environmental constraints. FACTS devices help to meet these requirements
with the existing transmission systems.
(B) Increased transmission system reliability and availability
Transmission system reliability and availability is affected by
many different factors. Although FACTS devices cannot prevent faults, they
can mitigate the effects of faults and make electricity supply more secure by
reducing the number of line trips. For example, a major load rejection results
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in an over voltage of the line which can lead to a line trip. SVC’s or
STATCOMs counteract the over voltage and avoid line tripping.
(C) Increased dynamic and transient grid stability
Long transmission lines, interconnected grids, impacts of
changing loads and line faults can create instabilities in transmission
systems. These can lead to reduced line power flow, loop flows or even to
line trips. FACTS devices stabilize transmission systems with resulting
higher energy transfer capability and reduced risk of line trips, which
increase quality of supply for sensitive industries.
Modern industries depend upon high quality electricity supply
including constant voltage, and frequency and no supply interruptions.
Voltage dips, frequency variations or the loss of supply can lead to
interruptions in manufacturing processes with high resulting economic
losses.
(D) Environmental benefits
FACTS devices are environmentally friendly. They contain no
hazardous materials and produce no waste or pollutions. FACTS help
distribute the electrical energy more economically through better utilization
of existing installations thereby reducing the need for additional
transmission lines.
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1.10 BENEFITS OF TCSC
The TCSC's high speed switching capability provides a
mechanism for controlling line power flow, which permits increased loading
of existing transmission lines, and allows for rapid readjustment of line
power flow in response to various contingencies. The TCSC also can
regulate steady-state power flow within its rating limits. Transmission
loading may be limited by system stability or transient stability of
generation. The TCSC is a powerful new tool to help relieve these
constraints.
Its controls can be designed to modulate the line reactance and
provide damping to system swing modes, with dramatic results. This
condition was simulated using the actual Slatt control system and GE's
power system simulator with damping deliberately reduced. (See Figure 3.)
In the top trace, large, undamped system swings are produced as the result of
a fault.
At six seconds the TCSC damping function is manually
engaged, at which point there is a dramatic increase in the damping. The
control dead band stops further action in this test case, but normally the
system's inherent damping would make the power swings die out entirely.
the level of damping introduced by the TCSC, per dollar invested, is
unequaled by any other device and the fast action permits increased line
loading for the same stability constraint. The output of generating plants
may also be limited by transient instability under certain contingency
conditions. The fast-acting TCSC can provide the means of rapidly
increasing power transfer upon detection of the critical contingencies,
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resulting in increased transient stability.Finally the TCSC provides a
mechanism for greatly reducing a potential sub-synchronous resonance
problem at thermal generators electrically close to transmission lines with
series compensation. In some cases, the inability to mitigate SSR with
conventional series capacitors has limited line compensation to levels
between 20 and 40 percent. With even a small percentage of TCSC, the total
compensation can be increased significantly.
1.11 ADVANTAGES OF FACTS DEVICES
FACTS devices provides concurrent real and reactive control of
series line compensation with out an external electric energy source. It also
provides the following benefits:
1. Voltage Stability
2. VAR Compensation
3. Static & Dynamic stability
FACTS devices has needed control facilities for voltage
stability improvement it obvious choice for selecting for the project that
demonstrates the voltage stability improvement. Also it helps to improve
voltage stability, sub synchronous resonance, and damping of power swings
etc. It also provides us to make better power flow control and also helps to
minimize the real and reactive power losses.
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CHAPTER 2
POWER FLOW MODEL WITHOUT TCSC
2.1 INTRODUCTION
Generally, the power flow model without TCSC is nothing but
the normal Newton Raphson method. It is used to calculate power flow of
the system. And also it is used to calculate the load flow, line losses, and
total cost of function. Here load flow, line losses are calculated using
jacobian matrix compared with the specified accuracy.
Many calculation methods have been proposed to solve this
problem Among them Newton-Raphson method and fast decoupled load
flow method are two very successful methods. In general, the decoupled
power flow methods are only valued for weakly loaded network with large
X/R ratio network. By using Newton-Raphson method, we can over come
the general weakly loaded network. So Newton-Raphson method may be
required. Therefore when the AC power flow calculation is needed in system
with FACTS devices. So Newton-Raphson method is a suitable power flow
calculation method because high accuracy is required.
2.2 PROBLEM STATEMENT
Complex current at bus-i,
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(2.1)
Where,
n = number of the buses
Y = Bus admittance matrix
I = Complex current in the bus
V = Complex voltage in the bus
Pi + jQi = Complex power in the bus
i,j= bus number
In polar form,
(2.2)
Complex power at bus-i,
Pi – jQi = Vi* Ii (2.3)
Therefore, (2.4)
Separating real and imaginary parts,
(2.5)
(2.6)
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Equation 2.3 and 2.4 constitute a set of nonlinear algebraic
equations in terms of the independent variables, voltage magnitude in pu,
and phase angle in radians.
We have two equations for each load bus, given by 2.5 and 2.6,
and one equation for each generator bus, given by 2.5.
Expanding 2.5 and 2.6 in Taylor’s series about the initial
estimate and neglecting all higher order terms results in the following set of
linear equations. In short form, it can be written as
(2.7)
∆P- Change in real power
∆Q- Change in reactive power
J1, J2, J3, J4- Jacobian elements
∆δ- Change in load angle
∆|V|- change in voltage
The terms ΔPi(k) and ΔQi
(k) are the difference between the
scheduled and the calculated values, known as the power residuals, given by
ΔPi(k) = Pi
sch – Pi(k) (2.8)
ΔQi(k) = Qi
sch – Qi(k) (2.9)
The new estimates for bus voltages are,
δi(k+1) = δi
(k) + Δ δi(k) (2.10)
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|Vi(k+1)|= |Vi
(k)| + |Δ Vi(k)| (2.11)
The diagonal and the off-diagonal elements of J1 are
(2.12)
(2.13)
The diagonal and the off-diagonal elements of J2 are
(2.14)
(2.15)
The diagonal and the off-diagonal elements of J3 are
(2.16)
(2.17)
The diagonal and the off-diagonal elements of J4 are
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(2.18)
(2.19)
2.3 PROBLEM FORMULATION FOR SYSTEM WITHOUT TCSC
Excitation Voltage Source real power (PE)
PE = Real { VE IE* } (2.20)
PE = ( - VE VET Sin (E - ET) / XE) (2.21)
Boosting Voltage Source real power (PB)
PB = Real {VB IB* } (2.22)
PB = ( - VB VET Sin (B - ET) / XB) + (VB VBT Sin (B - BT) / XB) (2.23)
TCSC Power Flow constraints are:
PB = PE (2.24)
The superscript T indicates transposition. ∆X is the solution
vector and Jac is the Jacobian matrix. The TCSC linearized power equations
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are combined with the lineralized system of equations corresponding to the
rest of the network,
F(X) = Jac ∆X (2.25)
Where,
F(X) = [∆P ∆Q ∆PBT ∆PET ∆QBT ∆QET]T (2.26)
∆X = [∆ ∆V ∆B ∆E ∆VB ∆VE]T (2.27)
(2.28)
H = ∂P / ∂ (2.29)
N = ∂P / ∂V (2.30)
J = ∂Q / ∂ (2.31)
L = ∂Q / ∂V (2.32)
2.4 ALGORITHM FOR LOAD FLOW WITHOUT TCSC
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1) For load buses, where Pisch and Qi
sch are specified, voltage
magnitude and phase angle are set equal to the slack bus values or 1.0 and 0.
For generator buses voltage phase angle is set as 0. In addition to this one
more fictitious bus B’ (load bus) is included. The exciter bus E is converted
into generator bus.
2) For load buses Pi(k) and Qi
(k) are calculated from equations 2.5 and
2.6 and ΔPi(k) and ΔQi
(k) are calculated from equations 2.7 and 2.8.
3) For generator buses, Pi(k) and ΔPi
(k) is calculated from equations 2.5
and 2.6 respectively
4)The elements of the Jacobian matrix H, N, J and L are calculated
form equations 2.29 to 2.30.
5) The linear simultaneous equation 2.28 is solved.
6) The new voltage magnitude and phase angle are computed from
equation 2.27.
7) The process is continued until the residual of f(X) from equation
2.26 are less than the specified accuracy (F(X) =Jac (∆X)).
CHAPTER 3
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POWER FLOW MODEL WITH TCSC
3.1 INTRODUCTION TO TCSC
Thyristor Controlled Series Capacitors (TCSCs) are finding
increasing application in power systems, mainly as series compensators in
transmission lines, designed to control power flow, to increase transient
stability, or to dampen power oscillations and sub synchronous resonances.
TCSCs represent an example of a flexible AC transmission system (FACTS)
component.
To understand and control the interactions between FACTS
devices such as TCSCs and the utility system, there is an increasing need for
convenient models that will allow fast and accurate transient simulations and
facilitate control design. However, it is difficult to analyze the dynamics of
TCSC systems, because they incorporate both continuous time dynamics
(associated with the voltages and currents on capacitors and inductors) and
discrete events (associated with the switching of the thyristors). The standard
quasi-static approximation models the TCSC as a variable fundamental-
frequency reactance; the line and TCSC dynamics are omitted. This
approach is widely used because of its simplicity, but relies on the
assumption that the transmission system is operating in sinusoidal steady-
state, with the only dynamics being that of the generators. This quasi-static
approximation lacks accuracy when voltage stability, transient stability or
other dynamic phenomena are of interest; several authors have already
shown that TCSC dynamics cannot always be neglected.
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In an attempt to characterize the dynamic behavior of the TCSC
system , sampled-data models for TCSCs have been developed. They predict
system behavior very well, and also offer an analytical basis for control
design. However, this approach presents some disadvantages: the model
derivation is relatively complicated; the model structure has no clear relation
to the system configuration; and the model does not interface well with the
standard phasor-based models of generator dynamics. This project work
derives TCSC model based on time-varying Fourier coefficients that capture
the phasor dynamics of the TCSC.
Restricting our focus to the representation of the fundamental
frequency components, we obtain a natural dynamic extension of the usual
quasi-static phasor analysis. Various simplifications and refinements of the
fundamental model are possible; some of these variations are developed in
this project work. The same approach has been used in order to analyze the
line dynamics together with the generator dynamics, and earlier in to
develop frequency selective averaged models for power converters. This
approach can be viewed as intermediate between detailed time domain
circuit representation (as used in EMTP) and the quasi-static sinusoidal
steady-state approximation.
Our models of TCSC phasor dynamics represent the dynamics
of the fundamental currents and voltages and their harmonics in a simple but
powerful way. The models are well suited to efficient simulation of
transients. As the models are modular, the inclusion of line dynamics and
generator dynamics is straightforward. In this chapter we introduce the time-
varying Fourier coefficient representation that will be used in this project
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work. Also we review the basic operation of a TCSC and derives our
fundamental phasor dynamics model for the TCSC, and from that only we
obtained a reduced-order version of it. Finally it presents numerical
simulations that validate the model and we show the results of embedding
the TCSC model in a larger model that incorporates line dynamics.
3.2 TCSC OPERATION
Figure 3.1 shows the circuit diagram of a single-phase TCSC.
Fig 3.1 TCSC module
The system is composed of a fixed capacitor in parallel with a
Thyristor Controlled Reactor (TCR). The switching elements of the TCR
consist of two anti-paralleled thyristors, which alternate their switching at
the supply frequency. The system is controlled by varying the phase delay of
the thyristor firing pulses relative to the zero crossings of some reference
waveform. The effect of such variation can be interpreted as a variation in
the value of the capacitive/inductive reactance at the fundamental frequency.
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In our analysis, the thyristor2 will be ideal, so that
nonlinearities due to the thyristor turn on and turn-off are neglected. We also
assume that the line current is essentially sinusoidal, and take it to be the
reference waveform for the synchronization of the firing pulses a slight
modification of our analysis is needed if the synchronization instead done
with the capacitor voltage Vc. The waveforms of the TCSC are shown in
Fig. 3.1, with I denoting the current through the TCR branch.
The thyristor is turned on after a delay relative to the zero-
crossing the line current, and keeps conducting until here the inductor
current becomes zero. In steady-state, the conductor angle becomes
symmetrical with respect to the peak value of the line current, and the angle
between the negative peak of the inductor current (which occurs at the zero-
crossing of the capacitor voltage) and the positive peak of the line current
goes to zero. In this project work we will assume firing angle control for the
TCSC minor modifications are needed for conduction angle control.
3.3 POWER FLOW MODEL OF TCSC
Complex current at bus-i,
(3.1)
Where,
n = number of the buses
Y = Bus admittance matrix
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I = Complex current in the bus
V = Complex voltage in the bus
Pi + jQi = Complex power in the bus
i,j – bus number
In polar form,
(3.2)
Complex power at bus-i,
Pi – jQi = Vi* Ii (3.3)
Therefore,
(3.4)
Separating real and imaginary parts,
(3.5)
(3.6)
Equation 3.5 and 3.6 constitute a set of nonlinear algebraic
equations in terms of the independent variables, voltage magnitude in p.u,
29
and phase angle in radians. We have two equations for each load bus, given
by 3.5 and 3.6, and one equation for each generator bus, given by 3.5.
Expanding 3.5 and 3.6 in Taylor’s series about the initial
estimate and neglecting all higher order terms results in the following set of
linear equations. In short form, it can be written as
(3.7)
The terms ΔPi(k) and ΔQi
(k) are the difference between the
scheduled and the calculated values, known as the power residuals, given by
ΔPi(k) = Pi
sch – Pi(k) (3.8)
ΔQi(k) = Qi
sch – Qi(k) (3.9)
The new estimates for bus voltages are,
δi(k+1) = δi
(k) + Δ δi(k) (3.10)
|Vi(k+1)|= |Vi
(k)| + |Δ Vi(k)| (3.11)
The diagonal and the off-diagonal elements of J1 are
(3.12)
(3.13)
30
The diagonal and the off-diagonal elements of J2 are
(3.14)
(3.15)
The diagonal and the off-diagonal elements of J3 are
(3.16)
(3.17)
The diagonal and the off-diagonal elements of J4 are
(3.18)
(3.19)
3.4 PROBLEM FORMULATION FOR SYSTEM WITH TCSC
Let the power flow equation of TCSC are solved as follows
Excitation Terminal real and reactive power flow
PET = Real { VET IE* + VET IB* } (3.20)
31
PET = ( - VET VB Sin (ET - B) / XB) + (VET VBT Sin (ET - BT) / XB) +
(VET VE Sin (ET - E) / XE) (3.21)
QET = Img { VET IE* + VET IB* } (3.22)
QET = (VET 2 / XB) + ( VET
2 / XE) + (VET VB Cos (ET - B) / XB) -
(VET VBT Cos (ET - BT) / XB) - (VET VE Cos (ET - E) / XE) (3.23)
Boosting Terminal real and reactive power flow
PBT = Real { VBT IB* } (3.24)
PBT = ( - VBT VET Sin (BT - ET) / XB) - (VBT VB Sin (BT - B) / XB)
(3.25)
QBT = Img { VBT IB* } (3.26)
QBT = ( - VBT 2 / XB) + (VBT VET Cos (BT - ET) / XB) +
(VBT VB Cos (BT - B) / XB) (3.27)
Excitation Voltage Source real power
PE = Real { VE IE* } (3.28)
32
PE = ( - VE VET Sin (E - ET) / XE) (3.29)
Boosting Voltage Source real power
PB = Real {VB IB* } (3.30)
PB = ( - VB VET Sin (B - ET) / XB) + (VB VBT Sin (B - BT) / XB) (3.31)
TCSC Power Flow constraints are:
Boosting voltage source real power=Excitation voltage source real power
PB = PE (3.32)
To find TCSC control variables (VE VB E B) Taylors series expansion
method is used as follows
f(PET) = ( - VET VB Sin (ET - B) / XB) + (VET VBT Sin (ET - BT) / XB) +
(VET VE Sin (ET - E) / XE) - PET (3.33)
f(QET) = (VET 2 / XB) + ( VET
2 / XE) + (VET VB Cos (ET - B) / XB) -
(VET VBT Cos (ET - BT) / XB) - (VET VE Cos (ET - E) / XE) - QET
(3.34)
f(PBT) = ( - VBT VET Sin (BT - ET) / XB) - (VBT VB Sin (BT - B) / XB) – PBT
(3.35)
f(QBT)=(VBT VET Cos (BT - ET) / XB) + (VBT VB Cos (BT - B) / XB) –
VBT 2/XB -QBT (3.36)
33
The superscript T indicates transposition. ∆X is the solution
vector and Jac is the Jacobian matrix. The TCSC linearized power equations
are combined with the linearized system of equations corresponding to the
rest of the network,
F(X) = Jac ∆X (3.37)
Where,
F(X) = [∆P ∆Q ∆PBT ∆PET ∆QBT ∆QET]T (3.38)
∆X = [∆ ∆V ∆B ∆E ∆VB ∆VE]T (3.39)
(3.40)
H = ∂P / ∂ (3.41)
N = ∂P / ∂V (3.42)
J = ∂Q / ∂ (3.43)
34
L = ∂Q / ∂V (3.44)
3.5. ALGORITHM FOR LOAD FLOW WITH TCSC
1) For load buses, where Pisch and Qi
sch are specified, voltage
magnitude and phase angle are set equal to the slack bus values or
1.0 and 0. For generator buses voltage phase angle is set as 0. In
addition to this one more fictitious bus B’ (load bus) is included.
The exciter bus E is converted into generator bus.
2) For load buses Pi(k) and Qi
(k) are calculated from equations 3.5 and
3.6 and ΔPi(k) and ΔQi
(k) are calculated from equations 3.7 and 3.8.
3) For generator buses, Pi(k) and ΔPi
(k) is calculated from equations 3.5
and 3.8 respectively
4) The elements of the Jacobian matrix H, N, J and L are calculated
form equations 3.41 to 3.44. In addition to these four rows
corresponds to equations 3.32 – 3.36 and four columns
corresponds to four TCSC control variables (B E VB VE) are
augmented as shown in equation 3.40
5) The linear simultaneous equation 3.39 is solved.
6) The new voltage magnitude and phase angle are computed from
equation 3.39.
7) The process is continued until the residual of f(X) from equation
3.38 are less than the specified accuracy (F(X) = Jac ∆X).
35
CHAPTER 4
APPLICATION OF TCSC FOR WARD AND HALE
6 BUS SYSTEM
36
4.1. WARD AND HALE 6 BUS SYSTEM
The line data and bus data of the 6-bus system are given in
Tables A-2 and A-3 respectively given in appendix. The Table A-2
corresponds to the resistance, tap ratios and half line charging. The Limits of
bus voltages, load side real and reactive power and generator side real and
reactive power are given in table A-3. The single line diagram of Ward and
Hale System is given in appendix A-1.
4.2. RESULTS AND OBSERVATION
4.2.1. Results of the system without TCSC:
-------------------------------------------------------------------------- Base Case Details . . . -------------------------------------------------------------------------- Power Flow Solution by Newton-Raphson Method Maximum Power Mismatch = 0.000898141 No. of Iterations = 3-------------------------------------------------------------------------- Bus Voltage Angle ------Load------ ---Generation--- Injected No. Mag. Degree MW Mvar MW Mvar Mvar -------------------------------------------------------------------------- 1 1.005 0.000 150.000 0.000 517.446 -36.418 0.000 2 1.010 0.622 0.000 0.000 243.000 -35.527 0.000 3 1.020 -8.297 0.000 0.000 74.000 207.772 0.000 4 1.000 -8.397 200.000 0.000 42.000 -47.650 0.000 5 0.993 -7.752 500.000 0.000 0.000 0.000 0.000-------------------------------------------------------------------------- Total 850.000 0.000 876.446 88.178 0.000
Table 4.1- Output of load flow without TCSC
Line Flow & Losses
Line Power at bus Line flow Line lossfrom to MW MVAR MVA MW MVAR1 367.446 -36.418 369.246
37
25
-56.828424.386
35.591-72.055
67.053430.460
0.66818.529
0.44555.037
2134
243.00057.49679.506105.998
-35.527-35.146-4.7844.458
245.58367.38779.650106.092
0.6680.9331.655
0.44512.43816.550
3245
74.000-78.57354.62597.949
207.77217.22261.039129.629
220.55780.43981.912162.473
0.9330.9673.806
12.4381.2902.537
423
-158.000-104.343-53.657
-47.65012.092-59.749
165.029105.04180.306
1.6550.967
16.5501.290
531
-500.000-94.143-405.857
0.000-127.092127.092
500.000158.162425.291
3.80618.529
2.53755.037
Total Loss 26.558 88.298
--------------------------------------------------------------------------
The Lambda Values @ Various Nodes In The Generator Buses Are . . .
--------------------------------------------------------------------------
Node 1 --> 61.74
Node 2 --> 60.60
Node 3 --> 64.40
Node 4 --> 65.40
--------------------------------------------------------------------------
The Total Production Cost Of All Generators = 32248.08 Fr
--------------------------------------------------------------------------
4.2.2. Results of the system with TCSC:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
After TCSC connection
38
TCSC connected between Bus 2 and Bus 4 Exciter Bus=2 and Booster Bus=6
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Power Flow Solution by Newton-Raphson Method
Maximum Power Mismatch = 0.000259387
No. of Iterations = 7
Bus Voltage Angle ------Load------ ---Generation--- Injected ----TCSC--- No. Mag. Degree MW Mvar MW Mvar Mvar MW MVAR 1 1.005 0.000 150.000 0.000 504.098 -6.303 0.000 0.000 0.000 2 1.010 0.415 0.000 0.000 243.000 20.347 0.000 -96.461 -11.454 3 1.020 -7.109 0.000 0.000 67.000 148.887 0.000 0.000 0.000 4 1.000 -7.616 200.000 0.000 38.000 -79.871 0.000 0.000 0.000 5 1.012 -7.340 500.000 0.000 0.000 0.000 0.000 0.000 0.000 6 1.010 0.590 0.000 0.000 0.000 0.000 0.000 96.461 10.413 Total 850.000 0.000 852.098 83.060 0.000
Table 4.2- Output of load flow with TCSC
Line Flow & Losses line Power at bus Line flow Line lossFrom to MW MVAR MVA MW MVAR1 354.098 -6.303 354.154
39
25
-78.835432.933
-10.5664.263
79.540432.954
0.3130.186
0.62655.677
213
146.53979.14867.391
8.89311.192-2.300
146.80979.93667.430
0.3130.223
0.6268.914
3245
67.000-67.16866.56567.603
148.88711.21485.559359.114
163.26868.098108.40385.358
0.2230.5650.350
8.9142.2590.700
463
-162.000-96.000-66.000
-79.8713.429-83.300
180.62096.061106.278
0.4610.565
13.8422.259
531
-500.000-67.253-432.747
0.000-51.41451.414
500.00084.654435.791
0.3500.186
0.70055.677
64
96.46196.461
10.41310.413
97.02197.021 0.461 13.842
Total Loss 2.098 82.018
------------------------------------------------------------------------ The Lambda Values @ Various Nodes In The Generator Buses Are . . .
--------------------------------------------------------------------------
Node 1 --> 61.00
Node 2 --> 61.00
Node 3 --> 61.00
Node 4 --> 61.00
--------------------------------------------------------------------------
The Total Production Cost Of All Generators = 30744.72 Fr
--------------------------------------------------------------------------
Observation:
The above tables 4.1& 4.2 represent the output of load flow for
system without and with TCSC respectively. From these tables, the line
40
losses are calculated and compared. From the above results, the power flow
solution for the system with TCSC is relatively lesser than for the system
without TCSC. In above table 4.1, the real and reactive power losses are
calculated as 26.558 MW, 88.298 MVAR respectively which is relatively
higher as compared to system with TCSC having real and reactive power
losses as 2.098 MW, 82.018 MVAR respectively. From these above tables
4.1& 4.2 real power loss is reduced higher than reactive power loss and the
total production cost for system without TCSC is 32248.08 Fr which is
reduced upto 30744.72 Fr for system with TCSC. The load flow with TCSC
possesses low real and reactive power losses as compared to load flow
without TCSC. Comparing the above two programs, we can conclude that in
load flow with TCSC real power loss, reactive power loss are reduced.
Hence the total cost of production function will be reduced in load flow with
TCSC compared without TCSC.
4.3. CONCLUSION
41
This project work has presented a Newton-Raphson load flow
algorithm to solve power flow problems in power systems with Thyristor
controlled series capacitor (TCSC). This algorithm is capable of solving
power networks very reliably. The IEEE 6 Bus system has been used to
demonstrate the proposed method over a wide range of power flow
variations in the transmission system. It has also been observed that in IEEE
30 Bus system the results upto Ybus formation has been achieved and in
future load flow can also be determined. It has also been observed that it is
possible to place TCSC in specific transmission lines to improve the system
performance. The proposed algorithm has been tested for IEEE-6 Bus
system and it provides better power flow control. We have implemented and
tested the TCSC scheme on 6 bus Ward-Hale system. We have illustrated
how the proposed allocation scheme of TCSC, in which the voltage stability
margin is improved and losses are reduced after connecting the TCSC into
the power system.
4.4. FUTURE WORK
42
IEEE 30-BUS SYSTEM
To test the performance of the proposed algorithm IEEE 30 bus
test system as shown in figure A-5 with 230kv and 100 MVA base has been
considered. Specified power flow control over the transmission lines has
been achieved by determining the converged firing angle ‘α’.
The TCSC has been designed with an inductance of 5.2 mH and
capacitor of 200F.The firing angle of TCSC corresponding to the desired
increase in power in the line without TCSC.
Out of 41 lines available in the system, solutions for specific
lines have been presented. Unfortunately rest of the lines fails to provide
better power flow control with TCSC. Even though the solutions for positive
increase in power flow are demonstrated but the same procedure can be
repeated for negative power flow control also.
Data for sample problem
Tables A-6, A-7, A-8, A-9 gives the details of the generator,
transformer, load and bus data respectively for the IEEE 30 bus system is
given in appendices. The figure A-5 shows the line diagram of IEEE 30 bus
system also given in appendices.
APPENDICES
A-1 Bus diagram for Ward and Hale 6 bus system:
43
Fig A-1 Six Bus Ward and Hale System
TABLE A2-Line Data of Six bus system
Bus Bus R X ½ B Tap Max
44
nl nr P.U P.U P.U Setting Rating1 6 0.123 0.518 0.0 0 65.0
1 4 0.080 0.370 0.0 0 75.0
1 4 0.080 0.470 0.0 0 75.0
2 5 0.282 0.640 0.0 0 65.0
2 3 0.723 1.050 0.0 0 40.0
4 6 0.087 0.407 0.0 0 30.0
4 3 0.0 0.133 0.0 0.956 60.0
6 5 0.0 0.300 0.0 0.981 70.0
TABLE A3 Bus Data of Six bus system
BusNo
Bus Code
Voltage Mag
Angle deg
load Generator Injected Qsh
MW MVAR MW MVAR Q min Q max
1 1 1.05 0.0 0.0 0.0 0.0 0.0 -50.0 100.0 0.0
2 2 1.05 0.0 0.0 0.0 50.0 0.0 -25.0 50.0 0.0
3 0 1.00 0.0 55.0 13.0 0.0 0.0 0.0 0.0 0.0
4 0 1.00 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
5 0 1.00 0.0 30.0 18.0 0.0 0.0 0.0 0.0 3.0
6 0 1.00 0.0 50.0 5.0 0.0 0.0 0.0 0.0 1.5
A-4 Program coding
45
Load Flow without TCSC
clear allclc
basemva = 100; accuracy = 0.001; maxiter = 50;sigma = 0.001;kl = 1;kg = kl;lamda = 0.1;
% 1- Slack Bus; 2- PV Bus: 0- PQ Bus% Bus Bus Voltage Angle ----Load---- --------Generator------- Injected% No Code Mag. Degree MW Mvar MW Mvar Qmin Qmax Mvarbusdata = [1 1 1.005 0.0 150.0 0.0 0.0 0.0 -30.0 100.0 0.0 2 2 1.010 0.0 0.0 0.0 243.0 0.0 -40.0 90.0 0.0 3 2 1.020 0.0 0.0 0.0 74.0 0.0 0.0 0.0 0.0 4 2 1.000 0.0 200.0 0.0 42.0 0.0 0.0 0.0 0.0 5 0 0.986 0.0 500.0 0.0 0.0 0.0 0.0 0.0 0.0];k = 0.01; % Line Data% Bus bus R X 1/2 B tap Line% nl nr p.u. p.u. p.u. setting Rating linedata=[1 2 0.005+k 0.01 0.0 1 200 2 3 0.005+k 0.20 0.0 1 300 2 4 0.005+k 0.15 0.0 1 150 3 4 0.005+k 0.02 0.0 1 150 3 5 0.005+k 0.01 0.0 1 400 1 5 0.0001+k 0.03 0.0 1 500]; fprintf('\n\n--------------------------------------------------------------------------');fprintf('\n Base Case Details . . . \n');fprintf('--------------------------------------------------------------------------\n');
lfybus; % Forms The Bus Admittance Matrixlfnewton; % Newton Raphson Power Flow
46
busout; % Prints The Power Flow Solution on the Screenlineflow; % Computes & Displays the Line Flow and Losses
cost = zeros(nbus, 1)';costg = zeros(nbus, 1)';costl = zeros(nbus, 1)';lamda = zeros(nbus - 1, 1)';
cof_a = [10 10 0.05 12 12 0.10 12 20 0.30 20 15 0.60];cof_b = zeros(5,2);
% Load datafload = [1 130 170 85 20 4 180 220 95 30 5 480 520 120 15]; slp = zeros(3,1);
% Find Slope for i = 1 : 3 slp(i)= (fload(i,3) - fload(i,2)) / (fload(i,5) - fload(i,4));end
fload = [fload slp];
% Find b - Coffecients Of Loadfor i = 1 : nbus for inl = 1:3 if (i == fload(inl)) cof_b(i,1) = fload(inl,3) * fload(inl,6) + fload(inl,4); cof_b(i,2) = fload(inl,6) / 2; end endend
% % Find Cost Of Load% for i = 1 : nbus
47
% costl(i) = cof_b(i,1)*Pd(i) + cof_b(i,2)*Pd(i)*Pd(i);% if i ~= 1 & i ~= 4% lamda(i)= lamda(i) + cof_b(i,1) + 2*cof_b(i,2)*Pd(i);% end% end
% Find Cost Of Generationfor i = 1 : nbus if busdata(i,2) ~= 0 costg(i)= cof_a(i,1) + cof_a(i,2) * Pg(i) + cof_a(i,3) * Pg(i) * Pg(i); lamda(i)= lamda(i) + cof_a(i,2) + 2 * cof_a(i,3) * Pg(i); endend
fprintf('--------------------------------------------------------------------------\n');fprintf(' The Total Production Cost Of All Generators = %8.2f Fr', sum(costg));fprintf('\n--------------------------------------------------------------------------\n');
Load Flow with TCSC
48
clear allclcbasemva = 100; accuracy = 0.001; maxiter = 50;k = 1;PBT = 96.461; QBT = 10.413;PET = -PBT;QET = -1.1 * QBT;
% 1- Slack Bus; 2- PV Bus: 0- PQ Bus% Bus Bus Voltage Angle ----Load---- -------Generator------- Injected% No Code Mag. Degree MW Mvar MW Mvar Qmin Qmax Mvarbusdata = [1 1 1.005 0.0 150.0 0.0 0.0 0.0 -30.0 100.0 0.0 2 2 1.010 0.0 0.0 0.0 243.0 0.0 -40.0 90.0 0.0 3 2 1.020 0.0 0.0 0.0 67.0 0.0 0.0 0.0 0.0 4 2 1.000 0.0 200.0 0.0 38.0 0.0 0.0 0.0 0.0 5 0 0.986 0.0 500.0 0.0 0.0 0.0 0.0 0.0 0.0]; busdata_base = busdata;
% Bus bus R X 1/2 B = 1 for lines% nl nr p.u. p.u. p.u. > 1 or < 1 tr. % tap at bus nl linedata=[1 2 0.005 0.01 0.0 1 200 2 3 0.005 0.20 0.0 1 300 2 4 0.005 0.15 0.0 1 150 3 4 0.005 0.02 0.0 1 150 3 5 0.005 0.01 0.0 1 400 1 5 0.0001 0.03 0.0 1 500]; % FACT - TCSC Data % n sb rb Xse Xsh Vse Vsh facts=[1 2 4 0.1 0.1 0.2 1.1];
% Extract Facts detail
49
fmax = max(facts(:,1));fet = facts(:,2); fbt = facts(:,3);fxb = facts(:,4);fno = facts(:,1);fxe = facts(:,5);fvb = facts(:,6);fve = facts(:,7);%---------------------- Update TCSC connected Bus data--------------------------------------------
% Connection between Exciter Terminal & Booster Terminal is broken% New bus is connected to Booster terminal Bus - PV Bus fiter = 1;Ebus = facts(:,2);Bbus = facts(:,3);nl = linedata(:,1); nr = linedata(:,2);nbr = length(linedata(:,1)); nbus = max(max(nl), max(nr));for L = 1:nbr if nl(L) == Ebus & nr(L) == Bbus % Line data Update linedata(L,1) = nbus + 1; else,endend
% Exciter Terminal of TCSC - Power drawnbusdata_base(Ebus,12)= PET;busdata_base(Ebus,13)= QET;busdata_base(Ebus,2)= 2; busdata_base(Ebus,3)= 1.01; busdata_base(Ebus,4)= 0.0; Bbus = nbus + 1;% Booster Terminal of TCSC - Power Injection% New Bus is created - PV Bus - generation = line flow% Bus No. PV -- Voltage-- |--Load--| |-- Gen --| |-- Q limit-| Capacitor ---TCSC--% Bus Mag. Angle | P Q | | P Q | | Qmin Qmax| Bank(Var) P Qbusdataf=[Bbus 0 1.01 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 PBT QBT];
50
% Updated busdata with FACTS -Adding rowbusdata=[busdata_base;busdataf];
fprintf('\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~')fprintf('\n After TCSC connection ')fprintf('\n\n TCSC connected between Bus %g and Bus %g ',fet(fiter),fbt(fiter))fprintf('\t\t Exciter Bus=%g and Booster Bus=%g',Ebus,Bbus)fprintf('\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n')
lfybus % form the bus admittance matrixVmat=zeros(nbus+2,1);Amat=zeros(nbus+2,1);%----------------------------------------------------------------------------------------% Modified Newton Raphson Methodns=0;ng=0;Vm=0; delta=0; yload=0; deltad=0;
for k=1:nbus n = busdata(k,1); kb(n) = busdata(k,2); Vm(n) = busdata(k,3); delta(n) = busdata(k,4); Pd(n) = busdata(k,5); Qd(n) = busdata(k,6); Pg(n) = busdata(k,7); Qg(n) = busdata(k,8); Qmin(n) = busdata(k,9); Qmax(n) = busdata(k,10); Qsh(n) = busdata(k,11); PTCSC(n) = busdata(k,12); QTCSC(n)=busdata(k,13); if Vm(n) <= 0 Vm(n) = 1.0; V(n) = 1 + j * 0; else delta(n) = pi/180*delta(n);
51
V(n) = Vm(n)*(cos(delta(n)) + j*sin(delta(n))); P(n)=(Pg(n)-Pd(n)+PTCSC(n))/basemva; Q(n)=(Qg(n)-Qd(n)+ Qsh(n)+QTCSC(n))/basemva; S(n) = P(n) + j*Q(n); endendk=nbus;Vm(k+1)=fvb(fiter); Vm(k+2)=fve(fiter);delta(k+1)=0.0; delta(k+2)=0.0;
Vm_first=Vm;delta_first=delta;
for k=1:nbus if kb(k) == 1, ns = ns+1; else, end if kb(k) == 2 ng = ng+1; else, end ngs(k) = ng; nss(k) = ns;endYm = abs(Ybus);t = angle(Ybus);rowb=2*nbus-ng-2*ns;colb=rowb;m=2*nbus-ng-2*ns + 4*fmax;maxerror = 1; converge=1;iter = 0;
% Start of iterationsclear A DC J DX% maxiter=1;while maxerror >= accuracy & iter <= maxiter % Test for max. power mismatchfor i=1:mfor k=1:m A(i,k)=0; %Initializing Jacobian matrixend, endDC=zeros(1,m); DX=zeros(m,1);iter = iter+1;
52
for n=1:nbusnn=n-nss(n);lm=nbus+n-ngs(n)-nss(n)-ns;J11=0; J22=0; J33=0; J44=0; for i=1:nbr if nl(i) == n | nr(i) == n if nl(i) == n, l = nr(i); end if nr(i) == n, l = nl(i); end J11=J11+ Vm(n)*Vm(l)*Ym(n,l)*sin(t(n,l)- delta(n) + delta(l)); J33=J33+ Vm(n)*Vm(l)*Ym(n,l)*cos(t(n,l)- delta(n) + delta(l)); if kb(n)~=1 J22=J22+ Vm(l)*Ym(n,l)*cos(t(n,l)- delta(n) + delta(l)); J44=J44+ Vm(l)*Ym(n,l)*sin(t(n,l)- delta(n) + delta(l)); else, end if kb(n) ~= 1 & kb(l) ~=1 lk = nbus+l-ngs(l)-nss(l)-ns; ll = l -nss(l); % off diagonal elements of J1 A(nn, ll) =-Vm(n)*Vm(l)*Ym(n,l)*sin(t(n,l)- delta(n) + delta(l)); if kb(l) == 0 % off diagonal elements of J2 A(nn, lk) =Vm(n)*Ym(n,l)*cos(t(n,l)- delta(n) + delta(l));end if kb(n) == 0 % off diagonal elements of J3 A(lm, ll) =-Vm(n)*Vm(l)*Ym(n,l)*cos(t(n,l)- delta(n)+delta(l)); end if kb(n) == 0 & kb(l) == 0 % off diagonal elements of J4 A(lm, lk) =-Vm(n)*Ym(n,l)*sin(t(n,l)- delta(n) + delta(l));end else end else , end end Pk = Vm(n)^2*Ym(n,n)*cos(t(n,n))+J33; Qk = -Vm(n)^2*Ym(n,n)*sin(t(n,n))-J11; if kb(n) == 1 P(n)=Pk; Q(n) = Qk; end % Swing bus P if kb(n) == 2 Q(n)=Qk; if Qmax(n) ~= 0 Qgc = Q(n)*basemva + Qd(n) - Qsh(n)-(QTCSC(n)); if iter <= 7 if iter > 2 if Qgc < Qmin(n), Vm(n) = Vm(n) + 0.01; elseif Qgc > Qmax(n),
53
Vm(n) = Vm(n) - 0.01;end else, end else,end else,end end if kb(n) ~= 1 A(nn,nn) = J11; %diagonal elements of J1 DC(nn) = P(n)-Pk; end if kb(n) == 0 A(nn,lm) = 2*Vm(n)*Ym(n,n)*cos(t(n,n))+J22; %diagonal elements of J2 A(lm,nn)= J33; %diagonal elements of J3 A(lm,lm) =-2*Vm(n)*Ym(n,n)*sin(t(n,n))-J44; %diagonal of elements of J4 DC(lm) = Q(n)-Qk; endend%-------------------------------------------------------------------------------------
% PBT -----------------------l=Ebus;ll = l -nss(l); % Deltalk = nbus+l-ngs(l)-nss(l)-ns; % VoltA(rowb+1,ll)=(Vm(Bbus)*Vm(Ebus)*cos(delta(Bbus)-delta(Ebus)))/fxb(fiter);A(rowb+1,lk)=-((Vm(Bbus)*sin(delta(Bbus)-delta(Ebus)))/fxb(fiter));
l=Bbus;ll = l -nss(l); % Deltalk = nbus+l-ngs(l)-nss(l)-ns; % VoltA(rowb+1,ll)=-((Vm(Bbus)*Vm(Ebus)*cos(delta(Bbus)-delta(Ebus)))/fxb(fiter))... -((Vm(Bbus)*Vm(nbus+fiter)*cos(delta(Bbus)-delta(nbus+fiter)))/fxb(fiter));A(rowb+1,lk)=-((Vm(Ebus)*sin(delta(Bbus)-delta(Ebus)))/fxb(fiter))... -((Vm(nbus+fiter)*sin(delta(Bbus)-delta(nbus+fiter)))/fxb(fiter)); % PET -----------------------l=Ebus;ll = l -nss(l); % Deletlk = nbus+l-ngs(l)-nss(l)-ns; % Volt
% Modified flow
54
A(rowb+2,ll)=((Vm(Ebus)*Vm(nbus+fiter)*cos(delta(Ebus)-delta(nbus+fiter))/fxb(fiter)))... -((Vm(Ebus)*Vm(Bbus)*cos(delta(Ebus)-delta(Bbus))/fxb(fiter)))... -((Vm(Ebus)*Vm(nbus+2*fiter)*cos(delta(Ebus)-delta(nbus+2*fiter)))/fxe(fiter)); A(rowb+2,lk)=((Vm(nbus+fiter)*sin(delta(Ebus)-delta(nbus+fiter)))/fxb(fiter))... -((Vm(Bbus)*sin(delta(Ebus)-delta(Bbus))/fxb(fiter)))... -((Vm(nbus+2*fiter)*sin(delta(Ebus)-delta(nbus+2*fiter)))/fxe(fiter));
l=Bbus;ll = l -nss(l); % Delbtlk = nbus+l-ngs(l)-nss(l)-ns; % Volt
% Modified flowA(rowb+2,ll)=((Vm(Ebus)*Vm(Bbus)*cos(delta(Ebus)-delta(Bbus))/fxb(fiter))); A(rowb+2,lk)=-(Vm(Ebus)*sin(delta(Ebus)-delta(Bbus))/fxb(fiter));
% QBT -----------------------l=Ebus;ll = l -nss(l); % Deltalk = nbus+l-ngs(l)-nss(l)-ns; % VoltA(rowb+3,ll)=((Vm(Bbus)*Vm(Ebus)*sin(delta(Bbus)-delta(Ebus)))/fxb(fiter));A(rowb+3,lk)=(Vm(Bbus)*cos(delta(Bbus)-delta(Ebus)))/fxb(fiter);
l=Bbus;ll = l -nss(l); % Deltalk = nbus+l-ngs(l)-nss(l)-ns; % Volt
A(rowb+3,ll)=-((Vm(Bbus)*Vm(Ebus)*sin(delta(Bbus)-delta(Ebus)))/fxb(fiter))... -((Vm(Bbus)*Vm(nbus+fiter)*sin(delta(Bbus)-delta(nbus+fiter)))/fxb(fiter));
A(rowb+3,lk)=((Vm(Ebus)*cos(delta(Bbus)-delta(Ebus)))/fxb(fiter))... +((Vm(nbus+fiter)*cos(delta(Bbus)-delta(nbus+fiter)))/fxb(fiter))-(2*Vm(Bbus)/fxb(fiter)); % QET -----------------------l=Ebus;ll = l -nss(l); % Deletlk = nbus+l-ngs(l)-nss(l)-ns; % Volt
55
% Modified flowA(rowb+4,ll)=((Vm(Ebus)*Vm(nbus+fiter)*sin(delta(Ebus)-delta(nbus+fiter)))/fxb(fiter))... -((Vm(Ebus)*Vm(Bbus)*sin(delta(Ebus)-delta(Bbus)))/fxb(fiter))... -((Vm(Ebus)*Vm(nbus+2*fiter)*sin(delta(Ebus)-delta(nbus+2*fiter)))/fxe(fiter));
A(rowb+4,lk)=-(2*Vm(Ebus)/fxb(fiter))- (2*Vm(Ebus)/fxe(fiter))... -((Vm(nbus+fiter)*cos(delta(Ebus)-delta(nbus+fiter)))/fxb(fiter))... +((Vm(Bbus)*cos(delta(Ebus)-delta(Bbus)))/fxb(fiter))... +((Vm(nbus+2*fiter)*cos(delta(Ebus)-delta(nbus+2*fiter)))/fxe(fiter));
l = Bbus;ll = l -nss(l); % Delbtlk = nbus+l-ngs(l)-nss(l)-ns; % Volt
% Modified flowA(rowb+4,ll)=((Vm(Ebus)*Vm(Bbus)*sin(delta(Ebus)-delta(Bbus)))/fxb(fiter));A(rowb+4,lk)=((Vm(Ebus)*cos(delta(Ebus)-delta(Bbus)))/fxb(fiter));
% --------------------------- J5 Updation -----------------------------------------------% PBT--------A(rowb+1,colb+1)=(Vm(Bbus)*Vm(nbus+fiter)*cos(delta(Bbus)-delta(nbus+fiter)))/fxb(fiter);A(rowb+1,colb+2)=0;A(rowb+1,colb+3)=-((Vm(Bbus)*sin(delta(Bbus)-delta(nbus+fiter)))/fxb(fiter));A(rowb+1,colb+4)=0;
% Modified flowA(rowb+2,colb+1)=-(Vm(Ebus)*Vm(nbus+fiter)*cos(delta(Ebus)-delta(nbus+fiter)))/fxb(fiter);A(rowb+2,colb+2)=((Vm(Ebus)*Vm(nbus+2*fiter)*cos(delta(Ebus)-delta(nbus+2*fiter)))/fxe(fiter));A(rowb+2,colb+3)=((Vm(Ebus)*sin(delta(Ebus)-delta(nbus+fiter)))/fxb(fiter));A(rowb+2,colb+4)=-(Vm(Ebus)*sin(delta(Ebus)-delta(nbus+2*fiter)))/fxe(fiter);
% QBT-------- A(rowb+3,colb+1)=(Vm(Bbus)*Vm(nbus+fiter)*sin(delta(Bbus)-delta(nbus+fiter)))/fxb(fiter);A(rowb+3,colb+2)=0;
56
A(rowb+3,colb+3)=(Vm(Bbus)*cos(delta(Bbus)-delta(nbus+fiter)))/fxb(fiter);A(rowb+3,colb+4)=0;
% Modified flowA(rowb+4,colb+1)=-(Vm(Ebus)*Vm(nbus+fiter)*sin(delta(Ebus)-delta(nbus+fiter)))/fxb(fiter);A(rowb+4,colb+2)=((Vm(Ebus)*Vm(nbus+2*fiter)*sin(delta(Ebus)-delta(nbus+2*fiter)))/fxe(fiter));A(rowb+4,colb+3)=-(Vm(Ebus)*cos(delta(Ebus)-delta(nbus+fiter)))/fxb(fiter);A(rowb+4,colb+4)=((Vm(Ebus)*cos(delta(Ebus)-delta(nbus+2*fiter)))/fxe(fiter)); % DC column matrix update----------FPBT=-((Vm(Bbus)*Vm(Ebus)*sin(delta(Bbus)-delta(Ebus)))/fxb(fiter))... -((Vm(Bbus)*Vm(nbus+fiter)*sin(delta(Bbus)-delta(nbus+fiter)))/fxb(fiter));
% Modified flowFPET=((Vm(Ebus)*Vm(nbus+fiter)*sin(delta(Ebus)-delta(nbus+fiter)))/fxb(fiter))... -((Vm(Ebus)*Vm(Bbus)*sin(delta(Ebus)-delta(Bbus)))/fxb(fiter))... -((Vm(Ebus)*Vm(nbus+2*fiter)*sin(delta(Ebus)-delta(nbus+2*fiter)))/fxe(fiter));
FQBT=((Vm(Bbus)*Vm(Ebus)*cos(delta(Bbus)-delta(Ebus)))/fxb(fiter))... +((Vm(Bbus)*Vm(nbus+fiter)*cos(delta(Bbus)-delta(nbus+fiter)))/fxb(fiter))... -(Vm(Bbus)*Vm(Bbus)/fxb(fiter));
% Modified flowFQET=-(Vm(Ebus)*Vm(Ebus)/fxb(fiter))- (Vm(Ebus)*Vm(Ebus)/fxe(fiter))... -((Vm(Ebus)*Vm(nbus+fiter)*cos(delta(Ebus)-delta(nbus+fiter)))/fxb(fiter))... +((Vm(Ebus)*Vm(Bbus)*cos(delta(Ebus)-delta(Bbus)))/fxb(fiter))... +((Vm(Ebus)*Vm(nbus+2*fiter)*cos(delta(Ebus)-delta(nbus+2*fiter)))/fxe(fiter));
DC(colb+1)=(PBT/basemva)-FPBT; % DELBDC(colb+2)=(PET/basemva)-FPET; % DELEDC(colb+3)=(QBT/basemva)-FQBT; % VBDC(colb+4)=(QET/basemva)-FQET; % VEQET=Q(Ebus)*basemva + Qd(Ebus) - Qsh(Ebus)-QTCSC(Ebus);%--------------------------------------------------DX=A\DC';for n=1:nbus nn=n-nss(n); lm=nbus+n-ngs(n)-nss(n)-ns;
57
if kb(n) ~= 1 delta(n) = delta(n)+DX(nn); end if kb(n) == 0 Vm(n)=Vm(n)+DX(lm); end end%--------------------------------------------------%TCSC DELE,DELB,VE,VB Updatedelta(nbus+1)= delta(nbus+1)+DX(rowb+1); % DELBdelta(nbus+2)= delta(nbus+2)+DX(rowb+2); % DELEVm(nbus+1)= Vm(nbus+1)+DX(rowb+3); % VBVm(nbus+2)= Vm(nbus+2)+DX(rowb+4); % VEVmat=[Vmat Vm'];Amat=[Amat delta'];maxerror=max(abs(DC));if iter == maxiter & maxerror > accuracy fprintf('\nWARNING: Iterative solution did not converged after ') fprintf('%g', iter), fprintf(' iterations.\n\n') fprintf('Press Enter to terminate the iterations and print the results \n') converge = 0; pause, else, end end
if converge ~= 1 tech= (' ITERATIVE SOLUTION DID NOT CONVERGE'); else, tech=(' Power Flow Solution by Newton-Raphson Method');end
V = Vm.*cos(delta)+j*Vm.*sin(delta);deltad=180/pi*delta;i=sqrt(-1);k=0;
for n = 1:nbus if kb(n) == 1 k=k+1; S(n)= P(n)+j*Q(n); Pg(n) = P(n)*basemva + Pd(n)-PTCSC(n); Qg(n) = Q(n)*basemva + Qd(n) - Qsh(n)-QTCSC(n); Pgg(k)=Pg(n); Qgg(k)=Qg(n); elseif kb(n) ==2
58
k=k+1; S(n)=P(n)+j*Q(n); Qg(n) = Q(n)*basemva + Qd(n) - Qsh(n)-QTCSC(n); Pgg(k)=Pg(n); Qgg(k)=Qg(n); end yload(n) = (Pd(n)- j*Qd(n)+j*Qsh(n))/(basemva*Vm(n)^2);end
for k=1:nbus busdataVm(k)=Vm(k); busdatadeltad(k)=deltad(k);endbusdata(:,3)=busdataVm'; busdata(:,4)=busdatadeltad';Pgt = sum(Pg); Qgt = sum(Qg); Pdt = sum(Pd); Qdt = sum(Qd);Qsht = sum(Qsh);
%-----------------------------------------------------------------------------------------------------busout_TCSC % Prints the power flow solution on the screenlineflow % Computes and displays the line flow and losses
cost = [0 0 0 0 0];costg = [0 0 0 0 0];costl = [0 0 0 0 0];lamda = [0 0 0 0 0];cof_a = [10 10 0.05 12 12 0.10 12 20 0.30 20 15 0.60];cof_b = zeros(5,2);% Load datafload =[1 130 170 85 20 4 180 220 95 30 5 480 520 120 15];slp = zeros(3,1);% Find Slope
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for i = 1 : 3 slp(i)= (fload(i,3) - fload(i,2)) / (fload(i,5) - fload(i,4));endfload = [fload slp];
% Find b - coffecients for loadfor i = 1 : nbus - 1 for inl = 1 : 3 if (i == fload(inl)) cof_b(i,1) = fload(inl,3)*fload(inl,6) + fload(inl,4); cof_b(i,2) = fload(inl,6) / 2; end endend% % Find cost of load% for i = 1 : nbus - 1% costl(i)= cof_b(i,1)*Pd(i) + 2*cof_b(i,2)*Pd(i)*Pd(i);% if i~=4 & i~=1% lamda(i)= lamda(i) + cof_b(i,1) + cof_b(i,2)*Pd(i);% end% end
% Find cost of generationfor i = 1 : nbus - 1 if busdata(i,2) ~= 0 costg(i)= cof_a(i,1) + cof_a(i,2)*Pg(i) + cof_a(i,3)*Pg(i)*Pg(i); lamda(i)= lamda(i) + cof_a(i,2) + 2*cof_a(i,3)*Pg(i); endendfprintf('--------------------------------------------------------------------------\n');fprintf(' The Total Production Cost Of All Generators = %8.2f Fr', sum(costg));fprintf('\n--------------------------------------------------------------------------\n');
A-5 Bus diagram for 30 bus system:
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Fig A-5 IEEE 30 BUS SYSTEM.
Table A-6 Generator data for 30 bus system
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Table A-7-Transformer Data for 30 bus system
Transformer no
Bus
Tap Setting From to
1 6 9 1.0155
2 6 10 0.9629
3 4 12 1.0129
4 28 27 0.9581
Table A-8 Load Data for 30 bus system
Bus
No
Load Bus
No
Load
MW MVAR MW MVAR
1 0 0 16 3.5000 1.8000
Bus no PG Min MW
PG Max MW
QG Min MVAR
QG Max MVAR
1 50 200 -20 150
2 20 80 -20 60
5 15 50 -15 62.5
8 10 35 -15 48.7
11 10 30 -10 40.0
13 12 40 -15 44.7
62
2 21.7000 12.7000 17 9.0000 5.8000
3 2.4000 1.2000 18 3.2000 0.9000
4 7.6000 1.6000 19 9.5000 3.4000
5 94.2000 19.0000 20 2.2000 0.7000
6 0 0 21 17.5000 11.2000
7 22.8000 10.9000 22 0 0
8 30.0000 30.0000 23 3.2000 1.6000
9 0 0 24 8.7000 6.7000
10 5.8000 2.0000 25 0 0
11 0 0 26 3.5000 2.3000
12 11.2000 7.5000 27 0 0
13 0 0 28 0 0
14 6.2000 1.6000 29 2.4000 0.9000
15 8.2000 2.5000 30 10.6000 1.9000
Table A-9 Bus Data for 30 bus system
Branch no
From bus To bus R (p.u) X (p.u) B (p.u) Rating MVA
1 1 2 0.0192 0.0575 0.0264 130
2 1 3 0.0452 0.1852 0.0204 130
63
3 2 4 0.0570 0.1737 0.0184 65
4 3 4 0.0132 0.0379 0.0042 130
5 2 5 0.0472 0.1983 0.0209 130
6 2 6 0.0581 0.1763 0.0187 65
7 4 6 0.0119 0.0414 0.0045 90
8 5 7 0.0460 0.1160 0.0102 70
9 6 7 0.0267 0.0820 0.0085 130
10 6 8 0.0120 0.0420 0.0045 32
11 6 9 0.0 0.2080 0.0 65
12 6 10 0.0 0.5560 0.0 32
13 9 1 0.0 0.2080 0.0 65
14 9 10 0.0 0.1100 0.0 65
15 4 12 0.0 0.2560 0.0 65
16 12 13 0.0 0.1400 0.0 65
17 12 14 0.1231 0.2559 0.0 32
18 12 15 0.0662 0.1304 0.0 32
19 21 16 0.0945 0.1987 0.0 32
20 14 15 0.2210 0.1997 0.0 16
21 16 17 0.0824 0.1932 0.0 16
22 15 18 0.1070 0.2185 0.0 16
23 18 19 0.0639 0.1292 0.0 16
24 19 20 0.0340 0.0680 0.0 32
64
25 10 20 0.0936 0.2090 0.0 32
26 10 17 0.0324 0.0845 0.0 32
27 10 21 0.0348 0.0749 0.0 32
28 10 22 0.0727 0.1499 0.0 32
29 21 22 0.0116 0.0236 0.0 32
30 15 23 0.1000 0.2020 0.0 16
31 22 24 0.1150 0.1790 0.0 16
32 23 24 0.1320 0.2700 0.0 16
33 24 25 0.1885 0.3292 0.0 16
34 25 26 0.2544 0.3800 0.0 16
35 25 27 0.1093 0.2087 0.0 16
36 28 27 0.0 0.3690 0.0 65
37 27 29 0.2198 0.4153 0.0 16
38 27 30 0.3202 0.6027 0.0 16
39 29 30 0.2399 0.4533 0.0 16
40 8 28 0.0636 0.2000 0.0214 32
41 6 28 0.0169 0.0599 0.0065 32
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