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Chapter-4: MODELING OF A SINGLE AREA POWER SYSTEM

4.1. Introduction

The single area power system model without the LMES unit is the same as that described in for convenience of the reader; both the original model and the changes due to addition of the LMES unit are described here. It is assumed that initially the system is in a steady state, characterized by a constant frequency fO, a constant prime mover value setting xOE, and constant generator output pOG. Initially the speed changer command signal is zero. In response to the step load change, the deviations of these parameters from their nominal steady-state values are ∆f, ∆XE, ∆PG and ∆PC respectively.

4.2.1. Mathematically Representation Single Area Power Pystem

The complete block diagram of the frequency control of an isolated power system is shown figure:

Fig: 4.1. Load Frequency control of an isolated power system

The governor adjusted XE, the value setting of the turbine depending on the frequency deviation signal ∆f and the speed changer command signal ∆PC. R denotes the speed regulation due to governor action. Thus the gain of the frequency feedback loop is 1/R. If KG and TG are the gain and time constant of the speed governing mechanism, then the transfer function of the speed governor is,

Thus the deviation in turbine value setting is given by,

………………………………………(4-1)

The prime mover considered here is a reheat type steam turbine, with one high pressure stage and one low pressure stage. There is reheating arrangement is between the two stages.

Suppose the high-pressure stage is rated at a fraction Kr of the total power developed by the turbine. Let KT and TT be the gain and time constant respectively of the individual turbine and Tr , the time delay due to the passage of steam in the reheater. So the power developed in the H. P. stage is,

…………………………….(4-2)

And that developed in the L. P. stage is,

The total power developed by the turbine is obtained by adding the power developed in the two stages.

∆PG = ∆PG, HP + ∆PG, LP

……………………(4-3)

When a sudden change in loading occurs, the governor turbine control system does not react instantaneously and the immediate requirement of extra power demand has to come from the inertia of the generator rotor. Consequently the generator decelerates and the frequency falls. As the frequency deviation is fed into the governor control system, the governor comes into action after its characteristic time delay and increases the output. At any point of time in the process the difference between load and generated power is made up in two ways.

  1. By decreasing the area kinetic energy Wkin at the rate 
  2. By a reduced load consumption caused by the reduction in frequency. This reduction is found empirically and is denoted by D, which is equal to   MW/HZ.

Thus the power balance equation can be written as,

……………………………….(4-4)

The kinetic energy of the rotor varies with the square of the speed or frequency of the area. Therefore,

……………………………………………(4-5)

Where f is the instantaneous frequency and WOkin is the initial kinetic energy of the area.

…………………………….(4-6)

Every term of the above equation is expressed in Megawatts. If we divide the equation by the area capacity PR, the power deviation terms and the term D will be given in per units.

Thus, equation becomes,

…………………………(4-7)

Or, ………………………………………(4-8)

Where, is the per unit inertia constant and has the dimension of seconds.

So equation (4-9) becomes,

……………………………………..(4-9)

To achieve a zero steady state error condition, an integral control has to be added to the system. This is achieved by making the speed changer signal ∆PC commanded by a signal obtained by first amplifying and then integrating the frequency error. Thus,

…………………………………………………..(4-10)

By derivative this equation we get,

………………………………………………….(4-11)

Where KI = 1/R is the integrator gain. The negative sign in equation (4-11) appears because a decrease in frequency gives rise to an increase command.

The two transfer function blocks for governor and turbine are in series, with a total gain of that portion of the block diagram equal to (KG. KT). It is convenient to set the two gains so that the combined gain KG. KT equals unity. Considering this setting we omit these two gain parameters from further analysis.

When the model of an LMES unit is added to the power system model, the input signal to the LMES unit is the frequency deviation of the power area and the power coming from the LMES unit is added in the summing junction of the signal area model.

The addition of the LMES power in the summing junction leads to the modification of equation (4-10) which becomes,

…………………………………(4-12)

Here ∆Pd , the per unit change in converter power is shown negative because the power into the inductor is assumed to be positive.

On manipulation (4-13) becomes,

………………………….(4-13)

Equation (4- 1), when written in state variable equation from becomes,

………………….(4-14)

As the turbine transfer function is not of first order, we define a dummy variable ∆DM as,

………………………………………(4-15)

This gives,

Or, ………………………..(4-16)

The dummy state variable brings equation (4-3) down to first order,

Or, ……………..(4-17)

The resulting systems of state variable equations defining the single area power system with an LMES unit are,

Setting K0 = 0 gives the system model without LMES unit.

4.2.2. The Uncontrolled Case of Single Area Power System

Uncontrolled case means ∆PC =0.So the integrator gain KI =0.That   means do not change the speed change portion.

 Speed regulation R=2.4;Hz/p.u MW

Power system gain KP=120

Power time constant TP=20s

Load increases ∆PL=.001(p.u) MW

Generator time constant TG=.08s

Turbine time constant TT=.3s

Integrating gain KT =0(uncontrolled case)

Fig: 4.2 frequency deviation following a step load change (uncontrolled case)

4.2.3. Controlled Case of Single Area Power System

It is necessary to achieve must better frequency constancy than is by the speed governor system itself, as demonstrated above. To accomplish this we must manipulate the speed changer in according with some suitable control strategy. It is necessary to settle for a set of control specifications the stringency of which will in the end determine the sophistication of the proposed control method.

We are presently discussing a single area power system and the control specifications for such a system are considerably simpler than those imposed upon a multiple area system.

Here follow some realistic specifications:

  1. The control loop must be characterized by a sufficient degree of stability.
  2. Following a step load change the frequency error should return to zero. This is referred to a specifications control. The magnitude of the transient frequency division should be minimizing.
  3. The integral of the frequency error should not exceed a certain maximum value.
  4. The individual generators of the control area should divide the load for optimum economy.

So, the equation of the frequency division of governor system is,

Chapter-3: MODELING OF ELECTRICAL POWER SYSTEM COMPONENTS

3.1. Introduction

 First consider the problem of controlling the real power output of electric generator within a prescribed area in response to changes in system frequency and the tie line loading or the relation of these to each other, so as to maintain the scheduled system frequency and the established interchange with other areas within pre-determined limits. This term automatic load frequency control (LFC) is also often used to identify this problem area [5, 6].

The operation objectives of the LFC are to maintain reasonably uniform frequency, to divide the load between generators, and to control the tie line interchanging schedules. The change in frequency and tie line power are sensed, which is measure of the change in rotor angle δ (i.e. the error ∆ δ) to be corrected. The error signal ∆f and ∆Ptie, are amplified, mixed and transformed into a real power command signal ∆PV, which is sent to the prime mover to call for an amount ∆PG which will change the values∆f and ∆Ptie within the specified tolerance.

The first step in analysis and design of control is mathematical modeling of the system. The two most common methods are the transfer function method and the state variable approach. The state variable approach can be applied to portray linear as well as nonlinear systems. In order to use the transfer function and linear state equation, the system must first be linearized. Proper assumptions and approximations are made to linearize the mathematical equations describing the system, and a transfer function model is for the following components [5].

3.2.1 Generator Model

Applying the swing equation of a synchronous machine given by to small perturbation, we have,

Or, in terms of small deviation in speed,

With speed expressed in per unit, without explicit per unit rotation, it can be written as,

The speed deviation is directly proportional to the system frequency deviation,

The above relation is shown in block diagram form:

Fig: 3.1 Generator block diagram [4]

3.2.2. Load model

The load on a system consists of a variety of electrical devices. For resistive loads, such as sighting and heating loads, the electrical power is independent of frequency [11]. Motor loads are resistive to changes in frequency. How sensitive it is to frequency depends on the composite of the speed-load characteristics of all the devices. The speed-load characteristic of a composite load is approximated by,

Where ∆PL is the nonfrequency-sensitive load change and D∆f is the frequency sensitive load change. D is expressed as the percent change in load divided by percent change in frequency [4, 5]. For example, if the load is changed by 1.6 percent a 1 percent change in frequency shown in figure below:

Fig: 3.2 (a) Combined Generator and Load block diagram

Fig: 3.2(b) Equivalent block diagram

The above model can also be modified by the following block diagram in fig: 3.2(b)

Fig: 3.2. (c) Modified Block diagram of the Combined Generator and Load
Fig: 3.2. Block diagram of the Combined Generator and Load

Where,

, is power system proportional gain value.

, is power system time constant.

 

3.2.3. Governor Model

When the generator electrical load is suddenly increased, the electrical power exceeds the mechanical power input. This power defiance is the kinetic energy store in the rotating system the reduction in kinetic energy causes the turbine speed and consequently, the generator frequency to fall. The change in speed is sensed by the turbine governor which acts to adjust the turbine input valve to change the mechanical power output to bring the speed to a new steady-state. The earliest governors were the watt governors which sense the speed by means of rotating fly balls and provide mechanical motion in response to speed changes. However most modern governors use electronic means to sense speed changes [4, 5]. Figure 3.3 shows mathematically the essential element of a conventional watt governor, which consist of the following major parts.

Fig: 3.3. Speed Governing System

Speed Governor: The essential parts are centrifugal fly balls driven directly or through gearing by the turbine shaft. The mechanism provides up ward and downward vertical movement proportional to the change in speed.

Linkage Mechanism: These are links for transforming the fly balls movement to the turbine valve through a hydraulic amplifier and providing a feedback from the turbine valve movement.

Hydraulic Amplifier: Very linkage mechanical forces are needed to operate the system valve. Therefore, the governor movements are transformed into high power factor via several stages hydraulic amplifiers.

Speed Changer:  The speed changer consists of servomotor that can be operated manually or automatically for scheduling load nominal frequency. By adjusting this set point, a desired load dispatch can be scheduled at nominal frequency.

For stable operation, the governors are designed to permit the speed to drop as the load is increased. The steady-state characteristic [4] of such a governor is shown in fig 3.3.

The slopes of the curve represent the speed regulation R. Governors have speed regulation of 5-6 percent from zero to full load. The speed governor mechanism acts as a comparator whose output ∆P is the difference between references set power ∆Pref  and the power, as given from the governor speed characteristics,

Define in s domain,

This can also written as,

The command ∆Pg is transformed through the hydraulic to the steam valve position command ∆Pv. Assuming a linear relationship and considering a simple time constantan Tg, it can be expressed by the following s-domain relation.

Fig: 3.4 .block diagram representation of speed governing system turbine

3.2.4. Turbine Models:

 The change in valve position ∆XE causes an incremental increase in turbine power, ∆PT, via the electromechanical interaction the generator, will result in an increased generator power ∆PG.

The overall mechanism is relatively complicated, particularly if the generator voltage simultaneously undergoes while swings due to major network disturbances. The voltage level is constant and the torque variations are of small size, then an incremental analysis of the type we performed for the speed governor, above will give a relatively simple dynamic relationship between ∆XE and ∆PG.   It use any type turbine single gain factor KT and a single time constant TT and thus write.