**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 f^{O}, a constant prime mover value setting x^{O}_{E}, and constant generator output p^{O}_{G}. 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, ∆X_{E}, ∆P_{G} and ∆P_{C} 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:

The governor adjusted X_{E}, the value setting of the turbine depending on the frequency deviation signal ∆f and the speed changer command signal ∆P_{C}. R denotes the speed regulation due to governor action. Thus the gain of the frequency feedback loop is 1/R. If K_{G} and T_{G} 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 K_{r} of the total power developed by the turbine. Let K_{T }and T_{T} be the gain and time constant respectively of the individual turbine and T_{r }, 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.

∆P_{G }= ∆P_{G, HP} + ∆P_{G, 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.

- By decreasing the area kinetic energy W
_{kin}at the rate - 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

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 W^{O}_{kin} 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 P_{R}, 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 ∆P_{C} 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 K_{I }= 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 (K_{G}. K_{T}). It is convenient to set the two gains so that the combined gain K_{G}. K_{T} 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 ∆P_{d }, 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 ∆D_{M} 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 K_{0 }= 0 gives the system model without LMES unit.

**4.2.2. The Uncontrolled Case of Single Area Power System**

Uncontrolled case means ∆P_{C }=0.So the integrator gain K_{I }=0.That means do not change the speed change portion.

Speed regulation R=2.4;Hz/p.u MW

Power system gain K_{P}=120

Power time constant T_{P}=20s

Load increases ∆P_{L}=.001(p.u) MW

Generator time constant T_{G}=.08s

Turbine time constant T_{T}=.3s

Integrating gain K_{T} =0(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:

- The control loop must be characterized by a sufficient degree of stability.
- 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.
- The integral of the frequency error should not exceed a certain maximum value.
- 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,