In this article we will discuss about:- 1. Introduction to Power System 2. Components of Power System Network 3. Representation 4. Generalised Circuit Constants.
Introduction to Power System:
Electric power systems may be of great complexity and spread over large geographical area. An electric power system consists of generators, transformers, transmission lines and consumer equipment (loads). The system must be protected against flow of heavy short circuit currents (which can cause permanent damage to major equipment) by disconnecting the faulty section of system by means of circuit breakers and protective relaying.
It is necessary to know the maximum short circuit currents that can occur at the different points of a system in order that circuit breakers may be selected that are adequate to withstand the currents and operate successfully to cut out the faulty section, and also in order that the protective relays may be selected for correct operation. The design of machines, bus-bars, isolators, circuit breakers etc., is based on the considerations of normal and short circuit currents.
It is also necessary to be able to calculate, approximately at least, the size of the protective reactors which must be inserted in the system to limit the short circuit currents to a value which is not beyond that capable of being withstood by the circuit breakers.
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The short circuit currents in an ac system are determined mainly by the reactance of the alternators, transformers and lines up to the point of the fault in the case of phase-to-phase faults. When the fault is between phase and earth, the resistance of the earth path plays an important role in limiting the currents.
In case of circuit breakers their rupturing capacities are based on the symmetrical short circuit current which is the most simple calculation among all types of short circuits. However, for determination of settings of relays it is absolutely necessary to know fault current due to unsymmetrical fault condition too for which knowledge of symmetrical components etc., is required.
Components of Power System Network:
Power system can be subdivided into three major components namely:
(i) Generation System,
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(ii) Transmission System, and
(iii) Distribution system.
Nowadays, mostly, power is generated by three phase synchronous generators, transmitted by three phase lines and then distributed through three phase distribution networks. Three phase synchronous generators are always designed to produce three phase balanced excitation i.e. to generate three phase voltages which are equal in magnitude and displaced from each other by an equal angle of 120 electrical degrees.
The three phase transmission lines and three phase distribution network are either arranged symmetrically or if arrangement is un-symmetric, they are transposed at regular intervals to balance their electrical characteristics resulting in balanced three phase network comprising transmission and distribution networks.
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Thus three phase generation or supply as well as the three phase transmission and networks are balanced and, therefore, so far as the calculations are concerned, they can be treated as a single phase system for the analysis.
However, if there is a fault on the system, the excitation becomes unbalanced i.e. the generated three phase voltages may not be equal in magnitude or may not be displaced by an equal angle of 120 electrical degrees but the three phase transmission and distribution networks are balanced.
Thus, it becomes a case of unbalanced supply connected to the three phase balanced network. Such situations are usually handled by transforming the actual phasor quantities to component quantities by linear time dependent power invariant transformations because after transformation, these phasor quantities, become uncoupled and the differential equations, describing the dynamic behaviour of the system, which have time dependent i.e. variable coefficients, after transformation, become differential equations with constant coefficients. In practice, a large number of such transformations are available but all of them have the same common features.
In case, in addition to the unbalanced supply or excitation, the three phase network is also unbalanced, their transformation into component quantitates will not serve the purpose because even after transformation, the phasor quantities remain coupled. But, such situations are not very common.
Power System Representation:
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For the purpose of the power system studies such as transient stability studies, load flow studies or short-circuit studies, there is a need for the development of mathematical model of the power system network. For the development of fairly accurate models of the power system network, it is necessary that model reflects correctly the terminal behaviour of each component of the network for the purpose of study for which the model has been developed.
This is because the terminal behaviour of the power system components differ from the normal i.e. steady state conditions to abnormal i.e. transient conditions. The different components of power system are – synchronous machines, transmission network, transformers, distribution network including static loads, composite loads and dynamic loads such as induction machines etc.
Now, the representation of different components (synchronous machines, transformers and transmission lines) of the power system are discussed below:
1. Representation of Synchronous Machines:
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The synchronous machine is the most important component of a power system. It converts mechanical power into electrical form and supplies it into the power system network or, in the case of a motor; it draws electrical power from the network and converts it into the mechanical form.
Most simplified model i.e. representation of a synchronous generator for the purpose of transient stability studies is a constant voltage source behind proper reactance. The voltage source may be sub-transient, transient or steady-state voltages and the reactance may be the corresponding reactances.
However, for understanding this model, let us consider a synchronous generator operating at no load before occurrence of a 3-phase short circuit at its terminals. The current flowing in the synchronous generator just after occurrence of the three phase short circuit at its terminals is similar to the current that flows in an R-L circuit upon which sudden an ac voltage is applied.
Hence the current will have both ac (i.e. steady state) component as well as the dc (i.e. transient) component which decays exponentially with time constant τ equal to L/R. However, if the dc component is neglected the oscillograph of the ac component of the current that flows in the synchronous generator just after the fault occurs, will have the wave shape as shown in Fig. 2.1 for analysis of short circuit oscillogram.
2. Representation of Transformer:
The transformer is a static machine which is used to transform electric power from one circuit to another without change in frequency. It may raise or lower the voltage in a circuit but with a corresponding decrease or increase in current. Thus transformer plays a vital role in power systems.
Practically every transformer is provided with taps for ratio control, and thus it is also used for controlling secondary voltage level. Most of the transformers are provided with off-load tap-changers while some of the transformers are provided with on-load tap-changers.
A power transformer with turn ratio K = E1/E2 = N2/N1 can be represented as illustrated in Fig. 2.2.
The induced emf in primary winding E1 is primary applied voltage V1 less primary voltage drop. This voltage causes iron loss current Ie and magnetizing current lm and we can, therefore, represent these two components of no-load current by the current drawn by a non-inductive resistance R0 and pure reactance X0 having the voltage E1 or (V1—primary voltage drop) applied across them, as shown in Fig. 2.2.
Terminal voltage V2 across load is induced emf E2 in secondary winding less voltage drop in secondary winding.
The equivalent circuit can be simplified by transferring the voltage, current and impedance to the primary side. After transferring the secondary voltage, current and impedance to primary side equivalent circuit is reduced to that shown in Fig. 2.3.
The equivalent circuit diagram can further be simplified by transferring the resistance R0 and reactance X0 towards left end, as shown in Fig. 2.4. The error introduced by doing so is very small and can be neglected.
No-load current I0 is hardly 3 to 5 per cent of the full-load rated current, the parallel branch consisting of resistance R0 and reactance X0 can be omitted without introducing any appreciable error in the behaviour of the transformer under loaded condition. The equivalent circuit referred to primary side (neglecting no-load current I0) is shown in Fig. 2.5.
3. Representation of Transmission Lines:
The three phase transmission system is either symmetric (i.e. balanced) in arrangement or if it is un-symmetric, the lines are transposed in order to balance the electrical characteristics of different phases so as to have the same parameters for all the three phases. With such an arrangement the 3-phase network consisting of transmission system and also the distribution system are assumed to be symmetric or balanced.
The performance of a transmission line is governed by its four parameters—series resistance R, and inductance L, shunt capacitance C and the shunt conductance G. The resistance R is due to the fact that every conductor offers opposition to the flow of current. The inductance L is due to the fact that the current carrying conductor is surrounded by the magnetic lines of force.
The capacitance of the line is due to the fact that the conductor carrying current forms a capacitor with the earth which is always at lower potential than the conductor and the air between them forms a dielectric medium. The shunt conductance G is mainly due to flow of leakage currents over the surface of the insulators especially during bad weather.
However, the conductance is normally neglected in the case of transmission line calculations since leakage at normal frequency are negligible.
Generalised Circuit Constants:
In any passive, bilateral and linear network with two input and two output terminals, the input voltage and the input current can be expressed in terms of output voltage and output current. Incidentally, a transmission line is a 4-terminal network; two input terminals where power enters the network and two output terminals where the power leaves the network. Such a circuit is passive as it does not contain any source of emf, linear as its impedance is independent of current flowing and bilateral as its impedance is independent of direction of current flowing.
The input voltage per phase (voltage at sending end) and the input current (current at sending end) of a transmission line can be expressed as –
Vs = A VR+B IR …(2.1)
Is = C VR + D IR …(2.2)
where Vs is the sending-end voltage per phase, Is is the sending-end current, VR is the receiving-end voltage per phase, IR is the receiving-end current, and A, B, C and D (generally complex numbers) are the constants, called the generalized circuit constants of a transmission line.
The values of these constants A, B, C and D depend upon the particular method of solution employed. Once the values of these constants for a particular transmission line are known, the performance of the line can be determined easily.
The values of A, B, C and D can be determined as follows:
With receiving end open circuited-
Receiving-end current IR is zero, therefore, from Eqs. (2.1) and (2.2) –
A = VS/VR …(2.3)
and C = IS/VR …(2.4)
With receiving end short circuited-
Receiving-end voltage VR is zero, therefore, from Eqs. (2.1) and (2.2) –
B = VS/IR …(2.5)
and D = IS/IR …(2.6)
From the above expressions A, B, C and D can be defined as follows:
A can be defined as the ratio of sending-end voltage Vs to receiving-end voltage VR when the line is open circuited on receiving end. A is dimensionless and usually a complex number.
B is defined as the ratio of sending-end voltage Vs to receiving-end current IR when the line is short circuited on receiving end. B is also the complex number and has the unit of ohm.
C is defined as the ratio of sending-end current Is to receiving-end voltage VR when the line is open circuited on the receiving end. C is also the complex number and has the unit of siemen.
D is defined as the ratio of sending end current Is to receiving-end current IR when the line is short circuited on the receiving end. D is dimensionless and usually a complex number –
For passive network AD – BC = 1 …(2.7)
and in symmetrical network A = D …(2.8)