Symmetrical Components of Power System | Electrical Engineering

In this article we will discuss about:- 1. Introduction to Symmetrical Components 2. The Phase Operator “a” 3. Evaluation 4. Properties 5. Three-Phase Power 6. Physical Significance of Sequence Components 7. Sequence Impedances and Sequence Networks  

Introduction to Symmetrical Components of Power Systems:

The method of symmetrical components is very powerful approach and has simplified the procedure for solving problems on unbalanced polyphase systems. The method of symmetrical components was proposed by CL Fortesque in the year 1918. Although this method is valid for any number of phases, the three-phase system is of main interest.

According to Fortesque theorem, any unbalanced three phase system of currents, voltages or other sinusoidal quantities can be resolved into three balanced systems of phasors which are called symmetrical components of the original unbalanced system.

Such three balanced systems constitute three sequence networks which are solved separately on a single phase basis. Once the problem is solved in terms of symmetrical components, it can be transferred back to the actual circuit condition by superposition or phasor additions of these quantities (currents or voltages) easily.

Symmetrical Components of 3-Phase Systems:

The symmetrical components differ in phase sequence, that is, the order in which the phase quantities go through a maximum. There may be a positive phase sequence, a negative phase sequence and a zero or uniphase sequence. Thus the balanced sets of components can be given as positive sequence component, negative sequence component and zero or uniphase sequence component.

The positive phase sequence system is that system in which the phase or line currents or voltages attain a maximum in the same cyclic order as those in a normal supply, e.g., assuming the conventional counter-clockwise rotation, then the positive phase sequence phasors are those shown in Fig. 3.1 (a). A balanced system corresponding to normal conditions contains a positive phase sequence only. It is also the condition for a 3-phase fault.

The positive phase-sequence components are marked by subscript 1. The three phasors of the positive phase sequence system are of equal magnitude, spaced 120° apart and may be represented in complex form as follows:

 

 

 

The complex number e^J2/3 in above expression is called the operator of the three phase system and is denoted by letter ‘a’. Multiplication of a phasor by a or e^j2/3 turns it through 120° in counter-clockwise direction. Multiplying a phasor by a² or e^-j2/3 turns it through 240° in the counter-clockwise direction and the position of the phasor remains unchanged if multiplied by a³.

The negative phase sequence system is that system in which the phasors still rotate anti­clockwise but attain a maximum in reverse order i.e. A –  C –  B as shown in Fig 3.1 (b). This sequence only arises under conditions of unbalance as when an unsymmetrical fault occurs.

Such faults contain also the positive sequence system and, in the case of faults to earth, a zero phase sequence, explained as following:

The negative phase-sequence components are marked by subscript 2. The three phasors of the negative phase-sequence, alike positive phase-sequence are of equal magnitude and spaced 120° apart.

They are represented in complex form as follows:

 

 

 

The zero phase sequence system is a single phase phasor system combining three equal phasors in phase, as illustrated in Fig. 3.1 (c), and represents the residual current or voltage present under fault conditions on a 3-phase system with a fourth wire or earth return present.

Clearly the zero phase sequence embraces the ground, therefore, in addition to the three line wires and represents a fault condition to ground or to a fourth wire if present. Its presence arise only where fault to earth currents can return to the system via the star point of that system or via an artificial neutral point provided to earth a delta system. In an earth fault, positive and negative phase sequences are also present.

The zero phase-sequence components are marked by subscript 0. The three phasors of zero or uniphase-sequence system are expressed as – 

A₀ = B₀ = C₀

Figure 3.1 illustrates the phasor diagram for all the three phase sequences of the symmetrical components.

The Phase Operator “a”:

For convenience in notation and manipulation a phasor operator is introduced. Through usage it has come to be known as the phasor a and is defined as a = – 0.5 + j 0.866 = e^j2/3 It indicates that the phasor a has unit length and is oriented 120° (or 2/3 radians) in a positive (counter-clockwise) direction from the reference axis.

A phasor operated upon by a is not changed in magnitude but is simply rotated in position 120° in the forward direction. For example V’ = a V is a phasor having the same length as the phasor V, but rotated 120° forward from the phasor V. This relationship is illustrated in Fig. 3.2. The square of the phasor a is another unit phasor oriented 120° in a negative (clockwise) direction from the reference axis, or oriented 240° forward in positive direction.

 

 

 

As illustrated in Fig. 3.2, the resultant a² operating on a phasor V is the phasor V” having the same length as V, but located 120° in a clockwise direction from V.

The three phasors (1 + j 0), a² and a (taken in this order) form a balanced, symmetrical set of phasors of positive phase-sequence rotation, since the phasors are of equal length, displaced by equal angles of 120° from each other, and cross the reference line in the order 1, a² and a (following the usual convention of counter-clockwise rotation for the phasor diagram).

The phasor 1, a and a² (taken in this order) form a balanced, symmetrical, set of phasors of negative phase-sequence, since the phasors do not cross the reference line in the order named, keeping the same convention of counter-clockwise rotation, but third name following the first etc.

Fundamental properties of the phasors are given below and are shown on phasor (Fig. 3.3):

a = – 0.5 + j 0.866 = e ^j2/3

a² = – 0.5 – j 0.866 = e^-j2/3

a³ = 1 + j 0 = e ^{j 0}

a⁴ = e^4(j2/3) = a

a⁵ = e^5(j2/3) = a² and so on

Thus a is numerically equivalent to the cube root of unity.

a + a² + 1 = (– 0.5 + j 0.866) + (– 0.5 – j 0.866) + 1 = 0

a + a² =  – 1 + j 0 = e ^j

a² – a = ( – 0.5 – j 0.866) – ( – 0.5 + j 0.866

= 0  – j 1.732 = √3 e ^{j 3/2}

a – a² = 0 j 1.732 = √3 e ^j/2

a – 1 = √3 e ^{j 5/6}

1 – a = √3 e ^{j 11/6}

a² – 1 = √3 e ^j7/6

1 – a² = √3 e ^{j /6}

Evaluation of the Symmetrical Components:

Let us express the phasor of an unbalanced three-phasor system in terms of their symmetrical components –

A = A₁ + A₂ + A₀                                                   …(i)

B = B₁ + B₂ + B₀                                                   …(ii)

C = C₁ + C₂ + C₀                                                  …(iii)

Expressing all the phasors of the symmetrical system in the  above equations in terms of A₁, A₂ and A₀ we get –

A = A₁ + A₂ + A₀                                                 …(iv)

B = a² A₁ + aA₂ + A₀                                           …(v)             since B₁  = a² A₁ and B₂ = a A₂

and C = a A₁ + a² A₂ +A₀                                  …(vi)             since C₁ = a A₁ and C₂ = a² A₂

Adding equation (iv), (v) and (vi) we get –

A + B + C = A₁(1 + a² + a) + A₂ (1 + a + a²) + 3 A₀

or A₀ = A + B + C/3                                             …(vii)             since 1 + a + a² = 0

Multiplying equation (v) by a and equation (vi) by a² we get –

a B = a³ A₁ + a² A₂ + a A₀                                …(viii)

a² C = a³ A₁ + a⁴ A₂ + a² A₀                             …(ix)

Adding equations (iv), (viii) and (ix) we get –

A + a B + a² C = A₁ (1 + a³ + a³) + A₂ (1 + a² + a⁴) + A₀ (1 + a + a²) = 3 A₁ + A₂ (1 + a² + a) + A₀ (1 + a + a²) since a⁴ = a and a³ = 1

or A₁ = A + a B + a² C/3                                     …(x)      since 1 + a + a² = 0

Multiplying equation (v) by a² and equation (vi) by a we get –

a²B = a⁴ A₁ + a³ A₂ + a² A₀                              …(xi)

a C = a² A₁ + a³ A₂ + a A₀                                …(xii)

Adding equation (iv), (xi) and (xii) we get –

A + a² B + a C = A₁ (1 + a⁴ + a²) + A₂ (1 + a³ + a³) + A₀ (1 + a² + a) = A₁ (1 + a + a²) 3 A₂ + A₀ (1 + a + a²) since a⁴ = a and a³ = 1

or A₂ = A + a² B + a C/3                                    …(xiii)      since 1 + a + a² = 0

In matrix form we can express as –

 

 

 

 

 

 

Resolution of unbalanced phasor system into a symmetrical component is essentially a mathematical technique used for solving three phase circuit problems under unbalanced conditions which may be due to unbalanced loads, unequal phase voltages, single and two phase short circuits, line-wire-opens etc.

Some Properties of 3-Phase Circuits with Regard to Symmetrical Components of Currents and Voltages:

In a 3-phase, 4-wire system, current in the neutral wire is equal to the sum of the line currents and is consequently equal to three times the zero phase sequence component of currents.

The sum of the line currents in a 3-phase, 3-wire delta-or star-connected system is known to be zero. It follows from equation (vii) that line currents of such a system do not carry any zero-phase, component.

The above applies equally to any 3-phase system in which the sum of line voltages is zero. It means that a set of unbalanced 3-phase line-to-line voltages may be represented by a positive system and a negative system of balanced voltages.

The degree of unbalance is given by di-symmetry coefficient in terms of the ratio of the negative phase sequence component to the positive phase sequence component i.e. –

 

 

The system of line voltages is considered to be balanced when this coefficient is less than 5 per cent.

The absence of current in one or two phases under unbalanced condition means the sum of the three symmetrical components of the currents in these phases is equal to zero.

The above statement will be better understood from examination of the following examples. Let in a system the phases B and C be open (Fig. 3.4).

 

Then,

I_B = I_c = 0 

In accordance with equations (x), (xiii) and (vii) the symmetrical components of the system are as follows:

(a) The component of the positive phase-sequence current,

 

 

(b) The component of the negative phase-sequence current,

 

 

(c) The component of the zero phase-sequence current,

 

 

 

 

The geometric addition of symmetrical components of positive, negative and zero phase gives –

 

 

 

 

 

 

 

 

Three-Phase Power in Terms of Symmetrical Components:

The power consumed in a 3-phase circuit can be determined directly from the symmetrical components provided the currents and voltages are given.

Total complex power in a 3-phase circuit is given as –

 

 

where V_a, V_b and V_c are the voltages to neutral and l_a, I_b and I_c are the currents flowing into the circuit in the three lines, the 3-phase circuit being assumed to be star-connected. The superscript T stands for transposition.

 

 

 

 

 

 

Now the dot product of two phasors does not change when both are rotated through the same angel.

For example a² V_a1 . a²I_a1 = V_a1I_a1; aV_a1 . aI_a1 = V_a1 I_a1; a² V_a1 . aI_a2 = aV_a1 . I_a2. and so on.

Thus the expression for S becomes – 

 

 

 

 

 

 

 

or P = 3 [V _a1 I _a1 cos θ₁ + V_a2 I_a2 cos θ₂ + V_a0 I_a0 cos θ₀]            …[3.4(a)]

The same power expression can be derived very easily using matrix manipulations.

 

 

 

 

 

Now from Eq. (3.1) –

 

 

 

 

 

 

 

 

 

 

 

Now substituting for phase current the corresponding symmetrical components –

 

 

 

 

 

 

 

 

 

 

 

 

 

The above expression for power does not have any cross-terms (such as V_a1, V_a2, V_a1 V_a0 etc.) because there is no mutual coupling between the three sequence networks. The factor of 3 arises due to the total number of 9 components (3 for each phase). It is seen that three phase power is equal to the sum of powers of the three symmetrical components.

Physical Significance of Sequence Components:

A physical significance of sequence voltages can be well understood by considering the fields developed on application of such sequence voltages to the stator of a 3-phase ac machine (a synchronous machine or an induction machine). If a set of positive sequence voltages is applied to the stator winding of a 3-phase induction motor, a magnetic field revolving in a certain direction will be developed.

Now if two of the three phase supply leads to the stator winding are interchanged, the direction of revolution of the field developed would be reversed. It is seen that for this condition the relative phase positions of the voltages applied to the stator of the motor are the same as for the negative sequence set.

Thus the field developed by the set of negative sequence voltages will rotate in a direction opposite to that of the field developed by the set of positive sequence voltages. Phase sequence should not be confused with the rotation of phasors.

For both sets of positive and negative sequence voltages, the standard convention of counter-clockwise rotation is followed. The zero sequence voltages are single phase voltages and, therefore, they give rise to an alternating field in space.

The positive and the negative sequence sets are balanced ones i.e., if only positive and/or negative sequence currents are flowing, the phasor sum of each will be zero and there will be no residual current. However, the zero sequence components of currents in the three phases are equal in magnitude and are in phase and therefore, the residual current will be three times the zero sequence current of one phase. In case of an earth fault, the positive and the negative sequence currents balance between themselves but the zero sequence currents flow through the ground and ground wires.

Sequence Impedances and Sequence Networks:

So far we have discussed the symmetrical components for the currents, voltages and power. Now we will discuss something about the sequence impedances of the power system. The sequence impedances of a network or a component of power system are the positive-, negative-, and zero-sequence impedances.

They are defined as below:

The impedance of the network offered to the flow of positive sequence currents is called the positive- sequence impedance. Similarly if only negative sequence currents flow the impedance of the network offered to these currents is called the negative-sequence impedance. Also the impedance offered to zero sequence currents is called the zero-sequence impedance of the network.

If Z_a, Z_b and Z_c are the impedances of the load between phases a, b and c to neutral n, then sequence impedances are given as –

 

 

 

 

 

For a 3-phase, symmetrical static circuit without internal voltages like transformers and transmission lines, the impedances offered to the currents of any sequence are the same in the three phases; also the currents of a particular sequence will cause voltage drop of the same sequence or a voltage of a particular sequence will give rise to current of the same sequence only, which means there is no mutual coupling between the sequence networks.

Since in case of a static device, the sequence has no significance, the positive-and negative-sequence impedances are equal; the zero-sequence impedance which includes impedance of the return path through the ground is usually different from the positive- and negative- sequence impedances. The impedances offered by rotating machine to positive-sequence components of currents, differ from those offered to negative-sequence components of currents.

The single phase equivalent circuit composed of the impedances to current of any one sequence only is called the sequence network for that particular sequence. Hence corresponding to positive-, negative- and zero-sequence currents we have positive-, negative- and zero-sequence networks.

Thus, for every power system three sequence networks can be formed and these sequence networks, carrying currents l_a1, I_a2 and I_a0 are then interconnected in different ways to represent different unbalanced fault conditions. The sequence currents and voltages during the fault are then calculated from which actual fault currents and voltages can be determined.

The positive-sequence network is considered in the analysis of symmetrical faults. Positive-sequence network is the same as impedance or reactance diagram.

The negative-sequence network is in general quite similar to the positive-sequence network but differs in the following aspects:

(i) Normally there are no negative sequence emf sources.

(ii) Negative-sequence impedances of rotating machines are generally different from their positive sequence impedances.

The phase displacement of transformer banks for negative sequence is of opposite sign to that of positive sequence.

The zero-sequence network likewise will be free of internal voltages, the flow of current being caused by the voltage at the fault point. The impedances to zero-sequence currents are very frequently different from the positive-or negative-sequence currents.

Zero-sequence reactance of transmission lines is higher than that for positive sequence. The impedances of transformers and generators will depend upon the type of connections [delta or star (grounded or isolated)].