CHAPTER 5 STEADY-STATE ANALYSIS OF THREE-PHASE SELF-EXCITED INDUCTION GENERATORS

6 CHAPTER 5 STEADY-STATE ANALYSIS OF THREE-PHASE SELF-EXCITED INDUCTION GENERATORS 5.. INTRODUCTION The steady-state analysis of six-phase SEIG has been discussed in the previous chapters. In this chapter, the proposed method is applied for the steady-state analysis of three-phase SEIG in order to investigate the validity of the proposed method. The analytical technique suggested by Murthy et al. [22] involves manual separation of real and imaginary parts of the equivalent loop impedance. Then, the unknown quantities such as magnetizing reactance (X M ) and frequency (F) are computed by solving the real and imaginary polynomials by Newton-Raphson method. The core loss component is considered in the analysis by Malik et al. [25] and they used loop impedance model along with Newton-Raphson method to solve the unknown quantities X M and F. The capacitance requirements to maintain constant terminal voltage under varying load and speed were calculated by Singh et al [3] using loop impedance model. They used Newton-Raphson method for solving the unknown quantities like capacitive reactance (X C ) and frequency (F). The improvements in the performance of SEIG through series compensation using loop impedance model along with Newton-Raphson method was presented by Singh et al. [38]. A nodal admittance technique is used by Quazene et al. [39] to obtain a nodal equation. Then, the nodal equation is manually separated it into its real and imaginary

6 parts so as to solve them first for F and then for X M by substituting the value of F. The simulated annealing approach is used by Singh et al. [2] to solve voltage regulation optimization problem. The solution should satisfy the equality constraints on loop impendence and inequality constraints on generator and motor parameters without violating the specified upper and lower limits of shunt and series capacitances. The loop impedance approach on the per phase approximated equivalent circuit of the SEIG with static VAr compensator was applied by Tarek Ahmed et al. [3]. By equating the real and imaginary parts to zero manually, two nonlinear simultaneous equations of the magnetizing reactance X M are obtained. A th degree polynomial equation is derived by using these two equations, and the frequency is calculated. It is observed that the disadvantage of the SEIG is its poor voltage regulation characteristics under varying load conditions. To improve the performance of a threephase SEIG, capacitive series compensation is employed as shown in Fig. 5.. Both short shunt and long shunt configurations of capacitive series compensation of the SEIG are also considered in this chapter. Fig. 5.. Schematic diagram of three-phase SEIG with series compensation.

62 In order to maintain good voltage regulation, capacitive VAr has to be increased with load [22, 25, 3]. To achieve this aim, several voltage regulating schemes [6, 25, 84,, 6] have been reported. But, these schemes mostly utilize variable inductor or saturable core reactor or switched capacitor based closed loop schemes or semiconductor switches. The drawbacks of such schemes are complicated control circuit design, complex system configuration, and operational problems like harmonics and switching transients, associated with voltage regulators. This makes recommendation of induction machine to autonomous power generation ineffective. To improve voltage regulation of SEIG, additional series capacitance can be included [29, 43] to provide additional VAr with load. Long shunt compensation was proposed by Chan [43, 89] to examine the effect of parameter k on the performance of an induction generator under different loading conditions. The parameter k is defined as the ratio of the series capacitive reactance to shunt capacitive reactance. From the results shown in [43], it is observed that the long shunt compensation can be employed to maintain load voltage under various load currents. The steady-state performance of short shunt SEIG was presented by Shridhar et al. [36] and the comparative results of both long shunt and short shunt configurations on steady-state voltage variation of a SEIG were presented by Wang et al. []. The sensitivity of the performance of the SEIG was presented by Singh et al. [38] to select suitable values of the capacitances for a desired voltage regulation. Most of the methods available in literature [22, 25, 29, 3, 36, 38, 43, 89,, 2, 3] need manual separation of real and imaginary component of complex

63 impedance/admittance and applications of Newton-Raphson method to compute steady-state performance evaluation of a SEIG with and without series compensation. Also, the mathematical model is found to be different for different types of loads, and capacitor configurations. Moreover, change in load and capacitance configurations results in the change in the coefficients of mathematical model. Also, the model becomes more complicated if the core loss of the machine is considered. Singaravelu et al. [5] suggested a steady-state model of SEIG using graph theory approach. This mathematical model reduces the manual separation of real and imaginary parts of equivalent loop impedance or nodal admittance. But the graph theory based approach also involved the formation of graph, tree, co-tree, tie-set or cut-set, etc. which makes the modeling complicated. Therefore, a novel mathematical model is proposed [45] using nodal admittance method based on inspection. The proposed mathematical model using inspection method completely avoids the tedious manual work involved in separating the real and imaginary components of the complex impedance/admittance of the equivalent circuit. The proposed model is a simplified approach in which the nodal admittance matrix can be formed directly from the equivalent circuit of SEIG rather than deriving it using graph theory approach. Since the model developed using inspection method results in matrix form any element of the equivalent circuit can be easily included or eliminated. The proposed mathematical model is more generalized such that the same model can be used to obtain the steady-state performance of SEIG with/without series compensation, resistive/resistive-inductive loads and any combination of unknown parameters.

64 To carry out the steady-state analysis of three-phase SEIG, a genetic algorithm approach is used instead of Newton-Raphson method [22] or nonlinear optimization method [88, 7, 9]. The major difficulty in applying the Newton-Raphson method is the need to compute the Jacobian matrix which involves partial differentiation and finding inverse of Jacobian matrix to obtain the solution. Unconstrained nonlinear optimization techniques such as Hooke and Jeeves method (pattern search method) [7-9] and 88] generally involve more number of function evaluations which may lie in the range 3 to 45 [7] and 4 to 35 [8, 9] with the practical range of load impendence. These methods make use of the gradient vector directly or indirectly in finding the solution. Further, these methods demand very high mathematical ability and higher computational time. Genetic algorithm is used to carry out the steady-state analysis of three-phase SEIG because, it is simple and robust. The analytical results are validated by comparing the experimental results. Further, genetic algorithm provides a fruitful direction with good prospects for having global solutions even when start with bad initial conditions. The developed genetic algorithm uses general mathematical model of SEIG and the same model can be implemented for any type of load and any combinations of unknown quantities such as magnetizing reactance (X M ) and frequency (F) or capacitive reactance (X C ) and frequency (F). 5.2. PROPOSED MATHEMATICAL MODELING A mathematical model using nodal admittance method by inspection is proposed for the steady-state analysis of three-phase SEIG from the equivalent circuit

65 of generator. The developed model results in a matrix form which is convenient for computer solution irrespective of any combinations of unknowns. Fig.5.2 shows steady-state equivalent circuit of a three-phase SEIG. Fig. 5.2. Steady-state equivalent circuit of three-phase SEIG. The various elements of equivalent circuit are given below. Y R = / {R r / (F - j X lr }; Y M = / {j X M }; Y S = / {R S / F + j X ls - j X clo / F 2 }; Y C = / {- j X C / F 2 }; Y L = / {R L / F + j X L - j X csh / F 2 }; The matrix equation based on nodal admittance method for the equivalent circuit can be expressed as [Y] [V] = [I S ] (5.) where [V] is the node voltage matrix, [I S ] is the source current matrix, and [Y] is the nodal admittance matrix. The [Y] matrix can be formulated directly from the equivalent circuit (Fig.5.2) by inspection [47] as [Y] = Y R + Y M + Y S - Y S - Y S Y S + Y C + Y L (5.2)

66 where Y ii Y ij = - th node th node and j th node Since [Y] is symmetric, Y ji = Y ij. If there is no branch between any two nodes, then the corresponding element in the matrix is zero. Since, the equivalent circuit does not contain any current sources, [I S ] = [] and hence Eq. (5.) is reduced as [Y] [V] = (5.3)! " # $ Eq. (5.3), [Y] should be a singular matrix i.e., determinant of [Y] =. It implies that both the real and the imaginary components of det [Y] should be independently zero. Therefore to obtain required parameter which results det [Y] =, genetic algorithm based approach is implemented which is discussed in section 2.3 of Chapter 2. 5.3. EXPERIMENTAL SETUP AND MACHINE PARAMETERS A three-phase induction machine is selected for conducting the experiment. The view of experimental setup is shown in Fig. 5.3. Fig.5.3. View of the laboratory experimental setup.

67 The machine details and parameters of the induction generator are: Rating=3.7kW, Number of poles=4, Delta connected, Line voltage = 23Volts (Base Voltage), Line current = 2.5Ampere, Base Current = 7.23Ampere, Base Impedance=3.87%, Base Power=.66 kw, Stator resistance, R S = 2.68%, Rotor resistance, R r = 2.458%, and Leakage reactance, X ls = X lr = 3.837%. The variation of air gap voltage (V g /F) with magnetizing reactance (X M ) is non linear due to magnetic saturation. To simplify the analysis, this variation under saturated region can be linearized and it can be expressed in the form of equation (in p.u) as V g /F =.69 & 34 X M (5.4) 5.4. STEADY-STATE PERFORMANCE ANALYSIS WITHOUT SERIES COMPENSATION To obtain the steady-state performance of three-phase SEIG without series compensation, the long shunt component- j X clo /F 2 from Y S term and the short shunt component - j X csh /F 2 from Y L term are eliminated. Then, Eq. (5.3) is solved to find the unknown quantities X M and F using proposed genetic algorithm discussed in a).4.2 V T = V L V L (p.u).6.4 Speed = p.u C=33 µf C=38 µf C=43 µf ' ( ), x - Expt..5.5 2

68 b).7.6 Speed = p.u I S (p.u).5.4.3. C=33 µf C=38 µf C=43 µf ' ( ), x - Expt..5.5 2 c).7.6 Speed = p.u I L (p.u).5.4.3 C=33 µf C=38 µf C=43 µf. * +,, x - Expt..5.5 2 Fig. 5.4. Variation of: (a) Load voltage, (b) Stator current and (c) Load current of three-phase SEIG with output power under unity power factor load. section 2.3. Fig. 5.4 shows the variation of load voltage, stator current and load current with output power for different fixed capacitances at constant per unit speed =.p.u. From Fig. 5.4(a), it is observed that the load voltage drops with increase in load. The maximum output power is considerably higher for higher value of C.

69 Figs. 5.4(b) and 5.4(c) indicate the variation of stator current and load current with output power for different fixed capacitances. It is observed that the currents are within their rated values. A close agreement between calculated and measured values can be seen. 5.5. STEADY-STATE PERFORMANCE ANALYSIS WITH SERIES COMPENSATION The variable capacitor excitation scheme gives good performance in terms of good terminal voltage. However, it makes the three-phase SEIG system costly and complex. To make the three-phase SEIG system simple and cost-effective, an investigation is carried out as part of this research work, to study the effect of series capacitor excitation on regulating the load terminal voltage. Short shunt and long shunt configurations of capacitive series compensation for the three-phase SEIG are analyzed for the selection of capacitive elements in this section using the proposed method. The steady-state analysis of a three-phase SEIG without series compensation has been presented in section 5.4. But, it is observed that the uncompensated SEIG suffers from poor voltage regulation. In order to improve the voltage regulation, short shunt and long shunt configurations of capacitive series compensation are considered in this section. Inclusion of series capacitance to provide additional VAr is one of the attractive options to improve voltage regulations of SEIG. Therefore the steady-state analysis of short shunt and long shunt configurations is presented in this section.the proposed mathematical model which is discussed in section 5.2 along with genetic

7 algorithm (section 2.3) can be used to obtain the steady-state performance of SEIG with series compensation. 5.5.. Steady-State Performance Analysis of SEIG with Short Shunt Configuration To obtain the model for steady-state analysis of three-phase SEIG for short shunt configuration (fixed shunt capcitance C and different constant values of C sh are a).2. V T (p.u).9 C=3 µf Speed= p.u Csh=8 µf Csh=22 µf Csh=26 µf.5.5 2 2.5 3 b). V L (p.u).9 C=3 µf Speed= p.u.5.5 2 2.5 3 Csh=8 µf Csh=22 µf Csh=26 µf

7 c) I S (p.u).2.6.4 C=3 µf Speed= p.u Csh=8 µf Csh=22 µf Csh=26 µf.5.5 2 2.5 3 d) C= 3 µf Speed= p.u I L (p.u).6.4 Csh=8 µf Csh=22 µf Csh=26 µf.5.5 2 2.5 3 Fig. 5.5. Effect of series capacitance on the characteristics of the short shunt SEIG. considered (Fig.5.)), Eq. (5.3) can be used after eliminating the long shunt component -jx clo /F 2 from admittance Y S. Solution is obtained by genetic algorithm process proposed in section 2.3 using matrix [Y] to predict the unknown quantities X M and F of the equivalent circuit.

72 Figs. 5.5 show the effect of series capacitance on the characteristics of the short shunt SEIG under fixed shunt excitation capacitance C of 3 F and rotor speed of pu. Three different capacitance values, 8 F, 22 F and 26 F for short shunt configuration were employed to study the steady-state performance of the short shunt SEIG. Fig. 5.5(a) shows the terminal voltage versus output power for short shunt configuration. The terminal voltage decreases under lower output powers and increases under higher output powers. Fig. 5.5(b) shows the load voltage versus output power for short shunt configuration. Load voltage variations are less from no load to rated load conditions. Figs. 5.5(c) and 5.5(d) show the characteristics of stator current and load current versus output power for short shunt configuration respectively. The currents are well below the rated value at rated output power..2 V T V T, V L, I S and I L (p.u).6.4 C =3 µf C sh =22 µf Speed= p.u V L I S I L.5.5 2 2.5 3 Fig. 5.6. Typical characteristics of the short shunt SEIG (UPF). Characteristics of the short shunt SEIG feeding power to resistive load are shown in Fig. 5.6. Shunt capacitance C is selected as 3 F as it results in a no-load

73 terminal voltage of.pu (base voltage is 23V). It is observed from Fig. 5.6 that the variation of the load voltage with output power is marginal. Further the inclusion of series capacitance results in higher overload capability of the system. Fig. 5.7 shows the predicted characteristics of the short shunt SEIG with an inductive load power factor of. The system observes voltage sag at light load as shown in Fig. 5.7 nevertheless, the system displays self-regulation when loaded to higher load. V T,,V L,I S and I L (p.u).4.2.6.4 I S.5.5 2 2.5 3 Fig. 5.7. Typical characteristics of the short shunt SEIG (PF lag). 5.5.2. Steady-State Performance Analysis of SEIG with Long Shunt Configuration At this juncture, it is worth comparing the performance of the short shunt SEIG with that of the long shunt SEIG. To obtain the model for steady-state analysis of three-phase SEIG with long shunt configuration (fixed capcitance C and different constant values of C lo are considered (Fig. 5.)), Eq. (5.3) can be used after eliminating the short shunt component & jx csh /F 2 from admittance Y L. The solution can V L V T I L C=3 µf C sh =22 µf Speed= p.u pf= lag

74 be obtained by genetic algorithm proposed in section 2.3 using matrix [Y] to compute the unknown quantities X M and F of the equivalent circuit. Fig. 5.8 shows the predicted characteristics of long shunt configuration under the fixed value of shunt capacitance (C=44 F) and rotor speed of pu. Three different capacitance values, 24 F, 27 F and 3 F for long shunt configuration were a).3.2 V T (p.u). C=44 µf Speed= p.u.5.5 2 2.5 3 Clo=24 µf Clo=27 µf Clo=3 µf. b) V L (p.u).9 C= 44 µf Speed= p.u Clo=24 µf Clo=27 µf Clo=3 µf.5.5 2 2.5 3

75 c) I S (p.u).2.6.4 C= 44 µf Speed= p.u Clo=24 µf Clo=27 µf Clo=3 µf.5.5 2 2.5 3 d) I L (p.u).6.4 C= 44 µf Speed= p.u Clo=24 µf Clo=27 µf Clo=3 µf.5.5 2 2.5 3 Fig. 5.8. Effect of series capacitance on the characteristics of the long shunt SEIG. employed to study the steady-state performance of the long shunt SEIG. Fig. 5.8(a) shows the terminal voltage versus output power for long shunt configuration. When the output power is below.4p.u, the smaller the long shunt capacitance value, the lower the terminal voltage is. When the output power is greater

76 than.4p.u, the smaller the long shunt capacitance value, the higher is the terminal voltage. Fig. 5.8(b) shows the load voltage versus output power for long shunt configuration. It is observed that the voltage variations are low for different values of series capacitance at rated load conditions. Figs. 5.8(c) and 5.8(d) show the characteristics of stator current and load current versus output power for long shunt configuration respectively. From this, it is observed that the currents are well below.4.2 V T V T, V L, I S and I L (p.u).6.4 V L I S I L C =44 µf C lo =27 µf Speed= p.u.5.5 2 2.5 3 Fig. 5.9. Typical characteristics of the long shunt SEIG (UPF). the rated value at rated output power. The predicted characteristics of the long shunt SEIG feeding power to resistive load are shown in Fig. 5.9. The shunt capacitance C and long shunt capacitance C lo are selected as 44 F and 27 F respectively as it results in a no-load terminal voltage of.6pu (base voltage is 23V). It is observed from Fig. 5.9 that the variation of the

77 load voltage with output power is marginal. Further the inclusion of series capacitance results in higher overload capability of the system..4 V T, V L, I S and I L (p.u).2.6.4 V T V L.5.5 2 2.5 3 I S I L C=44 µf C lo =27 µf Speed= p.u pf= lag Fig. 5.. Typical characteristics of the long shunt SEIG (PF lag). Fig. 5. shows the predicted characteristics of the long shunt SEIG with an inductive load power factor of. The system observes voltage sag at light load as shown in Fig. 5. nevertheless, the system displays self-regulation when loaded to higher load. 5.6. CONCLUSION Generalized mathematical model using nodal admittance method based on inspection and genetic algorithm based computation are proposed for the steady-state performance of three-phase self-excited induction generator. Experimental and predicted performances agree closely which validates the proposed method and solution technology. The proposed mathematical model is flexible so that the same

78 model which is used for uncompensated SEIG can be extended for compensated SEIG. Both short shunt and long shunt configurations are considered to improve the performance of SEIG. It is observed that the value of total capacitance, i.e. sum of series and shunt capacitances and VAr requirement to achieve the desired performance, is much higher in the case of long shunt configuration. The excellent performance of the short shunt SEIG system demonstrates the usefulness of the series capacitance in improving the performance of the three-phase self-excited induction generator. Even though three-phase generation can be used for isolated operations, rural electrification is often based on single-phase generation and distribution systems. Therefore, the proposed method has been extended in Chapter 6 for the steady-state analysis of single-phase self-excited induction generators with and without series compensation.