Smart Grid and Renewable Energy
Vol.2 No.2(2011), Article ID:4959,10 pages DOI:10.4236/sgre.2011.22014

Investigation on the Excitation Capacitor for a Wind Pumping Plant Using Induction Generator

Manel Ouali, Mohamed Ben Ali Kamoun, Maher Chaabene


University of Sfax, Unité de Commande de Machines et Réseaux de Puissance CMERP-ENIS, Sfax, Tunisia.


Received February 27th, 2011; revised March 28th, 2011; accepted April 2nd, 2011.

Keywords: SEIG, IM, Centrifugal Pump, Optimal Excitation Capacitor, Wind Rotor Speed


This paper presents a SEIG-IM system using a self excited induction generator driven by wind turbine and supplying an induction motor which is coupled to a centrifugal pump. A method to describe the steady state performance based on nodal analysis is presented. Furthermore, a dynamic analysis and performance characteristics are examined. The proposed methodology is discussed in order to optimize the quantity of the pumped water. Therefore an optimal excitation capacitor for a given wind rotor speed is determined and a suitable operation mode of the system is established.

1. Introduction

Owing the increase of the depletion of conventional energy and the degradation of environmental conditions, the development of suitable isolated power generators driven by renewable energies as solar, geothermal, biomass, hydro and wind energy resources are envisaged to replace conventional resources [1]. Wind energy is considered the most adopted, attractive and practical of all renewable energies as it is available in most sites. It consists of the conversion of kinetic energy into a useful form of energy such as mechanical or electrical [2]. The main application is the wind water pumping using turbines [3]. Many types of generator are used for stand alone power generation in remote and windy areas such as permanent magnet, brushless, synchronous, asynchronous, variable reluctance. Self Excited Induction Generator (SEIG) is the most used for low and medium power generation thanks to its lower cost, easy operation, brushless construction and easy maintenance. It can sustain self excitation and supply electric power while an appropriate value of a capacitor banc is properly connected across its stator terminals [4,5].

SEIG driven by regulated or unregulated turbine and supplying a static or dynamic load has been undertaken by many researchers and is well discussed in literature [6,7]. Investigation in steady state operation are essentially based on the analysis of the per phase equivalent circuit of the three phase induction machine and on either the loop impedance [8] or the nodal admittance concept [9].

As for the dynamic and transient operation, it was treated for no load and for different loads: resistive [10], inductive [11], capacitive [12], induction motor [13], DC load, etc. In this context, this work presents an investigation on a wind pumping plant (SEIG-IM) using a SEIG driven by wind turbine and supplying an Induction Motor (IM) which is coupled to a centrifugal pump. A SEIG-IM performance analysis is developed so as to determine the optimal excitation capacitor which offers maximum pumped water for a given wind rotor speed.

This paper is organized as follows: section two gives the system modeling in static and a dynamic state. Section three shows the computation procedure which provides the optimal excitation capacitor that maximizes the volume of the pumped water. Simulation results are discussed in section four. Finally a conclusion resumes the developed work and its features.

2. System Modeling

2.1. System Configuration

The pumping unit, shown by Figure 1, is composed of a wind turbine which converts wind kinetic energy into electricity by means of SEIG. The produced power is used to supply an induction motor coupled to a centrifugal water pump. As the SEIG requires reactive power for

Figure 1. Wind electric pumping system.

its excitation, a three phase capacitor bank is connected across its stator terminals.

The IM cannot be supplied unless the SEIG stator voltage build up process occurs. For this reason an operating mode switcher selects first the no load condition () until the voltage build up process is accomplished. Subsequently the switcher is turned on () so as to connect the IM to the SEIG.

In order to analyze the performances of self excited induction generator witch supplies an induction motor driving water pump, a system modeling is required. Following, a steady state and dynamic modeling are presented.

2.2. Steady State Modeling

For a given excitation capacitance and wind rotor speed, the steady state of SEIG-IM can be studied unless the stator angular frequency, the IM angular velocity and the magnetizing inductance of the induction generator are known. By examining per phase equivalent circuit (Figure 2) and considering the magnetization characteristics of both induction machines, two equations are determined. A third equation is deduced from the system power balance. All circuit’s parameters except the magnetizing inductances and are assumed to be constant and insensitive to saturation. In addition, core losses and effect harmonics in the machine have been ignored.

The loop equation involving the stator voltage Vs is written as:


where, Y is the total loop admittance giving by:


The expression of admittance capacitive is giving by:


The induction generator admittance is expressed as:


As a consequence of the symmetry of per phase equivalent circuit, the expression of induction motor admittance can be deduced from that of by replacing the index by and by.

Since under steady – state excitation, it implies that or both real and imaginary parts of are equal to zeros:


While the rotational loss torque, the windage losses and damping factor are ignored, the mechanical power of induction motor is equal to that of centrifugal pump, and it can be expressed as a function of

Figure 2. Per phase equivalent circuit of self excited induction generator feeding an induction pump motor.

the equivalent circuit parameters:



The mechanical power of the centrifugal pump is giving by [14]:


where the Tpump is the load torque of the pump and is a coefficient computed by:


is the maximum rated torque and is the maximum rated mechanical motor speed.

The IM rotor current expression can by obtained by the application of voltage divider theorem to the motor circuit model:


The stator voltage can also be expressed by the application of voltage divider theorem to the generator circuit model:



, , ,.

The motor impedances expressions are calculated by the same way as above by replacing the index by and by in expressions of.


The IM magnetizing current is formulated as a function of:


Equations (5) and (6) are nonlinear, where, , , and are unknown variables. Therefore, in order to solve these equations, two of these variables must be specified. Two approaches are considered in literature [8,9,15]:

• To determine the steady state performance of the SEIG-IM system, and must be specified and the equations must be solved for.

• To determine the minimum capacitor value required for the onset of self-excitation process, and must be specified and the equations must be solved for.

The above mentioned equations can be solved by adopting the method of Newton Raphson and calculating the corresponding Jacobian matrix.

Since saturation phenomenon is a significant factor in the processes of self excited induction generator, the variation of magnetic generator inductance should be taken into consideration to find the correct voltage build up. Therefore, at each sampling step, the value of the magnetic generator current is adjusted according to thanks to a spline interpolation of the generator magnetization measured characteristic. Then, is computed using Equations 12 and 13. Finally, is adjusted according to the value thanks to a spline interpolation of the motor magnetizing measured characteristic.

2.3. Dynamic State Modeling

In order to analyze the transient response of SEIG-IM, a dynamic system modeling of the SEIG, the IM and the centrifugal pump is necessary.

2.3.1. Induction Machines Models

Using the generator convention, the voltage equations of the SEIG are expressed as:


where, the fluxes of the SEIG are giving by:


represents the initial residual flux. The generator magnetizing flux is computed by:


In the same way and by adopting the motor convention, the voltage equations and fluxes of IM are respectively given by:



The motor magnetizing flux is computed by:


The stator voltage of both machines is expressed as:


represents the capacity current. It is given by:


To take account of the non linearity of the magnetic material, the magnetization curve of both machines SEIG and IM are implemented in their models: at each sampling step, the value of and are adjusted. The mechanical equation that describes this system is:


where, and represent respectively the electromagnetic torque produced by IM and the load torque of the pump.

2.3.2. Centrifugal Pump Model

Due to mechanical losses, the useful power of centrifugal pump coupled to the shaft of IM is:


where is the mechanical power given by Equation (8), represents the pump efficiency,  is the water density (),is the gravity acceleration (),is the height of rise () and is the flow (). For a pumping period, the volume of pumped water () is computed by:


3. Determination of the Optimal Capacitor

The work aims to maximize the amount of the pumped water by minimizing the excitation capacitance value. Referring to Equations 8, 23 and 24, the volume of the pumped water increases while raising the IM mechanical power (rotor speed). This is ensured by minimizing the excitation capacitance value. Therefore, the criterion defined by “minimize so that is maximum” must be checked using an iterative procedure. First, it is necessary to determine the minimum excitation capacitance value for which the SEIG stator voltage builds upoccurs at IM load operation. Then the maximum excitation capacitance is calculated considering the limitation of the current stator at 120% of its nominal value to avoid destructive over current which causes serious damage to the system equipments. Finally, the following steps are to proceed:

1) For a given wind rotor speed, the excitation capacitance is assumed equal to for which the parameters of the centrifugal pump, the parameters of the two machines and their magnetization characteristics are defined.

2) The Newton Raphson iterative method is used as follows:

a) The initial conditions are fixed and the substitutions of parameters adopted in the resolution of Equations 1-13 are calculated.

b) The vector, the corresponding Jacobian matrix and the vector are determined.

c) The value is estimated using value, magnetic characteristics of both machines and Equations 12-13.

d) Using the new obtained values, repeat steps a, b and c until the error on the vector   is less than a given precision. In this case, the convergence is considered reached.

3) The obtained value of is saved and then the excitation capacitance is incremented by.

4) Repeat steps 2 and 3 until value is equal or greater then.

Thus, for each value of, a database is established in order to study the IM wind rotor variation function of the excitation capacitance. The minimum value of for which becomes maximum and substantially constant is chosen. This value is considered as the optimal capacitor.

4. Results and Discussions

The proposed method of evaluating the steady and dynamic state characteristic of a SEIG-IM is tested referring to a three-phase induction generator:.

This later supplies an induction motor:.

The fixed parameters of both induction machines used in simulation are:

The magnetizing inductances of the both induction machines are measured as a function of the magnetizing currents by performing an open circuit test for which the machines are driven at synchronous speed and a variable voltage source is applied to the stator (Figure 3).

4.1. Steady State Results

The behaviours of the stator voltage, the IG stator current, the IM stator current and the frequency are plotted as a function of the excitation capacitance and the wind rotor speed. The curves network for and different values of (a step of 0.025 is considered) are shown by Figures 4 and 5. It is noticed that the increase in causes a gradually increase of, and. As for the frequencyit increases then remains constant.

A similar curves set of for different values of is shown by Figure 6. For each curve, it is clear that by increasing from to, increases gradually. Nevertheless, since a certain value of, becomes maximum and substantially constant. At this functional point, the obtained value of is considered as optimal.

Figure 3. Magnetization characteristics of both induction.

Figure 4., variation function of excitation capacitance for various values of.

Figure 5 ., variation function of excitation capacitance for various values of.

Figure 6. characteristics for various.

4.2. Dynamic Results

In order to determine the suitable operation mode of SEIG-IM system driven by wind turbine at, the effect of the capacitance value on the stator voltage waveform is studied. Firstly, C is equal to. It is shown that the self excitation process does not occur (Figure 7). This proves the necessity of functioning in two stages. The induction generator is self excited at no load (in first stage), is established and reaches its steady state value; then the motor pump starts operation after a delay of about (in second stage).

Secondly, when is equal to   during the two stages, the stator voltage is established in first stage and decreases rapidly to be eliminated during the second stage (at IM operation) (Figure 8).

Subsequently, is equal to during the two stages (Figures 9(a) and 9(b)).

The self excitation process is well established and maintained but and can reach peaks at no load operation. This phenomenon is especially important when working at high wind rotor speed.

Hence, the value of is fixed at during the first stage and respectively at, and during the second stage (Figures 10

Figure 7. Variation with time at

Figure 8. characteristic at during the two stages of operation.

Figure 9. and characteristics at during the two stages.

Figure 10. Variation with time at different modes of operation.

Figure 11. variation with time at different modes of operation.

Figure 12. and variation with time at adequate operation mode.

Figure 13. and variation with time at adequate operation mode.

Figure 14. and variation with time at adequate operation mode.

(a)-10(c)). The self excitation process is consequently established and maintained.

For the same scenario of capacitance values, the variation as a function of time is presented (Figure 11) and their steady state values during the second phase noted by, , are examined. It shows that and.

This proves the justification of the choice of the value   calculated in the static state steady.

Therefore, the appropriate operation mode of SEIG is that for which is equal to   during the first stage and equal to during the second stage. Figures 12-14 show the variation of, , , , and as a function of time.  The continuous line curves are the curves obtained in dynamic mode and those in broken line are obtained in static mode. Good agreements between the two results are confirmed.

5. Conclusions

In order to optimize the quantity of the pumped water, an effect of an excitation capacitor on the steady state behaviour of the SEIG-IM plant is studied to enable selection of optimal capacitance value for a giving wind rotor. A steady state analysis is examined using the nodal admittance concept and the per phase equivalent circuit which represent the model of this plant. The transient performance of SEIG-IM system is moreover presented and examined to determine the suitable operation mode of this system.


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,: Stator and rotor resistances,

,: Stator and rotor leakage inductances,

: Cyclic stator mutual inductance,

: Excitation capacitor,

: Stator angular frequency,

: Rotor electrical angular speed,

: Mechanical motor speed,

: Induction motor slip,

: Number of poles pairs of induction motor,

: Inertia moment of IM

: Imaginary operator

Superscript denotes the transformed rotor quantities based on the stator.

Indices and stand for stator and rotor respectively, while and stand for induction generator and induction motor respectively.