Evaluation of a solar powered water pumping system in Mutomo, Kenya: Comparison between a submersible induction motor and a PMSM system

Juni 2019

Evaluation of a solar powered

water pumping system in Mutomo, Kenya

Comparison between a submersible induction motor and a PMSM system

Gabriel Båverman

Edris Tavoosi

Teknisk- naturvetenskaplig fakultet UTH-enheten

Besöksadress:

Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0

Postadress:

Box 536 751 21 Uppsala

Telefon:

018 – 471 30 03

Telefax:

018 – 471 30 00

Hemsida:

http://www.teknat.uu.se/student

Evaluation of a solar powered water pumping system in Mutomo, Kenya

Gabriel Båverman, Edris Tavoosi

An existing solar-powered water pumping system located in Mutomo, Kenya has been evaluated in this paper. The requirement for this system is to produce a minimum of 25m³ water per day throughout the year.The aim of this thesis is to investigate the performance of the currently installed system and find a suitable replacement in terms of efficiency and economic viability. In order to acquire the necessary knowledge for this project, a literature study was carried out to analyse the research within the area.

Three simulation models were created which all include an electric motor driven by a photovoltaic array and are connected to a submersible groundwater pump. All models utilise space vector pulse width modulation. One model of an induction motor that represents the currently installed system, one induction motor that delivers a minimum of 25 m³ water per day, and one model of a permanent magnet synchronous motor for comparison. Simulations using weather data, representing an average day for each month of the year were carried out. It was shown that the currently installed system does not fulfil the requirement of producing 25 m³ water per day, and in addition produces a significant amount of energy that can not be utilised. It was also shown that the efficiency of the permanent magnet synchronous motor was superior to the induction motors. In order to compare the systems in terms of economic viability, price quotations from world leading manufacturers were acquired. The results of the economic comparison show that the superior efficiency of the permanent magnet synchronous motor was not enough to compensate for the higher investment cost.

ISSN: 1650-8300, UPTEC ES19 012 Examinator: Petra Jönsson

Ämnesgranskare: Rafael Waters Handledare: Kenneth Mårtensson

Sammanfattning

Ett vattenpumpsystem drivet av solpaneler bel¨aget i Mutomo, Kenya har utv¨arderats i denna rapport. Kravet f¨or systemet ¨ar att producera minimum 25 m³ vatten per dag ¨over hela ˚aret. Detta motsvarar ca 4 liter vatten per dag f¨or 6000 personer som bor i omr˚adet. Syftet med denna rapport ¨ar att utv¨ardera det befintliga systemets prestanda samt ta fram ett nytt l¨ampligt system med avseende p˚a ef- fektivitet och ekonomisk l¨onsamhet. En simuleringsmodell av en induktionsmotor driven av solpaneler, kopplad till en dr¨ankbar grundvattenpump, som represen- terar det befintliga systemet har tagits fram. V¨aderdata fr˚an Mutomo har anv¨ants i simuleringarna, d¨ar en genomsnittsdag f¨or varje m˚anad av ˚aret har tagits fram.

Resultaten visar att det befintliga systemet inte uppn˚ar det satta kravet samt att systemet ger upphov till en signifikant m¨angd outnyttjad energi. En modell av en induktionsmotor som uppn˚ar kravet p˚a 25 m³ vatten per dag har d¨arf¨or tagits fram och en modell av en permanent magnet synkronmotor har tagits fram i j¨amf¨orande syfte.

F¨or att f¨orv¨arva den n¨odv¨andiga kunskapen inf¨or detta projekt genomf¨ordes en litteraturstudie av forskningsomr˚adet som underlag f¨or den teori som kr¨avs f¨or att modellera systemen. Litteraturstudien visade att det inte ¨ar praktiskt att anv¨anda sig av en traditionell DC motor f¨or djupa borrh˚al och har d¨arf¨or exkluderats fr˚an j¨amf¨orelsen. Matematiska parametrar i det station¨ara trefas-referenssystemet in- neh˚aller tidsberoende induktanser, vilket leder till h¨ogre komplexitet vid matema- tiska ber¨akningar. Vid modelleringen av induktionsmotorn och synkronmotorn har Park-transformering d¨arf¨or anv¨ants f¨or att representera motorerna i ett roterande referenssystem. Detta medf¨or att de tidsberoende ekvationerna som beskriver motorerna blir tidsoberoende. De mest vanligt f¨orekommande styrsystemen f¨or en induktionsmotor och en synkronmotor har ocks˚a unders¨okts f¨or att kunna implementeras i modellerna. I litteraturstudien unders¨oktes ¨aven de vanligaste metoderna f¨or ”Maximum power point tracking” och kraven vid dimensionering av tillh¨orande kraftelektronik.

Offerter f¨or de framtagna systemen erh¨olls fr˚an flera av v¨arldens ledande motorpump- tillverkare i syfte att g¨ora en ekonomisk j¨amf¨orelse. Ink¨opsordern f¨or det befintliga systemet nyttjades ¨aven f¨or att avg¨ora kringliggande kostnader. Enligt offerterna s˚a har induktionsmotorsystemen relativt likv¨ardiga ink¨opspris medan synkronmo- torsystemet ¨ar ca 27 % dyrare. Det framgick ¨aven att de dr¨ankbara kablar som beh¨ovs st˚ar f¨or en betydande del av kostnaden i dessa system.

Om b˚ade prestanda och kostnad tas i beaktande visar resultaten att det korrekt dimensionerade induktionsmotorsystemet ¨ar ett b¨attre alternativ ¨an synkronmo-

torsystemet sett ¨over en livscykel p˚a 25 ˚ar. Om j¨amf¨orelsen utf¨ors ¨over en l¨angre tidsperiod skulle synkronmotorssystemet kunna vara mer f¨ordelaktigt.

Executive summary

Three solar-powered water pumping systems have been modelled and compared in this paper. The results acquired from the simulations show that the currently installed induction motor system is undersized and give rise to a significant amount of nonutilised energy. The adequately sized induction motor system proposed in this paper is sufficient to supply the required water demand. The developed model of a permanent magnet synchronous motor system has a higher efficiency than the adequately sized induction motor system, but it is not enough to compensate for the higher investment cost when considering a life-cycle of 25 years. A permanent magnet synchronous motor system could be economically viable if considering a life-cycle of 40 years.

Contents

Sammanfattning i

Executive summary iii

List of tables vii

List of figures viii

1 Introduction 1

1.1 Background . . . 1

1.2 Syunguni system . . . 1

1.3 Aim of thesis . . . 2

1.4 Limitations . . . 3

2 Theory 5 2.1 Submersible borehole pump . . . 5

2.1.1 System head . . . 5

2.1.2 Submersible vertical multi-stage centrifugal pump . . . 7

2.2 Photovoltaic array . . . 8

2.2.1 Temperature and irradiance dependence on a solar cell . . . 8

2.3 Buck converter control with MPPT . . . 11

2.3.1 Maximum power point tracking . . . 11

2.3.2 Buck converter topology . . . 13

2.4 Modelling of induction motor . . . 17

2.4.1 Mathematical parameters in the three-phase stationary ref- erence frame . . . 17 2.4.2 Mathematical parameters in the dq rotating reference frame 19

2.5 Modelling of permanent magnet synchronous motor . . . 22

2.5.1 Mathematical parameters in the three-phase stationary ref- erence frame . . . 22

2.5.2 Mathematical parameters in the dq rotating reference frame 23 2.6 Space vector pulse width modulation . . . 24

3 Method 30 3.1 PV array . . . 30

3.2 Weather data . . . 30

3.3 System sizing . . . 31

3.4 Buck converter control . . . 32

3.5 Motor control system . . . 32

3.5.1 Field oriented control of induction motor . . . 33

3.5.2 Field oriented control of PMSM . . . 36

3.6 Modelling of submersible pump . . . 38

3.6.1 Total dynamic head . . . 38

3.6.2 Pump Model . . . 40

3.7 Economy . . . 42

3.7.1 Investment cost . . . 43

3.7.2 Economic comparison . . . 43

4 Results 44 4.1 Motor efficiency . . . 44

4.2 Simulation results . . . 48

4.3 Economy . . . 56

5 Discussion & analysis 60 5.1 DC system . . . 60

5.2 Performance . . . 60 5.3 Economy . . . 61

6 Conclusion 63

References 64

List of tables

2.1 Space vectors for each switching state . . . 27

2.2 Switching state in every sector corresponding with the position angle 28 3.1 Approximation of the test result . . . 38

4.1 Motor efficiency at 50, 75 and 100 % load . . . 46

4.2 Energy and extracted volume for the IM_{3 kW} system . . . 54

4.3 Energy and extracted volume for the IM_{3.8 kW} system . . . 54

4.4 Energy and extracted volume for the PMSM system . . . 55

4.5 Aggregated result for the three systems . . . 55

4.6 Cost of 245 metre submersible drop-cables . . . 56

4.7 Investment costs [kSEK] . . . 56

4.8 Cost per cubic metre [SEK/m³] . . . 59

List of figures

1.1 Syunguni system . . . 2

2.1 Schematic figure of a bore-hole . . . 6

2.2 Vertical multi-stage centrifugal pump cross-section . . . 7

2.3 Temperature dependence on a solar cell . . . 10

2.4 Irradiance dependence on a solar cell . . . 11

2.5 Perturb & Observe flowchart . . . 12

2.6 Maximum power point of a PV array . . . 13

2.7 Progression of voltages and currents in an ideal buck converter op- erating in continuous conduction mode . . . 14

2.8 Buck converter in ON state . . . 15

2.9 Buck converter in OFF state . . . 16

2.10 Stationary reference frame and rotating reference frame axis . . . . 19

2.11 Equivalent circuit of the IM in the d-axis frame . . . 21

2.12 Equivalent circuit of the IM in the Q-axis frame . . . 21

2.13 Equivalent circuit of the PMSM in the D-axis frame . . . 23

2.14 Equivalent circuit of the PMSM in the Q-axis frame . . . 24

2.15 Equivalent circuit of VSI . . . 25

2.16 Inverter Space vectors for each switching state . . . 26

2.17 Reference space vector . . . 26

3.1 Weather data for an average day in July and October . . . 31

3.2 Buck converter control with MPPT . . . 32

3.3 Control system of induction motor . . . 35

3.4 Control system of permanent magnet synchronous motor . . . 37

3.5 Schematic diagram of the pump . . . 41

3.6 Total dynamic head and pump efficiency of the pumps . . . 42

4.1 Power measurement at 50, 75 and 100 % load for the PMSM system 44 4.2 Power measurement at 50, 75 and 100 % load for the IM3 kW system 45 4.3 Power measurement at 50, 75 and 100 % load for the IM_{3.8 kW} system 45 4.4 Stator current at rated load for the PMSM . . . 46 4.5 Stator current at rated load for the IM3 kW . . . 47 4.6 Stator current at rated load for the IM_{3.8 kW} . . . 47 4.7 Power measurement of an average day in July for the IM_{3 kW} system 48 4.8 Power measurement of an average day in July for the IM_{3.8 kW} system 49 4.9 Power measurement of an average day in July for the PMSM system 49 4.10 Pump measurement of an average day in July for the three systems 50 4.11 Power measurement of an average day in October for the IM_{3 kW}

system . . . 51 4.12 Power measurement of an average day in October for the IM3.8 kW

system . . . 51 4.13 Power measurement of an average day in October for the PMSM

system . . . 52 4.14 Pump measurement of an average day in October . . . 53 4.15 Net income based on investment cost and amount of energy con-

sumed by the pump . . . 57 4.16 Net income based on investment cost and the amount of extracted

water over a 40 year period . . . 58 4.17 Cost per cubic metre of water derived from investment cost and

amount of extracted water over a 40 year period . . . 59

1 Introduction

1.1 Background

Mutomo is a village of roughly 17 000 inhabitants located 230 km south-east of Nairobi. This part of Kenya is often considered one of the poorest in the country.

The infrastructure in and around Mutomo is substandard, and the few dirt roads leading to the area have been neglected in the past which have prevented devel- opment. The climate in Mutomo is semi-arid and the seasonal rainfall is unpre- dictable. When it occasionally rains, the rainfall is heavy and often causes erosion and flooding. People in Mutomo must carry water, often contaminated and green with algae, long distances from dams and seasonal rivers to their homes. Lack of clean water has for a long time been one of the biggest problems for people living in this area [1].

The project owner is Solel i Sala & Heby¹. Solel i Sala & Heby is an economic association which are the founders of a charity project named Solsk¨ankt. The goal of the charity project is to help people in the district of Mutomo by installing solar powered water pumping systems in the rural areas of Mutomo. These facilities produce renewable electricity primarily used for pumping clean drinking water for the locals.

The system evaluated in this paper is located in Syunguni and is driven by a traditional induction motor. The Syunguni system is connected to a pipeline, which is driven by a separate pump to distribute the extracted water to a water tank placed in the village of Mutomo. The stakeholders of the economic association are now interested in investigating if there is any gain in using a different motor type for their next water pumping system. A possible gain would be a simpler more maintenance-free system better adapted for use in rural areas far from any skilled technicians. Another gain would be a more efficient system, producing higher volumes of water, with a lower life-cycle cost.

1.2 Syunguni system

The borehole located in Syunguni has a diameter of 15.24 cm and is 250 m deep.

The current pump-system consists of 19 × 250 W solar panels and a 5 kW solar inverter connected to a submersible 3 kW induction motor, coupled with a 2.2 kW pump-end. The motor-pump is installed at 245 m depth, attached with 4 mm²

1https://www.solelisalaheby.se/

submersible drop-cables. A simplified diagram of the Syunguni system is visualised in figure 1.1.

Figure 1.1: Syunguni system

1.3 Aim of thesis

The most commonly used motor types for water pumping systems are the induction motor, permanent magnet synchronous motor and the brushed DC motor. Since the brushed DC motor requires regular maintenance due to mechanical contact between the brushes and the commutator segments, it is not a viable option for a submersible motor-pump at 245 m depth.

The demanded water supply for this project is 25 m³ per day, in order to provide

a minimum of 4 litres per day and person for 6 000 of the people living in the area.

The aim of this thesis is to evaluate the currently installed system in Syunguni, and present a better alternative in terms of efficiency, water supply and economic viability. In order to fulfil the aim of this thesis, the work is divided into the following objectives:

1. Create a a model of the currently installed induction motor system.

2. Create a simulation model of an induction motor system that meets the demanded water supply throughout the year.

3. Create a simulation model of a permanent magnet synchronous motor system for comparison.

4. Evaluate and compare the systems in terms of efficiency, water supply and economic viability.

1.4 Limitations

The mathematical model of the borehole, and therefore also the energy cost of pumping water, is based upon data acquired from an actual test pumping carried out by a third-party contractor. The setup and conditions during the test made the resulting data not directly applicable for this project. The data had to be adapted before it could be used under the specific conditions modelled in this paper. The mathematical model of the borehole used in the simulations is therefore an approximation of the actual energy cost of pumping water from this specific borehole.

The only data available from the PV array in the current system was the total power installed. Since the exact panel configuration of the current system was not available, preset models from Simulink libraries were used to model the solar panels according to the given power output in this project.

The pump efficiencies used in this paper are derived from pump curves given by manufacturers, where the total dynamic head is the only factor taken into consid- eration. A non-dynamic operating curve for the pumps is therefore implemented which may differ from the actual efficiency.

There is no obvious way of making a fair economical comparison between different charity built solar powered motor-pump systems. For these systems, the value lies in supplying clean drinking water for people in need, and is therefore hard

to quantify. This should be taken into consideration when interpreting the result from the economical comparison.

2 Theory

2.1 Submersible borehole pump

All types of pumps used for pumping of fluids follow the same basic principles; they need to provide enough pressure to overcome the operating pressure of the system, to move fluid at a specific flow rate. The height a fluid can be lifted is limited by the atmospheric pressure. It is therefore necessary to install a submersible pump at the bottom of a borehole in order to overcome the operational pressure and push the fluid upwards.

2.1.1 System head

Operating pressure, or total system head, is determined by the rate of flow and arrangement of pipe size, pipe length, kind of fittings used, number and angle of bends, elevation of pumped liquid and inlet/outlet pressure. System head is calculated in metres, and all pressure units are converted from pascal to metres for dimensional consistency using the conversion rate 1 kPa = 0.102 m. Total system head is defined as

Htotal= Hstatic+ Hdynamic+ (PRT − Preservoir) (1) where H_static is used to denote static pressure in a pipe when fluid is not moving i.e height difference between the surface of the reservoir and outlet to receiving tank. H_dynamic represents friction losses. P_RT − P_reservoir represents the pressure difference between reservoir and receiving tank. Since the difference in atmospheric pressure between inlet and outlet is negligible for most applications, equation 1 can be written as

H_total = H_static+ H_dynamic (2)

Dynamic head is calculated using the Darcy-Weisbach equation given by

H_dynamic = Kv²

2g (3)

where g is acceleration due to gravity, v is the velocity of the fluid and K is a loss coefficient given by materials and layout of the system [2].

Well recovery rate is the rate at which fluid returns into the well from cracks and openings in the surrounding rock into the lower portion of the well. The recovery rate of a well defines the rate at which fluid can be pumped out of a well before it starts to run dry [3].

Since the system head of a pumping system depends on the elevation of the reser- voir and receiving tank, which in themselves depend on the flow rate and well recovery rate, it can change over time. Best practice is therefore to calculate a maximum and minimum total system head. When choosing a pump for a specific application, these limits must be taken into consideration for best overall perfor- mance and to ensure the pump delivers the required flow over the entire operating range of the system.

A schematic figure of a bore-hole is shown in figure 2.1.

Figure 2.1: Schematic figure of a bore-hole

The static head is the water level when the pump is not running. Draw-down indicates how the water level drops when the pump is running. And the total dynamic head is the sum of static head, draw-down and the friction losses in the pipe.

2.1.2 Submersible vertical multi-stage centrifugal pump

There are a lot of different types of pumps with different working mechanics suited for different applications. In this paper, the focus will be on a specific pump type often used in applications where the goal is to pump clean water with a high head from narrow bore holes, i.e. the submersible vertical multi-stage centrifugal pump. This type of pump has two or more impellers, also called stages, which for higher flow output must be connected in parallel, and for higher pressure at the outlet should be connected in series. For deep bore holes, there may be a need for more serial connected stages, to achieve enough pressure to overcome the total system head. The fluid is directed towards the centre of each stage, before it moves towards the discharge at the outer diameter entering the next stage with increased pressure, as visualised in figure 2.2.

Figure 2.2: Vertical multi-stage centrifugal pump cross-section

To pump fluid, the mechanical energy driving the impeller is transferred to the fluid, resulting in increased pressure. The theoretical power consumption of a pump can be calculated with the following equation

P = ρgH_totalQ

η_pump (4)

where P is the input power, ρ is the fluid density, g is the standard acceleration of gravity, Q is the flow rate and η_pump is the efficiency of the pump.

2.2 Photovoltaic array

For many applications, multiple solar panels have to be connected through series or parallel connections to get the required power output. Total power output is derived by adding the power output from each individual panel and is not dependent on panel configuration.

When connecting multiple panels in series, the total output voltage is equal to the sum of each individual panel voltage, and the total current output is dictated by the panel with the lowest current rating. Similarly, when connecting multiple panels in parallel, the total output voltage is determined by the panel with the lowest voltage, and the total output current is the sum of the rated current from each panel. The most efficient configuration is system specific and depends on the voltage and current requirement of the application. A common term regarding characteristics and performance of a solar cell is the Standard Test Condition (STC). The parameters in STC can be seen in equation 5

ST C







G_{ST C} = 1000 [W/m²] T_{ST C} = 25 [^◦C]

AM = 1.5

(5)

where AM is unitless and refers to the amount of light that has to pass through the atmosphere. The power output from a PV array varies with solar irradiance and temperature.

2.2.1 Temperature and irradiance dependence on a solar cell The open-circuit voltage of a solar cell is given by

V_OC(G, T c) = V_{OC,ST C} +N_skT_cn

q ln(G) + µ_VOC(T c − T_{ST C}) (6) where T_cis the cell temperature, T_{ST C}is the temperature at STC, N_sis the number of series-connected cells, n is diode ideality factor, q is the electronic charge, µ is the thermal coefficient and k is Boltzmann’s constant. Since the dependence of

irradiance follows a logarithmic function it is not very significant and can often be omitted. A simplified version of equation 6 can be seen in equation 7.

V_OC(T_c) = V_OC(ST C) + dV_OC

dT (T_c− T_{ST C}) (7)

where ^dV_dT^OC is the temperature coefficient.

If the temperature coefficient is unknown, equation 8 can be used as an approxi- mation [4]

dVOC

dT = VOC − Eg0− γkbTc

T_c × number of cells in the module (8) where E_g0 is the energy gap and k_bT_c is thermal energy. The values for E_g0, k_bT_c and γ are dependent on properties of the materials used in the specific solar cell [4].

The cell temperature T_c can be estimated with equation 9

T_c= T_a+N OCT − 20°C

800 W/m² G (9)

where T_a is the ambient temperature, G is irradiance and N OCT is a parameter called Nominal Operating Cell Temperature. When N OCT is unknown, a reason- able value for the most commonly used PV modules is 48° C [4]. In most cases, the temperature dependence on the short-circuit current I_SC can be omitted since it is relatively small [4].

The temperature has a negative dependence on the open-circuit voltage in a solar cell, which can be seen in equation 7 and is visualised in figure 2.3 [4].

Figure 2.3: Temperature dependence on a solar cell

The short-circuit current of a solar cell is given by

ISC(G, Tc) = G

G_{ST C}[ISC,ST C+ µISC(Tc− TST C)] (10) I_SC is directly proportional to irradiance which can be seen in figure 2.4.

Figure 2.4: Irradiance dependence on a solar cell

2.3 Buck converter control with MPPT

In this section the maximum power point tracking method Perturb & Observe and the basic principle of a buck converter is explained. The MPPT algorithm is used to control the duty cycle of the buck converter, which in turn regulates the output voltage.

2.3.1 Maximum power point tracking

The optimal operating point on an I-V curve at which a PV array can produce maximum power is called maximum power point (MPP). Since MPP varies with solar radiation, ambient temperature, internal panel temperature and load varia- tions, it is essential to track MPP to ensure PV arrays work at the highest possible efficiency at all times [5].

Maximum power point tracking (MPPT) is an algorithm for extracting maximum available power under certain conditions from a PV array, by making it operate at MPP. The MPPT is connected to a DC/DC converter which regulates output voltage by adjusting the duty cycle. There are many MPPT algorithms but one of the most commonly used is the Perturb & Observe method [5]. A flowchart of the Perturb & Observe method is shown in figure 2.5.

Figure 2.5: Perturb & Observe flowchart

As can be seen in figure 2.5, the basic principle of this method is to perturb the PV voltage with a small value of the initial value and observe the resulting PV power. The algorithm compares recent power P_n with previous power P_n−1, based on measurements of the output voltage and current from the PV array at every instant of time. When P_n = P_n−1, maximum power is extracted from the PV array. The maximum power point of a PV array is visualised in figure 2.6.

Previous voltage V_n−1 and recent voltage V_n is then comared to decide wether to increase or decrease the voltage.

Figure 2.6: Maximum power point of a PV array

If the power increased from the initial value, the operating point is to the left of MPP, and if the power decreased, the operating point is to the right of MPP.

The main disadvantage of this method is oscillations around MPP, since pertur- bation is repeated periodically during steady state operation [6]. For this type of command a current and voltage sensor is needed to measure the PV power at every instant of time.

2.3.2 Buck converter topology

When the input voltage is higher than the needed output voltage at the load, a buck converter can be used to lower the output voltage to the required level. In a DC/DC converter, as in any switched-mode power supply, there is a switch that controls the transfer of energy. There are also inductors and capacitors used to store energy. The converter has two modes, ON and OFF, and it switches between these modes periodically, as can be seen in figure 2.7.

Figure 2.7: Progression of voltages and currents in an ideal buck converter oper- ating in continuous conduction mode

The time the switch is on is denoted t_ON and the time the switch is off is t_{OF F}, one period is given by T = t_ON + t_{OF F}. The duty cycle is defined as D = t_ON/T , where D is the fraction of the period the switch is on. A buck converter in the ON state is depicted in figure 2.8.

Figure 2.8: Buck converter in ON state

In this state, the current flowing through the inductor rises gradually with a slope given by

dIL

dt = Vin− Vout

L (11)

and the current increases with the amount given by

∆I_L⁺= V_in− V_out

L t_ON = V_in− V_out

L DT (12)

A buck converter in the OFF state is shown in figure 2.9, where the diode shunts the connection between the ground and inductor.

Figure 2.9: Buck converter in OFF state

In this state, the current through the inductor decreases with a slope given by dI_L

dt = 0 − V_out

L (13)

and the current decreases by

∆I_L⁻= −V_out

L t_{of f} = −V_out

L (1 − D)T (14)

In steady state operation, the current in the ON state needs to match the current in the OFF state [7], which leads to the following steady state equations























∆I_L⁺+ ∆I_L⁻ = 0

Vin−V_out

L DT −^Vout_L (1 − D)T = 0 V_inD − V_out = 0

V_out = V_inD

(15)

For a higher efficiency it is preferred that the converter is operating in continous conduction mode (CCM) [7]. In CCM the required size of the inductor is given by

L > (V_in− V_out)D

f_s∆I_L (16)

where ∆I_Lis the maximum current ripple allowed and f_sis the switching frequency of the converter [8]. To limit the peak-to-peak value of the output voltage ripple below a certain value V_r, the required size of the filter capacitance is given by

C_out > (1 − D)V_out

8VrLf_s² (17)

The switching frequency reduces the required size of L and C_out. The upper boundary on the switching frequency is limited by the type of semiconductor used and by the switching losses, which are proportional to the switching frequency.

2.4 Modelling of induction motor

In this section, a mathematical model of an induction motor is presented. Math- ematical parameters in the abc stationary three-phase reference frame contain inductance terms that vary as a function of rotor angle, which leads to more complexity when solving machine and power system problems [9]. To avoid time dependent parameters, the mathematical model is transformed into the dq rotating reference frame.

2.4.1 Mathematical parameters in the three-phase stationary reference frame

The stator voltage equations in the abc stationary reference frame is given by equation 18 [10].





 V_sa V_sb V_sc





=







R_s 0 0 0 R_s 0 0 0 R_s











 i_sa i_sb i_sc





+ d dt





 ψ_sa ψ_sb ψ_sc





 (18)

where R_s is stator winding resistance, V_sa, V_sb, V_sc are stator voltages, i_sa, i_sb, i_sc are stator currents and ψ_sa, ψ_sb, ψ_sc are stator flux linkages.

The rotor voltage equations in the abc stationary reference frame is given by equation 19 [10].





 Vra

V_rb V_rc





=







Rr 0 0 0 R_r 0 0 0 R_r











 ira

i_rb i_rc





+ d dt





 ψra

ψ_rb ψ_rc





 (19)

where R_r is rotor winding resistance, V_ra, V_rb, V_rc are rotor voltages, i_ra, i_rb, i_rc are rotor currents and ψ_ra, ψ_rb, ψ_rc are rotor flux linkage.

Stator and rotor flux linkages are given by equation 20 and 21 [10].





 ψ_sa ψ_sb ψ_sc





=







L_aa L_ab L_ac L_ab L_bb L_bc L_ac L_bc L_cc











 i_sa i_sb i_sc





+ M_srR(θ_e)





 i_ra i_rb i_rc





 (20)

where L_aa, L_bb, L_cc are self-inductances of the stator windings, L_ab, L_bc, L_ca are mutual inductances between each stator phase and θ_e is the electrical angle.





 ψ_ra ψ_rb ψrc





=







L_AA L_AB L_AC L_AB L_BB L_BC LAC LBC LCC











 i_ra i_rb irc





+ MsrR(θe)T





 i_sa i_sb isc





 (21)

L_AA, L_BB, L_CC are self-inductances of the rotor windings and L_AB, L_BC, L_CA are mutual inductances between each rotor phase.

M_sr is the maximal mutual inductance established between a stator and rotor phase when their axis are collinear [10], which is given by

Msr









cos( ~O_sa, ~O_ra) cos( ~O_sa, ~O_rb) cos( ~O_sa, ~O_rc)









= Msr







cos(θ_e) cos(θ_e− ^2π₃ ) cos(θe− ^4π₃ )





 (22)

and R(θ_e) is an operator that converts rotor quantities into the stator reference frame based on the rotor position relative to the stator position [10], given by

R(θ_e) =







cos(θe) cos(θe− ^4π₃ ) cos(θe−^2π₃ ) cos(θ_e− ^2π₃ ) cos(θ_e) cos(θ_e−^4π₃ ) cos(θ_e− ^4π₃ ) cos(θ_e− ^2π₃ ) cos(θ)





 (23)

Both self-inductances and mutual inductances are a function of the rotor angle that varies with time, depending on the rotational speed of the rotor [11].

2.4.2 Mathematical parameters in the dq rotating reference frame The angle-dependent inductances become constant when the three-phase station- ary reference frame is transformed into a rotating reference frame, which leads to lower complexity since the dynamic equations are now time independent [9]. The abc three-phase stationary reference frame and the dq rotating reference frame axis are shown in figure 2.10

Figure 2.10: Stationary reference frame and rotating reference frame axis

where θ_e= ω_et and ω_e is the rotational speed of the dq reference frame.

The transformation from stationary abc-variables to rotating dq0-variables can be written in matrix form given by the Park Transformation [9]





 id

i_q i₀





= 2 3







cos(θe) cos(θe− ^2π₃ ) cos(θe+ ^2π₃ )

−sin(θ_e) −sin(θ_e−^2π₃ ) −sin(θ_e+ ^2π₃ )

1 2

1 2

1 2











 ia

i_b i_c





 (24)

The inverse Park Transformation is given by





 ia

i_b i_c





=







cos(θe) −sin(θe) 1 cos(θ_e− ^2π₃ ) −sin(θ_e−^2π₃ ) 1 cos(θ_e+^2π₃ ) −sin(θ_e+^2π₃ ) 1











 id

i_q i₀





 (25)

The Park Transformation can also be applied to voltages and flux linkages for the stator and rotor equations. The zero sequence, i₀, is included to give a complete degree of freedom when transforming into the dq-reference frame. i₀ is equal to zero under balanced conditions [9].

When the Park Transformation is applied to equation 18 and 19, the voltage equations in the rotating dq reference frame is given by











 V_ds V_qs V_dr V_qr













=













dψds

dt + R_si_ds− ω_sψ_qs

dψqs

dt + R_si_qs+ ω_sψ_ds

dψ_dr

dt + R_ri_dr − ω_rψ_qr

dψqr

dt + R_ri_qr+ ω_rψ_dr













(26)

where V_ds, V_qs, V_dr, V_qr are stator and rotor voltages respectively, i_ds, i_qs, i_dr, i_qr are stator and rotor currents, ψ_ds, ψ_qs, ψ_dr, ψ_qr are stator and rotor flux linkages, ω_s is the electrical angular frequency of the stator and ω_r is the electrical angular frequency of the rotor. The relationship between the electrical speeds are

ω_s = ω_mech+ ω_r (27)

where ω_mech is the mechanical speed of the rotor [10].

The equivalent circuits of the induction motor in the rotating dq reference frame are shown in figure 2.11 and 2.12.

Figure 2.11: Equivalent circuit of the IM in the d-axis frame

Figure 2.12: Equivalent circuit of the IM in the Q-axis frame

By applying Kirchoff’s voltage law to figure 2.11 and 2.12, the flux linkage estab- lished in both the stator and rotor in the dq reference frame is given by











 ψ_ds ψ_qs ψ_dr ψqr













=











L_si_ds+ L_m(i_ds+ i_dr) L_si_qs+ L_m(i_qs+ i_qr) L_ri_dr+ L_m(i_ds + i_dr) L_ri_qr+ L_m(i_qs+ i_qr)











(28)

2.5 Modelling of permanent magnet synchronous motor

A mathematical model of a PMSM is presented in this section. Mathematical parameters will be presented in the abc stationary three-phase reference frame, and transformed into the rotating dq reference frame by using the Park Transformation, see section 2.4.2.

2.5.1 Mathematical parameters in the three-phase stationary reference frame

Space vector form of the stator voltage equation is given by [11]





 V_sa V_sb V_sc





=







R_s 0 0 0 R_s 0 0 0 R_s











 i_sa i_sb i_sc





+ d dt





 ψ_sa ψ_sb ψ_sc





 (29)

where R_s is stator winding resistance, V_sa, V_sb, V_sc are stator voltages, i_sa, i_sb, i_sc are stator currents and ψ_sa, ψ_sb, ψ_sc are stator flux linkages. Stator flux linkages are given by [11]





 ψ_sa ψ_sb ψsc





=







L_aa L_ab L_ac L_ab L_bb L_bc Lac Lbc Lcc











 i_sa i_sb isc





+





 ψ_ra ψ_rb ψrc





 (30)

where L_aa, L_bb, L_cc are self-inductances of the stator windings, L_ab, L_bc, L_ca are mutual inductances between each phase, θ_e is the electrical angle and ψ_ra, ψ_rb, ψ_rc are the flux linkages established in each stator winding caused by the permanent magnets, which are dependent on the rotor angle and given by [11]





 ψ_ra ψ_rb ψ_rc





= ψ_r







cos(θ_e) cos(θ_e− ^2π₃ ) cos(θ_e+ ^2π₃ )





 (31)

where ψ_r is the permanent magnet flux linkage.

2.5.2 Mathematical parameters in the dq rotating reference frame When the Park Transformation is applied to equation 29 and 30, as explained in section 2.4.2, the voltage equations in the rotating dq reference frame are given by

"

Vds

V_qs

#

=

"_dψ

ds

dt + R_si_ds− ω_eψ_qs

dψqs

dt + R_si_qs+ ω_eψ_ds

#

(32)

"

ψ_ds ψ_qs

#

=

"

L_di_ds+ ψ_r L_qi_qs

#

(33)

where V_ds, V_qsare stator voltages, i_ds, i_qs are stator currents, ψ_ds, ψ_qs are stator flux linkages, L_d, L_q are dq-axis inductances and ω_e is the electrical speed of the rotor.

Equation 32 and 33 can be used to derive the equivalent circuit of the PMSM in the rotating dq reference frame, seen in figure 2.13 and 2.14.

Figure 2.13: Equivalent circuit of the PMSM in the D-axis frame

Figure 2.14: Equivalent circuit of the PMSM in the Q-axis frame

2.6 Space vector pulse width modulation

Pulse width modulation (PWM) generates sinusoidal voltages of variable voltage and frequency. The most commonly used techniques for implementing PWM for multi-level inverters are sinusoidal pulse width modulation (SPWM) and space vector pulse width modulation (SVPWM). The SVPWM technique is considered advantageous over other PWM techniques since it provides a higher fundamental output voltage and produces less harmonic distortion when the switching frequency is approximately equal to the fundamental frequency [12].

The equivalent circuit of a three-phase voltage source inverter that consists of six IGBT switches on three legs is shown in figure 2.15.

Figure 2.15: Equivalent circuit of VSI

where each pair of series connected switches represent one leg. The phase-to-phase voltages are given by

V_a = V_scos(ωt) V_b = V_scos(ωt −2π

3 ) V_c= V_scos(ωt −4π

3 )

(34)

An abc three-phase system can be transformed into an equivalent αβ two-phase system by using the Clarke Transformation [13]

"

V_α Vβ

#

= 2 3

"1 ⁻¹₂ ⁻¹₂ 0

√3 2

−√ 3 2

#





 V_a V_b V_c





 (35)

The three-phase inverter aim to create a balanced three-phase AC voltage from a constant DC supply voltage. The switches have two different switching states, conductive and blocking state. Only one switch per leg is able to conduct at the same time, for example if S5 is in a conductive state, S6 is in a blocking state and so on. The switching states of the inverter are denounced either 1 or 0, where

1 means that the upper switch of the leg is conducting and the lower switch is blocking, 0 means that the upper switch of the leg is blocking and the lower switch is conducting. Combining these two states for each inverter leg leads to eight possible switching states in a two-level three-phase voltage source inverter, where each state is determined according to the switching sequence. Six of the switching states produce a non-zero voltage and two produce zero output voltage. Each switching state when the output voltage is non-zero correspond to a state vector, spatially separated by ^π₃ in six different sectors as shown in figure 2.16 [14].

Figure 2.16: Inverter Space vectors for each switching state

By applying SVPWM method on the output voltages of the inverter, a single vector with a fixed length is obtained, figure 2.17[12].

Figure 2.17: Reference space vector

Where α is the position of the desired voltage space vector V_s. V_s is given by

V_s(t) = V_α+ jV_β = 2

3(V_s1(t) + V_s2(t)e^j2π3 + V_s3(t)e^j4π3 ) (36) and the magnitude and position of the desired voltage space vector is calculated by

|V_s| =q

V_α²+ V_β² α = arctan(V_α

V_β)

(37)

Each space vector corresponding to each switching state is shown in table 2.1 [12]

Table 2.1: Space vectors for each switching state Voltage Vector Switching State Space Vector

V₀ [000] 0

V₁ [100] ²₃V_dce^j0

V₂ [110] ²₃V_dce^jπ3 V₃ [010] ²₃V_dce^j2π3 V4 [011] ²₃Vdce^j3π3 V₅ [001] ²₃V_dce^j4π3 V₆ [101] ²₃V_dce^j5π3

V₇ [111] 0

Each switching state in every sector according to the position of the desired space vector is shown in table 2.2, and can be used to determine which sector the desired space vector is residing in [12].

Table 2.2: Switching state in every sector corresponding with the position angle Sector Switching State Position angle α

1 V0, V1, V2 0 ≤ α < ^π₃ 2 V₀, V₂, V₃ ^π₃ ≤ α < ^2π₃ 3 V₀, V₃, V₄ ^2π₃ ≤ α < ^3π₃ 4 V₀, V₄, V₅ ^3π₃ ≤ α < ^4π₃ 5 V0, V5, V6 4π

3 ≤ α < ^5π₃ 6 V₀, V₁, V₆ ^5π₃ ≤ α < ^6π₃

By using the voltage second balance principle on the time intervals for each switch- ing state over half a PWM switching cycle, the time interval for each switching state in each sector can be determined [14]. Implementation on sector 1 according to table 2.2 gives

V_s∆T = V₀t₀+ V₁t₁+ V₂t₂

∆T = t₀+ t₁+ t₂ (38)

where t0, t1, t2 are the time interval for the corresponding voltage vector and ∆T is the sampling interval. Combining the space vectors corresponding to V₀, V₁, V₂ from table 2.1 with equation 38 results in

∆T V_s

"

cos(α) j sin(α)

#

= t₁2 3V_dc

"

1 0

# + t₂2

3V_dc

"

cos(^π₃) j sin(^π₃)

#

(39)

To simplify the calculation of the time intervals, equation 39 is divided into real and imaginary parts, where the real part correspond to the α axis and the imaginary part correspond to the β axis, see figure 2.17.

Real : ∆T V_scos(α) = t₁2

3V_dc+ t₂2

3V_dccos(π

3) = t₁2

3V_dc+ t₂1 3V_dc Imaginary : ∆T V_ssin(α) = t₂2

3V_dcsin(π

3) = t₂ 1

√3V_dc

(40)

Solving equation 40 results in the following time intervals [15]

t₀ = ∆T − t₁− t₂ t1 =

√3∆T Vs

V_dc sin(π 3 − α) t₂ =

√3∆T V_s

V_dc sin(α)

(41)

These calculations can be used to derive the time interval for each corresponding voltage vector in sector 1. A general calculation to derive time intervals in all sectors can be made by modifying the position angle α, since the sectors are spatially separated by the distance ^π₃. The modified angle is given by

α_m = α − (n − 1)π

3 (42)

where n is the sector number [15].

3 Method

The theory in section 2 is implemented and the simulation methods are presented in this section. All simulations were carried out in the simulation software MATLAB Simulink, from Mathworks.

3.1 PV array

The implementation of the PV array was done using a preset model. When choos- ing a preset model, maximum power and number of parallel and series-connected modules per string were taken into consideration to derive an installed power for the PV array. The input to the PV array is irradiance and cell temperature.

3.2 Weather data

To simulate real weather conditions at Mutomo Kenya, historical irradiance and ambient temperature data at coordinates 1.8464°S, 38.2085°E were collected from the HelioClim-3² archives. The data points were used to derive an average value for each one hour interval of each day of every month. The result was 12 pairs of irradiance and temperature vectors representing an average day for each month of the year 2006. Ambient temperature is converted into panel cell temperature by using equation 9, seen in section 2.2.1, and the vectors were used as input to the PV array. As can be seen in extracted data from the HelioClim-3 archives, the month with the highest respectively lowest energy potential is October and July.

Solar irradiance, ambient temperature and cell temperature for an average day in July and October between 03:00 and 17:00 are shown in figure 3.1

2http://www.soda-pro.com/web-services/radiation/helioclim-3-archives-for-free

Figure 3.1: Weather data for an average day in July and October

3.3 System sizing

The system should be sized after July in order to meet the water demand through- out the year. In order to find an induction motor system that meets the water demand, submersible pump manufacturers were contacted. Manufacturers assisted in choosing a correctly sized motor-pump for this specific application. The result was an induction motor rated at 3.8 kW coupled with a pump rated at 3 kW. The induction motor system is replicated in the simulations where the PV array sizing

has been changed in order to ensure that the system provides the demanded water supply in July. A permanent magnet synchronous motor-pump in similar size was then acquired from pump manufacturers, and the same PV array sizing was used for easier comparison.

3.4 Buck converter control

Maximum power point tracking is implemented by using the Perturb & Observe method seen in section 2.3. The input to the MPPT is output current and voltage from the PV array, which is used to generate a duty cycle. Pulse width modulation is then used to generate gating signals for the IGBT switching device in the buck converter, see figure 3.2.

Figure 3.2: Buck converter control with MPPT

3.5 Motor control system

The control systems used for the motors are presented in this section. The closed loop field oriented control system is implemented for both motor types.

3.5.1 Field oriented control of induction motor

The induction motor is modelled as an asynchronous machine with a squirrel cage rotor. The motor is being fed by a PWM inverter which contains 6 IGBTs and acts as a three-phase current source. The control feedback loop contains a speed controller modelled as a PI controller, and a current regulator that consists of two PI controllers. Actual speed ω and reference speed ω^∗ generates the torque command T_e^∗ through the speed controller, which in turn is divided by estimated rotor flux linkage |Ψ_rest| to generate the quadrature axis stator current i^∗_qs [16] [17].

i^∗_qs = 2 3

2 p

Lr

L_m T_e^∗

|Ψ_rest| (43)

where p is number of poles, L_m, L_r are mutual inductance and rotor inductance.

Estimated rotor flux linkage is given by

|Ψ_rest| = L_mi_ds

(1 + τrs) (44)

where τ_r = L_r/R_ris the rotor time constant [16] [17]. Since the rotor flux is usually constant, the stator direct-axis current can be obtained from the flux reference |Ψ^∗_r| by using equation

i^∗_ds = |Ψ^∗_r|

L_m (45)

The rotor flux position is needed to transform the rotating reference frame into stationary three-phase reference frame. The flux position is generated by

θe = Z

(ωr+ ωsl)dt (46)

where ωr is the rotor speed and ωsl is the slip frequency. The flux position is slipping with respect to the rotor at a frequency equal to ω_sl [16], which is given by

ω_sl= L_mR_r

|Ψ_rest|i_qs (47)

The dq-axis stator current reference values are compared to the measured dq-axis stator currents, and the error is passed through two PI controllers to generate dq- axis stator voltage reference values. The voltage reference values are transformed into α, β components by using Clarke Transformation and Park Transformation, seen in section 2.6. Space vector pulse width modulation is used to generate gating signals for the inverter.

Figure 3.3: Control system of induction motor

3.5.2 Field oriented control of PMSM

The PMSM is modelled with a sinusoidal back EMF waveform and round rotor.

The motor drive consists of two nested control loops. A PI controller, functioning as a speed controller in the outer loop, compares motor speed ω and the motor reference speed ω^∗ in order to generate the q-axis stator current reference value by using the torque equation.

T_e = 3 2

p

2Ψ_mi^ref_q (48)

Since the permanent magnets are providing the excitation flux in a PMSM, there is no need for the stator side to provide flux and the d-axis stator current reference is set to zero [17]. The dq-axis stator current reference values are compared to the measured dq-axis stator currents, and the error is passed through two PI controllers in the inner loop to generate dq-axis stator voltage reference values. The voltage reference values are transformed into α, β components as explained in section 3.5.1.

SVPWM is used to generate gating signals for the inverter.

Figure 3.4: Control system of permanent magnet synchronous motor

3.6 Modelling of submersible pump

A mathematical model of a submersible pump is presented in this section. In order to model the pump, the total dynamic head (TDH) of the system is first determined. Submersible pump manufacturers were then contacted in order to find suitable pumps matching the TDH and the required flow rate of the system. The pump efficiency at different TDH was calculated from the received pump curves from the manufacturers.

3.6.1 Total dynamic head

To determine the TDH of the system, data from an actual test pumping procedure of the borehole was used. The test was performed on-site by a third party contrac- tor with the objective to investigate the potential of the borehole by estimating the sustainable yield. The test consisted of water extraction from the borehole at variable rates over a 24 hour period where the water level was measured periodi- cally.

When using solar power, the irradiance level at the site restricts available pump time in July to approximately eight hours, according to the HelioClim-3 data. An average flow of ^∼3 m³/h would therefore be sufficient to meet the demand. An approximation of the test result, when the recovery rate of the borehole and a constant flow rate of 3 m³/h has been used over 14 hours of operation, is shown in table 3.1. Since the pipe is straight, the friction losses in the pipe are assumed to be 5 % of the required lift in order to determine the TDH of the pump [18].

Table 3.1: Approximation of the test result

Time [min] Water Level [m] TDH [m] Discharge [m³/h] Volume [m³]

0 23.25 24.41 0 0.00

1 26.15 27.45 3.0 0.05

2 28.00 29.40 3.0 0.10

3 30.21 31.72 3.0 0.15

4 32.32 33.93 3.0 0.20

5 34.41 36.13 3.0 0.25

6 36.46 38.28 3.0 0.30

Time [min] Water Level [m] TDH [m] Discharge [m³/h] Volume [m³]

7 38.50 40.42 3.0 0.35

8 40.49 42.51 3.0 0.40

9 42.61 44.74 3.0 0.45

10 44.82 47.06 3.0 0.50

12 46.61 48.94 3.0 0.60

14 48.51 50.93 3.0 0.70

16 51.19 53.74 3.0 0.80

18 53.33 55.99 3.0 0.90

20 55.44 58.21 3.0 1.00

25 59.16 62.11 3.0 1.25

30 62.22 65.33 3.0 1.50

35 66.52 69.84 3.0 1.75

40 70.63 74.16 3.0 2.00

45 73.17 76.82 3.0 2.25

50 75.31 79.07 3.0 2.50

55 78.32 82.23 3.0 2.75

60 81.06 85.11 3.0 3.00

70 86.93 91.27 3.0 3.50

80 91.35 95.91 3.0 4.00

90 95.34 100.10 3.0 4.50

100 99.16 104.11 3.0 5.00

110 103.26 108.42 3.0 5.50

120 107.04 112.39 3.0 6.00

135 113.02 118.67 3.0 6.75

150 117.86 123.75 3.0 7.50

165 123.44 129.61 3.0 8.25

180 128.51 134.93 3.0 9.00