Abstract
This paper presents an algorithm MPPT (Maximum Power Point Tracking) for a Magnus wind system with a DC servo drive system (DC drive and BLDC motor) to rotate the turbine cylinders. The optimal cylinders rotation is the one to deliver the maximum power extracted from the wind tracked by fixed and adaptive step HCC (Hill Climbing Control) acting on the servo drive. The proposed wind system consists of a PMSG (Permanent Magnet Synchronous Generator), a three-phase diode rectifier, a DC/DC (boost) converter, and a resistive load. Furthermore, the boost converter acts with the fixed step HCC algorithm to track the maximum power operating point. Therefore, the MPPT for a Magnus wind system requires both tracking for the optimal cylinder speed and the optimal generator speed.
1. Introduction
The Magnus wind turbine has rotating cylinders rather than the conventional blade wind turbines as shown in Figure 1. The angular velocity is the rotational speed of the turbine and is the rotational speed of the cylinders. Its operation is based on the physical principle known as “Magnus effect.”
Detail (a) in Figure 1 is the front portion of the hub where the DC servo drive system (DC servo drive and BLDC motor) is located. A gear system transmits the rotation from the BLDC motor to the cylinders, detail (b).
A numerical simulation by Imai and Kato [1] was used to compare the performance of a smooth cylinder and a cylinder with 6 spiral fins as shown in Figure 2 (CFD—Computational Fluid Dynamics).
(a)
(b)
(c)
Figure 2(b) represents the air flow around a smooth rotating cylinder with the same diameter of the spiral fin cylinder represented in Figure 2(a). The featured points called “points of separation” are regions where there is a cylinder wall detachment from air flow.
Figure 2(c) illustrates the air flow pattern around the cylinder with spiral fin arranged in accordance with Figure 2(a). It is noticeable that the air flow detachment at the bottom of the cylinder with spiral fin is in a postponed point. This fact provides a larger surface extension on the lower face of the rotating cylinder with relatively less pressure than the upper face, and thus the lift force on the upper face is stronger when compared to the cylinder without fins.
A cylinder with radius rotating with an angular velocity , immersed in a fluid of density , is shown in Figure 3, with a laminar fluid velocity . Theoretically, the wind rotor is subject to a lift force in the perpendicular direction to the fluid flow called lift (lift force) and drag (drag force), a phenomenon known as Magnus effect [2, 3]. Theoretically the drag force should be zero, but it has been proven experimentally that this value is not null and increases with the increasing relative velocity of the cylinder surface [3].
Registers of the first prototypes of Magnus wind turbines are not new [4]. In 1926 Flettner built a horizontal-axis wind turbine with four blades, where each blade was a tapered cylinder driven by an electric motor. The cylinders (blades) were 5 m long and 0.8 m in diameter at the midpoint. The experiment used a 20 m diameter rotor on a 33 m tower, with a rated power of 30 kW at a wind speed of 10 m/s.
In 1983, a private entrepreneur in California constructed a Magnus-type wind turbine of the barrel blade type, with a 17 m diameter, of a rated capacity of 110 kW. The unit was later moved to the wind test site of Southern California Edison, which was located in San Gorgonio Pass.
Recently, in 2007, a Japanese company by joint efforts between industry, government, and universities has developed and commercially launched a Magnus wind turbine [5]. The cylinders of this Magnus turbine have two important characteristics that became commercially feasible, one spiral fin and an end cap for an enhanced Magnus effect [6].
Experiments were conducted also in Russia with prototypes of Magnus wind turbines in a wind tunnel. It was shown that the Magnus turbine even at wind speeds of 2 m/s presents a power coefficient enough to generate a net power output and take advantage of these low wind speeds. This turbine can be operated up to 40 m/s wind speed well above the upper limit of the conventional turbines [7–9].
2. Magnus Effect Wind Turbine
2.1. Mechanical Power of the Magnus Wind Turbine
Figure 4 shows a Magnus effect wind turbine, where is the number of rotating cylinders; is radius of the rotor [m]; is turbine speed [rad/s]; is radius of the cylinder [m]; is cylinder’s speed [rad/s]; is laminar wind speed [m/s]; and is air density [kg/m3].
The tip speed ratio (TSR or ) is the ratio of the tangential velocity of the blade or cylinder (at the tip) under an undisturbed wind speed, according to [10]The relative speed of the rotating cylinders is the ratio of the tangential velocity of the cylinder under an undisturbed wind speed, according to [11, 12]Therefore, the mechanical power extracted from the wind by the Magnus turbine is represented by [10–14]The analytical equation (4) is related to the power coefficient of the Magnus turbine and it was obtained based on the principles of fluid dynamics and the theory of momentum of blade elements (BEM—Blade Element Momentum Theory) neglecting the influence of other relevant factors at the turbine power output such as the drag force, losses at the cylinder ends, interaction between cylinders, and effects of the hub diameter. Moreover (4) was obtained based on the values obtained experimentally from a model closer to reality [11, 12]. The theoretical limit of the power coefficient for Magnus wind turbines is :
2.2. Hill Climbing Control (HCC) Algorithm
The HCC algorithm with fixed step performs the tracking of the maximum point (MP) on the net power curve by incrementing or decrementing the voltage reference at fixed steps . The algorithm is always searching for the MP as shown in Figure 5. A relatively large step can result in a shorter time to reach the top of the curve but also results in larger oscillations around the MP [15, 16].
An HCC algorithm with adaptive step performs the tracking of the maximum point (MP) on the net power curve by incrementing or decrementing the voltage reference (5) with an adaptive value (6). The algorithm is always searching for the MP as shown in Figure 6. The value of decreases when the output power is closer to the MP, thus reducing the oscillations around this point according (6). In (6) the symbol is a constant infinitesimal parameter:
2.3. The MPPT for a Magnus Wind Turbine
The mechanical power produced by a Magnus wind turbine, as well as conventional wind turbines, is the product of the torque (N·m) and the rotation speed of the turbine (rads/s) according to (7). There is an optimal rotational speed producing the maximum mechanical shaft power. Therefore, it is necessary to track this optimal rotational turbine speed (MPPT) [17–19]:Figure 7 shows the typical mechanical power curves produced by a Magnus turbine by increasing (relative rotation of the cylinders) at a given constant (TSR of the turbine). For each wind speed is depicted the relative cylinder rotation to provide the maximum mechanical power from the wind [20]. The Magnus wind system tracks the net power (8) on the turbine shaft, discounting the power consumed by the driving system (servo drive DC and BLDC motor) of the rotating cylinder:
3. The Proposed Magnus Wind System
The wind system proposed in Figure 8 consists of a mathematical model of the Magnus turbine according to (3) and (4), the DC servo drive system (inverter bridge and BLDC motor), a PMSG generator, a three-phase rectifier, a boost converter DC/DC, and a resistive load. The goal of the proposed wind system is to demonstrate the maximum power extraction from the wind to be dissipated in a resistive load.
The is voltage source at the input of the three-phase bridge inverter with 6-IGBT (Insulated Gate Bipolar Transistor). This bridge inverter drives the BLDC motor.
The IGBT switches in the inverter bridge are closed in pairs in a sequence according to Table 1 consisting of 6 steps for a complete rotation of the BLDC motor.
The signals from the three hall effect sensors consist of square waves 120° out of phase applied to a circuit consisting of AND and NOT logic gates providing the switching sequence for the inverter bridge switches driving the BLDC motor rotation.
The torque and motor speed can be controlled by varying the average voltage across the terminals (ABC) of the BLDC motor coils. This variation is used for the pulse width modulation (PWM1). The average voltage delivered to the coils is driven by modulating the duty cycle. Consequently, the BLDC motor speed also changes.
The wind system proposed in this paper measures the electrical power delivered to the inverter bridge. This power is identified by , that is, the power consumed to rotate the turbine cylinders. The electric power measured across the diode rectifier output is the electrical power generated, called . The difference between the respective generated powers and is the net power (8).
The block “MPPT cylinders” shown in Figure 8 represents the HCC algorithm tracking the optimal cylinder speed to provide the maximum net power extracted from the turbine. Whenever the net power is negative, the algorithm performs a fixed step and when the net power is positive the algorithm performs an adaptive step.
The block “MPPT boost” represents the HCC algorithm with fixed step tracking the maximum electrical power by scanning the boost converter duty cycle.
Figure 9 shows the control algorithm flowchart for the rotating cylinders. Initially, the wind speed is read, which is compared to wind speed reference denoted by . For winds below this value, the cylinder rotation is kept static avoiding unnecessary waste of energy. When the wind speeds are above this value, the control generates a positive net power step which assures to the turbine the necessary starting interval. From this point onwards, the control algorithm reads the net current power and calculates the difference between the power and the previous . It also measures the difference between the voltage reference of the PWM1 resulting from the difference between (the current) and (the previous voltage). If (net power current) is negative, the controller commands an increment (fixed step) to the voltage reference . If remains negative for longer than the predefined period, , the cylinder driver is blocked with a in order to avoid wasting energy. If is positive, the HCC algorithm (adaptive step) increments and decrements with a step aiming at tracking the maximum power point.
3.1. DC Servo Drive
The DC servo drive system (inverter bridge and BLDC motor) drives the cylinder rotation through the gear train system (gear crown and gear pinion).
The electric power generated is measured across the diode rectifier output. The power consumed by the DC servo drive system is measured across the power supply output. The net power is obtained from (8) and transmitted to the turbine hub by special collector rings conducting power and control signals.
The “MPPT cylinders” block output generates the reference signal to the PWM1 in order to switch the inverter bridge driving the DC servo system. The switching sequence of the three-phase inverter is defined by 3 signals coming from the hall sensors fitted on the brushless motor and logic gates AND and NOT [21].
The torque produced by the turbine is transmitted to the PMSG generator through a gear box.
3.2. DC/DC Boost Converter
The static converter consists of an uncontrolled three-phase rectifier, the filter, a DC/DC boost converter, and a resistive load, to dissipate the maximum electrical power. The boost converter shown in Figure 8 comprises a switch , a diode , and an inductance . The main feature of this boost converter is to produce an output voltage higher than the input voltage . Across the PMSG generator output there is a three-phase uncontrolled bridge rectifier and a capacitor filter . The purpose of the capacitor is to produce a smoother output voltage. The filter is also coupled to make smoother input current and input voltage across the boost converter terminals.
The boost converter parameters are as follows: = input average voltage boost; = output average voltage boost; = line-to-line peak voltage across the rectifier; = boost inductor; = inductor current through the boost terminals; = load average current; = duty cycle; = switching period; = inductor ripple current; = boost power input; and = load power output.
The “MPPT boost” block in Figure 8 also represents the HCC algorithm (Hill Climbing Control with fixed step) tracking the optimal duty cycle “” of the boost converter delivering the maximum electrical power to the resistive load. The conversion ratio of the boost “” is given by [22]The boost converter operating in continuous conduction mode (CCM) has two operation stages. The first step is the magnetization of the inductor and the second is its demagnetization. During the magnetizing interval the current slope across the magnetizing inductance is given byDuring the magnetizing interval the current variation through the inductor is expressed by the current slope the multiplied by the time interval:where is the ripple current through the inductor .
Expression (12) determines the value of magnetizing inductance for the desired current ripple. When the converter is operating at CCM, that is, the current does not reach zero within the demagnetization range and to be valid the static gain of , it is necessary to have a current ripple smaller than the average current through the inductor [22]. That is,Considering lossless ideal components, it can be said, therefore, that the input and output powers from the boost converter are equal; that is,where is the input current of the boost converter.
The output current from the boost converter can be obtained from the output voltage and resistance asThe load resistor sets the output power range of the boost converter. Resistor defines a valid range for the duty cycle. As practical converters operate at duty cycles between 0.2 and 0.8, the resistance in (15) is replaced into (14) to obtainBy replacing (9) into (16), the following is obtained:The average input voltage is related to the peak input line-to-line voltage across the rectifier according to [23]Therefore,
4. Simulation Results
4.1. A. Simulations Parameters of the BLDC Motor and PMSG
Table 2 presents the main BLDC motor parameters used in the Magnus turbine prototype as shown in Figure 1. These parameters were extracted from the manufacturer’s manual and used along with the simulations in PSIM [24]. The parameters of the PMSG generator were also taken from the manufacturer’s manual according to Table 3.
4.2. Design Boost Converter and Load
Replacing the power coefficient (4) into the mechanical power (3) and using the values given in Table 4, one obtains the calculated mechanical power curves as displayed in Figure 10. These curves are related to the mechanical power generated by the Magnus turbine for three different wind speeds.
Figure 10 displays the maximum mechanical power and the respective turbine speed considering a gearbox ratio (1 : 5) to the PMSG generator rotation. Therefore, in Figure 10, the PMSG speed provides the maximum mechanical power and with the PMSG constant /kRPM in Table 3 is used to calculate . With from (18) can be calculated. Neglecting the mechanical losses, the PMSG power is equal to and therefore the average inductor current . Equation (19) is used to calculate the load resistance .
Calculation of the Load Resistance . The load resistance is calculated from (19) for intermediate values of wind speeds (5.5 m/s) and the corresponding duty cycle for an intermediate value . It is intended to obtain thereby the load resistance by varying the boost cycle ratio up or down from the generator point of view for the appropriate maximum power transfer within the upper and lower limits of wind speeds. Therefore, by increasing the duty cycle, the load resistance is seen by the generator output as a lower value, suitable for low powers (speeds lower than 5.5 m/s wind). By decreasing the duty cycle, the load resistance ratio would be seen by the generator as a larger value, suitable for higher power (wind speeds greater than 5.5 m/s).
Using the data listed in Table 5 regarding a wind speed 5.5 m/s results in = 100.4 Ω.
Rewriting (17) results inUsing (20) to calculate and for a wind speed 3.5 m/s in Table 5 results in . Using (20) for and and a wind speed 7.5 m/s results in .
Calculating the Boost Inductor . The minimum inductance must be calculated for a lower converter input current using (12). Considering an input current of 2.7 A, and assuming a ripple current of 0.5 A, the converter switching frequency would be 6 kHz, and the converter will operate at its minimum duty cycle of 0.2:In this case one can use a higher inductance to ensure a CCM, even at lower currents, say 3.0 mH. Note that the switching frequency affects directly the inductance values.
Table 6 shows the values of the static converter filter parameters.
4.3. Simulation Results
In the simulated results a wind speed of 3.0 m/s is initially applied which is insufficient to generate positive net power and therefore the turbine remains static. Because of this low wind speed, the simulated wind system awaits for a minimum wind speed required for its operation (4.2 m/s) so as to start up with a positive net power . At the instant s a wind speed step of 4.5 m/s is applied, until a wind speed of 7.5 m/s, which is able to produce a net power positive. Finally at the instant s a negative step of −2.0 m/s is applied letting the wind speed at 5.5 m/s.
Figure 11 represents the electrical power at the output of the three-phase diode rectifier. It can be observed that the values obtained from the electrical power generated in the simulation are quite close to the desired values of given in Table 5. Therefore the load resistance and the design methodology are correct. By varying the duty cycle PWM2 of the boost converter it is maximized the conversion of mechanical power into electrical power, adequating so the speed control of the PMSG generator. The algorithm “HCC Cylinders” tracks the optimal cylinder speed with larger step increments whenever the net power is negative . For such reason the electrical power reaches positive values in a relatively short time. The static converter components used in the simulation are ideal and therefore the electrical power in the load has the same waveform as that of .
Figure 12 is the graph of the net power .
Figure 13 represents the cylinder rotating speed. The algorithm “HCC Cylinders” increments the duty cycle PWM1 that does not translate the rotational speed immediately until the three-phase inverter bridge reaches the minimum voltage applied across the BLDC motor terminals. The HCC algorithm increases the duty cycle PWM1 to increase the cylinder speed until it reaches its maximum point and remains at this maximum value for a wind speed of 7.5 m/s.
Figure 14 shows the HCC curve tracking the optimal boost-converter duty cycle PWM2 to deliver the maximum electrical power to load .
Figure 15 represents the load current and the corresponding wind step responses with relative fewer oscillations due to the effectiveness of the filter.
Figure 16 represents the generator speed with the algorithm “HCC Boost” for the PWM2 duty cycle increments and decrements to scan the MPPT optimal PMSG speed.
5. Conclusions
The simulation results presented in this paper refer to the maximum mechanical power as provided in Table 5 for wind speeds between 5.5 m/s and 7.5 m/s to reach the electrical generated power and load electrical power . These results prove that the control methodology applied to the boost and load resistor designs is correct as well as the specified values. The algorithms “HCC Cylinders” and “HCC Boost” effectively track the optimal cylinder rotation to provide the maximum net generated power and the maximum load power , respectively.
The Magnus wind system as discussed in this paper performs correctly the wind speed steps passing from 3.0 m/s to 7.5 m/s in order to increase the turbine mechanical power and wind speed steps passing from 7.5 m/s to 5.5 m/s in order to decrease the mechanical power, during the instants when performing the energy conversion and transfer of the maximum electrical power to the load.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank CNPq (National Counsel of Technological and Scientific Development), CEESP-CT-UFSM, and IFSC for the financial support during the development of this work.
Supplementary Materials
The video is the prototype of Magnus effect wind turbine of 1 m in diameter with 6 cylinders of 4 cm in diameter made of polyacetal.