Performance evaluation and modelling of the Atir marine current turbine

In this study, the behaviour of a full-size prototype of a platform for tidal energy was analysed. The turbine of this platform is formed by two counter-rotating rotors aligned with the tidal current, such that one rotor is always under the wake of the upstream rotor. The platform was tested under real working conditions; the tidal current was emulated by towing. The results of these tests were used to develop and parameterize a turbine model, which is presented in this paper. The turbine was modelled in terms of power, torque, and thrust coefﬁcients. This model was used to estimate the power curve of the platform and design control strategies that, in addition to maximizing energy production, contribute to the stability of the platform.


INTRODUCTION
Tide is defined as the periodic and alternating rise and fall of the ocean waters caused by the gravitational attraction of the moon and sun. The cyclical nature of the tidal phenomenon causes the generation of marine currents that vary in direction and intensity during a tidal cycle. A crucial difference with respect to other renewable sources is that tides are predictable [1], which makes this energy source very attractive.
To harness that kind of energy, Magallanes Renovables developed a floating system based on a platform that incorporates a submerged part where rotors are installed [2][3][4]. The platform is designed to be anchored to the sea bottom by four mooring lines. Thus, it does not require any construction on the sea bottom, thereby reducing the typical installation and maintenance costs related to marine energy devices [4].
The platform has a horizontal-axis tidal current turbine, which is arguably the most mature technology in tidal energy systems [5][6][7]. The turbine is formed by two counter-rotating rotors aligned with the tidal current (from bow to stern or backward), as shown in Figure 1. Both rotors have a variable pitch with a high range of variation, which allows them to change their rotation direction so that it can harvest energy from both tide directions without moving the platform. writing this paper) [19]. Before these tests at EMEC, Magallanes set up the systems to check the correct operation of the platform and its capabilities. For this purpose, tidal currents were emulated by towing the platform using a tugboat [14].
The objective of this study was to model the MCT of the Atir platform using data obtained from tests under emulated real working conditions. Given that no data were available from laboratory tests (such as those using flume tanks), several limitations (e.g., the conditions of the tests were not completely controlled) had to be overcome to develop the model.
No previous works presented a model for a full-size tidal converter. The majority of studies focused on numerical simulations [20][21][22] and laboratory tests of prototypes at a reduced scale [13,[23][24][25]. In fact, a model setup using measurements in real working conditions for these types of devices has not been published to date. The focus of this study was on modelling the turbine formed by two counter-rotating rotors by using only data collected during the tests that took place at the Ria de Vigo (Spain).
A model that reasonably represents the behaviour of the MCT from the point of view of generation was developed. This model is useful for predicting the behaviour of the MCT in different conditions from those of the tests and for adjusting the control parameters (mainly blade pitch and torque control in the variable speed drives or AC/AC converters). Therefore, different control strategies can be designed to maximize the power generation without affecting the stability of the platform.
In the following sections, the Atir platform and its main components are described. Then, the tests conducted are discussed, and their main results are analysed. Finally, a model of the MCT, which was validated using the measured data, is introduced. Several control possibilities are also evaluated in this section.

FLOATING TIDAL ENERGY PLATFORM
The tidal energy platform, developed by Magallanes Renovables (Spain) and called Atir, consists of a floating platform joined to an MCT formed by two rotors (see Figures 1 and 2). Its main characteristics are listed in Table 1.  The fundamental advantages of being a floating platform are low maintenance (it has an accessible engine room) and lower installation cost [26]. Given that it is a floating facility, it is adaptable to all sea environments and has little environmental impact.
The platform was designed in such a way (e.g. it features symmetrical shape of the hull, identical rotors etc.) that its behaviour is not affected when the tidal current direction is reversed.

Turbine configuration
The MCT is formed by a pair of rotors, one at the bow side and the other one at the stern side. Both rotors have three blades and are designed to rotate in opposite directions (see Figure 1). The rotors can vary their blade pitch from 0 • to 270 • . Therefore, they can optimize the energy capture under different currents [27], and their rotation direction can be changed according to  Figure 3, thus avoiding the movement of the platform. The rotor hub is located 16 m below the sea level (see Table 1). The rotors are aligned with the tide direction such that when one rotor is upstream with respect to the tidal current direction, the other rotor is downstream.
The rotors are mounted in the same nacelle, which implies a separation of roughly one diameter such that the downstream rotor is strongly affected by the upstream rotor (see Figure 1).

Drivetrain
Both rotors have identical drivetrains consisting of a high-speed SCIG [28] mechanically coupled to the rotor through a gearbox (see Figure 4). The energy from the generator is delivered to the power network through a full-power back-to-back AC/AC converter. This type of configuration is widely used in European offshore wind farms [28]. Finally, the outputs of both AC/AC converters are connected to a transformer with four windings, which is shared with the ancillary services. The energy is generated at 690 V and deliv-ered to the distribution network at 11 kV; this MV was selected according to the network conditions at the EMEC site.

Clockwise and counterclockwise rotations
The upstream rotor has clockwise rotation, whereas the downstream rotor has counterclockwise rotation (see Figure 1). When the current goes to the bow, the bow rotor is the upstream rotor, whereas the stern rotor is the downstream rotor. When the direction of the current is reversed, the blades of the rotors must rotate in opposite directions. In particular, they must be re-oriented by 180 • upon changes in the tide direction.
The rotor blades can rotate by 270 • to vary between the fullfeathered positions and maximum attack for the two tidal current directions (see Figure 3).

Monitoring and control
The platform was monitored and controlled using an SCADA system in which the main variables measured were the following ones: • Marine current speed using a current speed meter installed on the hull of the platform. • Platform speed by a GPS (during the towing tests, this value represented the current speed seen by the rotors). • Torque in the high-speed shaft by means of a torque meter (this value was also estimated by the control software of the AC/AC converter). • Electrical variables (power, voltage, current etc.) through the control software of the AC/AC converter.
Furthermore, during the tests, the main variables controlled were the following ones: • Blade pitch position, from 0 • to 270 • , which allows even changing the rotors' sense of rotation. • Load torque in the AC/AC converter; when this value is zero, the generator is considered to be unloaded. The load torque was set to different values during the tests.
Data in SCADA were stored during periods of 1, 15, or 30 s depending on the variable. For example, the yaw, pitch, and roll of the platform were stored every second; measurements obtained from variable speed drives (torque, power, rotation etc.) were stored every 15 s; finally, the blade pitch was saved every 30 s.

Power network emulation
Owing to the nature of towing tests, it was not possible to establish a connection to the power network. Therefore, the network was emulated using an autotransformer (400 kVA; The objective was to maintain the diesel unit generation at the lowest possible value. To this end, the power injected by the platform matched the power consumed by the resistors by connecting or disconnecting resistor steps. According to the tests conducted on the system, the voltage variation satisfied the limits established in European standards [29]. The frequency variation was higher, but the frequency remained within limits allowed by the power converters.

Test conditions
In this paper, the results of the tests in which the tidal current was emulated by towing the platform with a tugboat (see Figure 7 and Table 2) are analysed. The tests were conducted in the Ria of Vigo (Northwest of Spain), where the seabed depth is  The main objective of these tests was to configure the platform to work in tide conditions and analyse its behaviour. The main differences between these tests and real working conditions under tidal currents are as follows: 1. During tides, the current speed decreases with depth [30]; however, the speed profile during towing is almost constant. 2. Waves act on the platform and tugboat, thereby limiting the test conditions (speed, towing force etc.). They induce oscillations in the towing, and consequently in the moving speed.
Note that the period of the waves seen by the platform is affected by its speed. 3. Waves, wind, and marine currents affect the test conditions. 4. During the tests, the towline length was greater than 100 m to minimize the impact of the wake produced by the tugboat (see Figure 7). 5. The emulated tidal current depends on the tugboat operation; therefore, it is affected by the blade pitch and TSR of the rotors. Under real conditions, the current depends only on the tide. 6. The platform showed a certain level of roll, pitch, and yaw during towing (see related discussion in Section 6).
Several variables were measured during the tests. For example, Figure 8 shows the evolution over time for the following variables: 1. GPS speed (knots): Speed of the platform during the tests.
This speed was used as freestream current speed because, according to the characteristics of the test site, the existing marine current can be neglected. The results show that a power of approximately 400 kW was generated during the tests with a platform speed of approximately 5 knots (2.6 m/s). Note that the achievable speed was limited by the capability of the tugboat, which was strongly dependent on the rotor behaviour and sea conditions.

Estimation of feathered and optimum blade pitch positions
The blades were mounted at sea. Then, it was necessary to check the real values of the blade pitch for the full-feathered position and the optimum one that maximizes the power coefficient.
The blade pitch for the optimum power coefficient is the pitch at which, for a given marine current, the maximum power is generated. It is difficult to obtain because the speed of the tugboat depends on the behaviour of the rotors for a given speed and blade pitch, so it is affected by the generated power. Therefore, high values of generated power cause a reduction in tugboat speed that complicates the test of the MCT at high current speed and high power.
The data from the tests conducted with only one rotor running and at different values of current speed, blade pitch, and generated power were used to obtain the map of pitch, TSR, and power coefficient values shown in Figure 9. The same information but changing the main axes can be seen in Figure 10. From these figures, the optimum pitch position can be approximately obtained; its value is 7 • . This is the mean value of the pitch angle values at which the highest power coefficient values at the generator side (c pg ) were achieved.
The full-feathered position refers to the blade position at which the rotor does not rotate at different current speeds. Its value is between 92 • and 94 • . This value was estimated through tests in which the rotor was not rotating and was unloaded. Initially, the blade pitch was set to a value that was slightly higher than the supposed full-feathered position. At this moment, the blade pitch was smoothly increased and the value prior to that where the rotor started to rotate was supposed to be the full-

ROTOR AND DRIVETRAIN MODELLING
The main relationships between power (P r ), torque (Q r ), and thrust (T) in a rotor can be expressed as follows [31]: where A and R are the area and radius of the rotor, respectively; ρ is the fluid density; v is the current speed; and c pr , c qr , and c t are the power, torque, and thrust coefficients, respectively. The previous equations can be used when only one rotor is running. When both rotors are working, the running conditions of the upstream rotor (referred to as rotor no. 1) affect the downstream rotor (referred to as rotor no. 2) due to wake. Furthermore, the current was measured at one point of the platform that was not affected by rotor turbulences. Therefore, it is considered representative of the freestream current. Accordingly, the parameters of rotor no. 2 were calculated using that current, so they are apparent values. The TSR for both rotors is calculated using the following equations: where v ∞ is the freestream current speed in m/s; R is the radius rotor in m; ω 1 and ω 2 are the rotor speeds of the upstream and downstream rotors in rad/s, respectively; TSR 1 is the tip speed ratio in rotor no. 1; and TSR 2a is the apparent TSR value in rotor no. 2. Similarly, the power coefficient in rotor no.1 (c pr1 ) and the apparent power coefficient in rotor no. 2 (c pr2a ) can be defined as follows: where P r1 and P r2 are the powers in rotors 1 and 2, respectively. Finally, another parameter of interest is the torque coefficient: where Q r1 and Q r2 are the torques in rotors 1 and 2, respectively; c qr1 is the torque coefficient in rotor no. 1; c qr2a is the apparent torque coefficient in rotor no. 2; and R is the rotor radius.
Regarding the thrust force, it can be defined from the thrust coefficient in rotor no. 1 (c t1 ) and the apparent thrust coefficient in rotor no. 2 (c t2a ) as follows: where T 1 and T 2 are the thrust forces in rotors 1 and 2, respectively.
The total power of the MCT at the rotor side can be related to a platform power coefficient (c prP ) defined as follows: In addition to the power, the total rotor torque over the platform affects the behaviour of the platform, for example through its impact on the platform roll (see Section 6). This can be related to the torque over each rotor (it must be considered that they have counter-rotation) by means of the following expression: where c prP is the platform torque coefficient at the rotor side. According to the previous definitions, the performance of the rotor in terms of maximum torque, maximum power, or minimum thrust is more easily analysed in terms of the power, torque, and thrust coefficients. Therefore, these coefficients, as previously defined, were used to model the rotor.
The measurement data from the tests were used to obtain the values of these coefficients. However, most of the measurements were conducted in the power converter and highspeed shaft. Consequently, to estimate the rotor behaviour, it was necessary to consider the performance in terms of the losses in the generators, gearboxes, and AC/AC converters [32], whose models are presented in the following sections.

Gearbox performance
The platform has two gearboxes (see Figure 4) mounted between each rotor (low-speed shaft) and the generator (highspeed shaft). The gearbox has a gear ratio of 1:98. The loss data provided by the manufacturer are shown in Table 3.
The power losses can be obtained from the torque losses at different rotor speeds. The resulting data were fitted using the Power losses in the gearbox following equation (see Figure 12): where ω is the rotor speed in rad/s and P gbl denotes the gearbox power losses in kW.

Generator performance
The platform uses one asynchronous generator for each rotor (see Figure 4); its data can be seen in Table 4. According to the data provided by the manufacturer, the generator performance depends on the speed and torque load, as shown in Table 5.

Performance of generator and gearbox
The power available in a rotor can be obtained from the generator power (measured by the AC/AC converter) and the performance values for the generator and gearbox as follows: where P r is the power in the rotor; P g is the power in the generator; η gb is the gearbox performance; and η g is the generator performance. From the power values in the rotor and generator, and the measurements in the marine current, the power coefficient in the rotor (c pr ) and the generator (c pg ) can be obtained using the following equation: where P g = 1 2 c pg Av 3 ∞ . The power coefficient values shown above were obtained using measurements from only the upstream rotor to overcome the influence over the downstream rotor. The resulting values are shown in Figure 13. Note that the tests were conducted at a power below 50% of the platform rated power. Accordingly, the rotor speed was also below 50% of the rated speed. In this situation, the performance values of the generator and gearbox were approximately 70% and 90%, respectively. The estimated The rotor torque (Q r ) can be obtained from the torque values Q H in the high-speed shaft, where the torque meter was installed, and from the estimation of the gearbox losses (Q bgl ): where Q H = 1 2 c qH Av 2 ∞ , and c qH is the coefficient torque obtained from the torque at the high-speed shaft. The torque coefficient at the rotor (c qr ) can be estimated using these data, yielding the values shown in Figure 14, where the maximum estimated value at the generator side is 0.12.
The rotor torque when the generator is running under no load is equal to the gearbox losses plus the no-load losses of the generator. Note that, for speeds below 1.5 rpm at the lowspeed shaft (see Table 3), the gearbox losses lie between 100 and 150 kNm. Nevertheless, other mechanical elements, such as seals and bearings, were not taken into account. Thus, to start the rotor, a torque higher than 150 kNm is necessary.
Similarly, the mechanical power at the rotor when the generator is running under no load is the no-load power at the generator plus the power losses at the gearbox. According to the measurements, the overall no-load power in the drivetrain is in the range 30-40 kW.

Losses at AC/AC drive
According to the measurements (see Figure 15), the converter losses are in the range 15-25 kW for generated power levels up to 340 kW (40% of the rated power) and a generator speed

Rotor modelling
From a dimensional analysis of the blades, in which eighteen blade sections were considered, the rotor was modelled using the blade element momentum (BEM) method [6,7,33,34,20] implemented in XFOIL [30,35,23] and MATLAB [36]. A detailed flowchart of the process is provided in Appendix 11.  [37] (see Figures 18 and 19), and thrust coefficient c t (see Figure 20). These coefficients were calculated at different values of the blade pitch, TSR, and Reynolds numbers [21]. The figures show coloured contour lines with numerical values indicated in the colour bars.
As an example of application of the resulting model, the estimated value for the maximum rotor power coefficient (c pr,max ) was 0.52 for a pitch angle of 7 • and TSR of 6.8. Under these conditions, the torque coefficient (c qr ) was 0.082 and the thrust coefficient (c tr ) was 1.1.
The rotor model was validated through measurement data from the upstream rotor and from tests with only one rotor running, so the wake effect between rotors did not have to be considered. For this purpose, the coefficients c pr , c qr , and c t were obtained by taking into account the models for the drivetrain components shown in the previous sections. The values obtained from the measurements are represented by coloured points Figures 16-20.    Figure 22 and fitted by means of linear regression (see the resulting equation in the figure). Note that there is a good agreement with respect to the model (coefficient of determination R 2 ≈ 0.97) with a relative error below 20%.
The most important conclusion of this section is that the measured values reasonably agree with the predicted values using the rotor model.
In the following sections, we analyse the model for the two rotors running simultaneously and the wake effect between them taken into consideration.

Counter-rotating turbine
As mentioned in previous sections, the turbine of the platform is composed of two counter-rotating rotors (the model for a single rotor is shown in Section 4.5) mounted on the same axis and having the same swept area. Depending on the direction of the tide current, while a rotor is upstream (or downstream), the other one is downstream (or upstream). In any case, to analyse the behaviour of the complete turbine, the wake effect between rotors must be considered [12]. By using the model of the upstream rotor, we calculated the wake effect at a distance of one rotor diameter (1 × D), at the location of the downstream rotor (see Appendix 10). Once the wake effect was obtained, the rotor behaviour of the downstream rotor could be calculated.
During the calculations, we considered a set of TSR values from 0 to 15 with a step equal to 0.2, and a set of pitch angles from −5 • to 40 • with a step of 1 • . This means solving 3 × 10 6 working scenarios for the turbine, taking into account that only generating situations were considered.
As an example of application of the resulting model, Figure 23 represents the achievable power coefficient values in rotor no. 2 (c p2a as a function of TSR 2a and β 2 ) for different working conditions of rotor no. 1 defined by TSR 1 , β 1 , and c p1 .
According to the values shown in Figure 23, when rotor no. 1 is running at its maximum power coefficient, the maximum power coefficient values achievable in rotor no. 2 are lower than 0.03. As an example, this is shown in case a for TSR 1 = 7 and β 1 ≈ β opt = 7 • , where β opt is the pitch angle at which the maximum power coefficient values can be reached. The main reason for this is that the current speed at rotor no. 2 is significantly affected by rotor no. 1. However, when rotor no. 1 is running at higher blade pitch values (see case b with β 1 ≈ 20 • , c p1 ≈ 0.25), the value of c p2a can reach up to 0.3. As an example of extreme situation (see case c), the power coefficient of rotor no. 2 can be maximized (c p2a > 0.5) for a blade pitch of rotor no. 1 of approximately 20 • (β 1 ≈ 20 • , TSR 1 = 5).
In any case, most of the previous studies analysed the wake effect only when the upstream turbine was at its optimum blade pitch and for the maximum power coefficient. However, the maximum platform power is achieved when both rotors share the total power [9]. This implies that neither the upstream rotor nor the downstream rotor work at its optimum pitch, as will be shown in the following sections.

CHARACTERIZATION OF THE TURBINE
In this section, the behaviour of the turbine of the MCT, which is formed by two counter-rotating rotors (see Section 2), is analysed. First, the influence between both rotors is assessed. Note that the downstream rotor is always affected by the wake of the upstream rotor. Next, the power, torque, and thrust coefficients are estimated considering this effect. Finally, several considerations of the control possibilities of the turbine are analysed.

5.1
Considerations of the effect of the upstream rotor on the downstream rotor Figure 24 shows the TSR for the upstream (TSR 1 ) and downstream (TSR 2a ) rotors when the downstream rotor was running under no load at a given current speed. For low blade pitch values (below 20 • ) of the upstream rotor (β 1 ), the TSR of the downstream rotor (TSR 2a ) is significantly reduced from 9 to 4. The lowest value was reached when the upstream blade pitch was at its optimum value.
The reduction in the apparent downstream TSR was due to the decrease in the rotor speed. However, the effective reduction could be compensated by the fact that the stream current seen by the downstream rotor was also lowered owing to the wake effect induced by the upstream rotor.
According to this behaviour, during the tests with a constant torque in the downstream rotor set to 1500 Nm, its delivered power was reduced by approximately 17%, from 217 to 178 kW, when the upstream blade pitch was modified. Concerning the downstream apparent power coefficient (c pr2a ), its value was lowered by approximately 30%, from 0.3 to 0.21, and the TSR (TSR 2a ) by approximately 20%, from 7.5 to 6. Consequently, it was necessary to consider the simultaneous behaviour of the two rotors to obtain the turbine model. This is explained in the following section.

Power coefficients of the upstream and downstream rotors and platform
The maximum generation of the platform was achieved when, for a given marine current, the total power extracted by the two rotors reached its maximum value. This situation can be evaluated using the platform power coefficient (c prP ) (see Equation (10)), which takes into account the overall energy extracted by the two rotors under certain marine current conditions.
The power coefficient of a turbine formed by two rotors is usually higher than that of a turbine formed by a single rotor. However, the advantages of this configuration are more closely related to other running characteristics, such as torque balancing [11,38,24]. In this section, the values of the platform power, torque, and thrust coefficients are estimated and compared with the results obtained during the tests.
To derive the model of the MCT of the platform in terms of power, torque, and thrust coefficients, the wake effect on rotor no. 2 had to be considered. Therefore, the following procedure was carried out: 1. The power coefficient for rotor no. 1 c p1 is calculated for different running conditions (defined by TSR 1 and β 1 ), as shown in Figure 25. 2. For every running condition on rotor no. 1, the working conditions on rotor no. 2 are calculated. This maximizes the apparent power coefficient of rotor no. 2 (c p2amax ). Therefore, for every TSR 1 -β 1 pair, the corresponding TSR 2a (see Figure 26) and β 2 (see Figure 27) are obtained. 3. As a result, for a given TSR 1 -β 1 pair, the maximum platform power coefficient values (c prP = c pr1 + c pr2a,max ) are obtained with their corresponding TSR 2a and β 1 values, as shown in Figure 28. Note that the maximum platform power coefficients are easily achieved when the upstream rotor operates at pitch angles higher than the optimum one [12].
A similar procedure was followed to obtain the platform torque coefficient, which, in this case, is defined as c qrP = c qr1 − c qr2 and represents the torque balance between the rotors (see Figure 29). Similarly, the thrust coefficient sum represents the overall thrust of the platform (see Figure 30).
In the figures mentioned above, a set of points (#1 to #5) obtained from measurements is represented as an example. The values of these points are listed in Table 6. Note that the coefficient values from the model are similar to those obtained from the measurements. For example, in point #2, the values for  Table 6. Finally, note that the maximum platform coefficient can be obtained in a wide range of working conditions of rotors no. 1 and no. 2. This enables the design of different control strategies to maximize the energy production.

Considerations of the platform control strategy
As shown in the previous section, for a given stream current, it is possible to achieve a platform power coefficient of approximately 0.5 (see Figure 28). This value can be reached under different running conditions of rotors no. 1 and no. 2 (see Table 6). In this context, it is possible to maximize the power delivered by the platform and achieve another objective simultaneously. For example, to minimize the roll of the platform, it would be desirable that both rotors have a similar torque, considering that they have counter-rotation. The torque can be balanced when both torque coefficients are similar.
As an example, Table 7 shows the running values estimated for point #2 in Table 6. The platform power coefficient is close to 0.5, and the torque coefficients of rotors no. 1 and no. 2 (c qr1 and c qr2a ) are similar.

5.4
Platform power curve The following values were obtained: the cut-in current speed was approximately 1 m/s; the rated current speed was approximately 2.9 m/s; and the rated power was 1500 kW at the MV side.
The energy produced by the MCT can be estimated using this power curve. For example, data for the East Coast of Florida (USA) available in the Southeast National Marine Renewable Energy Center [39,40] can be used. For this site, tidal current data were measured with an acoustic Doppler current profiler for 287 days in 2019. The recorded mean current speed in the depth range of operation of the MCT was 2.2 m/s, reaching a maximum speed of 5.9 m/s. At this site, the estimated energy production at the MV side is 6238 MWh/year with a value of equivalent full-load hours of 4159 h/year.

ROLL, PITCH, AND YAW DURING THE TOWING TEST
When the platform is operating, the torque of the rotor produces the roll of the platform. This roll is related to the torque of the rotor and the metacentric height according to the positions of the centres of buoyancy and gravity (see Figure 32).
The Atir platform has two counter-rotating rotors. When they have the same torque, the roll of the platform is cancelled. When they do not have the same torque, the roll is a func- tion of the torque difference between the rotors. When towed, the platform roll causes a misalignment of the traction forces that induces significant pitch and yaw (see Figure 33). Figure 34 shows different roll values during a test when only one rotor was running; in this case, the relationship between generator torque and roll is clear.
Apart from mechanical stress on the platform, the roll, pitch, and yaw affect the generation of the turbine (e.g. the effective area seen by the current is reduced when the rotor axis is misaligned with the marine current) or the towing effectiveness (e.g. the towing resistance of the platform is higher when the yaw increases).
The platform pitch is corrected by moving the water between the reservoir tanks placed in the stern and bow. However, this could affect the position of the centre of gravity and, therefore, the overall platform behaviour.
All of the abovementioned aspects must be considered when extrapolating the towing tests to real tidal current conditions.

CONCLUSIONS
The Atir platform is a full-size prototype tidal energy device whose turbine is formed by two counter-rotating underwater rotors aligned with the tidal current. This paper reports an analysis of this platform based on test results and a complete model for its generation system. The starting point for this study was the results of tests carried out at Ría de Vigo (Spain). During these tests, the platform was towed to emulate tidal currents. Under these conditions, the platform was operated in the same way that under real tidal currents, although with reduced rated power owing to the size of the dump loads.
To achieve the objectives of the study, a complete model for the generation systems was developed. It includes the following elements: 1. Losses in gearboxes, generators, and AC/AC converters. 2. Estimation of the rotor behaviour in terms of the TSR, blade pitch, and power coefficient using BEM techniques. Torque and thrust coefficients are also included. 3. Wake effect of the upstream rotor on the downstream rotor.
Model tuning was performed using real data from the abovementioned tests.
As a result, a model of the MCT of the platform is proposed. It was obtained from tests carried out in real conditions. The control and measured variables are not completely under control in this type of tests. This makes obtaining the model parameters more challenging than in a laboratory. Nevertheless, the proposed model accurately represents the platform in terms of generation and rotor behaviour.
Once the model was obtained, the performance of the platform in terms of the power coefficient was estimated. An approximate value of 0.5 for the MCT was achieved during the tests. In addition, the model allows the analysis of the torque balance between rotors, which affects the platform roll, and the thrust forces by means of their corresponding coefficients.
Furthermore, the behaviour of the Atir platform under different conditions from those of tests was analysed. It was concluded that the use of aligned dual rotors allows operation under a wide range of running conditions while the platform power coefficient is close to its optimum value. In this context, several control strategies can be established, for example, by compensating the roll of the platform through the balance of the torque in the rotors.
Overall, this paper reports an MCT model for a full-scale prototype of a tidal energy platform. This model was used to analyse the behaviour of the platform in terms of energy production, torque, and thrust. Furthermore, the viability of control strategies with multiple objectives (energy, platform stability etc.) was studied.

NOMENCLATURE
a axial flow induction factor a′ tangential flow induction factor c p power coefficient c q torque coefficient c t thrust coefficient v current speed (m/s) A rotor area (m 2 ) D rotor diameter (m) R rotor radius (m) P power (W) Q torque (Nm) T thrust force (N) TSR tip speed ratio To analyse the behaviour of the complete turbine, we had to consider the fact that the rotors are aligned with the tide current. Therefore, the downstream rotor, which can be the stern or bow rotor depending on the tide direction, is under the wake of the upstream rotor. This is usually approximated as an effective reduction of the current speed [41][42][43][44]. However, owing to the proximity of the two rotors (see Table 1), a more detailed model had to be used. That model took into account the wake rotation [27,45,25], that is, the axial and tangential current speeds induced by the wake in the downstream rotor [12,22,46]. If only a rotor is considered, the axial speed (v x,i ) and the tangential speed (v t,i ) caused downstream by the rotor blade section i at a distance equal to 1 × D is expressed as follows [12]:

Abbreviations and names
where r i is the radius of the rotor blade section i in m; ω is the rotor speed in rad/s; and a i and a' i are the axial and tangential flow induction factors, respectively, in m/s. When the downstream rotor is considered, the axial and tangential speeds in the rotor can be written as follows: [46]: where x is the distance to the downstream rotor; for the Atir platform, x = D.
Using the above equations, the axial and tangential speeds on the downstream rotor can be obtained, as shown in Figures A.1 and A.2 in which these speeds were calculated for different rotor distances (1D and 2D). To obtain them, the running conditions shown in Table A.1 were considered. for different values of parameters: • Attack angle (α) between ±60 • • Reynolds numbers (Re) between 10 5 and 10 7 • Blade profiles, which are defined by the distance r to the centre of the rotor, from 1.5 to 9.5 m.
The resulting coefficient values are stored in 3D matrixes. BEM model for profile blocks: From the results of the previous block, the following values are obtained: • Axial flow induction factor a • Tangential flow induction factor a' • Attenuation values F • Gradient factors: lift gradient ∂L/∂r, drag gradient ∂L/∂r, torque gradient ∂L/∂r, power coefficient gradient ∂c p /∂r etc.
for different values of parameters: • Relative current speed v between 0.5 and 17 m/s • Relative current speed φ direction between ±45 • The values of the axial and radial fluxes (a and a') are used to calculate the impact over the current speed seen by rotor no. 2 due to the wake caused by rotor no. 1.
Wake current speed: The values of the flow induction factors (a and a') are used to calculate the current speed seen by rotor no. 2, which is affected by the wake caused by rotor no. 1.
BEM model for blades of rotor no. 2: Using data from previous steps, different coefficients (c q , c p, c T , and c M ) are obtained for rotor no. 2. These values are obtained for values between 0 and 15 for TSR 2 , and between −15 • and +45 • for blade pitch β 2 . In this case, the relative speed of each profile is different. Thus, the angle φ is obtained using a and a' from rotor no. 1. This implies that, for each pair of TSR 1 and β 1 values, all running conditions of rotor no. 2 must be obtained (TSR 2 between 0 and 15, and β 2 between −15 • and 45 • ).