Analysis of the Wind System Operation in the Optimal Energetic Area at Variable Wind Speed over Time

Due to high mechanical inertia and rapid variations in wind speed over time, at variable wind speeds, the problem of operation in the optimal energetic area becomes complex and in due time it is not always solvable. No work has been found that analyzes the energy-optimal operation of a wind system operating at variable wind speeds over time and that considers the variation of the wind speed over time. In this paper, we take into account the evolution of wind speed over time and its measurement with a low-power turbine, which operates with no load at the mechanical angular velocity ωMAX. The optimal velocity is calculated. The energy that is captured by the wind turbine significantly depends on the mechanical angular velocity. In order to perform a function in the maximum power point (MPP) power point area, the load on the electric generator is changed, and the optimum mechanical velocity is estimated, ωOPTIM, knowing that the ratio ωOPTIM/ωMAX does not depend on the time variation of the wind speed.


Introduction
In the literature [1][2][3][4][5][6][7][8][9][10][11][12], the operation of the wind turbine (WT) in the maximum energy area is studied at a constant wind speed; using various mathematical models, data from the manufacturing company and/or obtained in laboratory conditions, which are quite different from those in real conditions operation [13][14][15].For this reason, the electrical energy obtained is less than the maximum value that was obtained in a maximum power point (MPP) operation, at optimum mechanical angular velocity (MAV), ω OPTIM .
At variable wind speeds over time, the problem of operation in the MPP area becomes complex and not always solvable in due time due to high mechanical inertia and rapid variations in wind speed over time.
The WT power characteristic, i.e., the function P WT (ω), at a certain constant velocity in time, shows a maximum at ω OPTIM .
The optimal energy is at the velocity ω OPTIM , where the power captured by the WT is the maximum, the MPP point on the power characteristic.
Important points on the WT power curve are: maximum power point-MPP-with MAV-ω OPTIM ; and, -null power point with maximum MAV-ω MAX .
The correct determination of these points, in real operating conditions, assures the operation in the MPP area.
At variable wind speeds between V MIN and V MAX , the WT power takes values in the hash area of Figure 1 between: P WT-MAX -the power at V MAX and P WT-MIN -the power at V MIN The power at permanent magnets synchronous generator (PMSG) takes values between P G-MAX and P G-MIN , the green hazel area of Figure 1.
Sustainability 2019, 11 FOR PEER REVIEW 2 The optimal energy is at the velocity OPTIM, where the power captured by the WT is the maximum, the MPP point on the power characteristic.
Important points on the WT power curve are: -maximum power point-MPP-with MAV-OPTIM; and, -null power point with maximum MAV-MAX.
The correct determination of these points, in real operating conditions, assures the operation in the MPP area.
At variable wind speeds between VMIN and VMAX, the WT power takes values in the hash area of Figure 1 between: PWT-MAX -the power at VMAX and PWT-MIN -the power at VMIN The power at permanent magnets synchronous generator (PMSG) takes values between PG-MAX and PG-MIN, the green hazel area of Figure 1.
PMSG power oscillations are much lower due to the high EOLIAN ELECTROENERGY SYSTEM (EES) equivalent moment of inertia.The equivalent moment of inertia attenuates PMSG's power oscillations from the WT as a shock absorber.In the present paper, the results that were obtained by simulations are based on the usual mathematical models of WT and PMSG.

Mathematical Models
The simulations that presented in the paper are based on the classical mathematical models of WT and PMSG [16][17][18][19].

The mathematical model of WT, (MM-WT)
We will use a classical turbine model [19], which allows for the estimation of the reference angular velocity ref.The mathematical model of the WT also allows for the calculation of the optimal velocity, so the captured energy will be a maximum one.
The power given by the WT can be calculated using the following equation: PMSG power oscillations are much lower due to the high EOLIAN ELECTROENERGY SYSTEM (EES) equivalent moment of inertia.The equivalent moment of inertia attenuates PMSG's power oscillations from the WT as a shock absorber.
In the present paper, the results that were obtained by simulations are based on the usual mathematical models of WT and PMSG.

Mathematical Models
The simulations that presented in the paper are based on the classical mathematical models of WT and PMSG [16][17][18][19].

The Mathematical Model of WT, (MM-WT)
We will use a classical turbine model [19], which allows for the estimation of the reference angular velocity ω ref .The mathematical model of the WT also allows for the calculation of the optimal velocity, so the captured energy will be a maximum one.
The power given by the WT can be calculated using the following equation: where: ρ-is the air density, R p -the pales radius, C p (λ)-power conversion coefficient, λ = Rω/V, V-the wind speed, and ω-mechanical angular velocity (MAV).
The power conversion coefficient, C p (λ), could be calculated, as follows: By replacing, we can obtain the power conversion coefficient, as follows: The power given by the wind turbine can be calculated, as follows: or where For the wind turbine WT, the producer gives the experimental power characteristics, P WT (ω, V), or torque characteristics T WT (ω, V), with the last ones being known as mechanical experimental characteristics.
The maximum value of the function P WT (ω, V) is achieved for a reference MAV ω ref , and it yields: ) where k 0 is the constructive constant of the turbine.This result proves the direct link between reference velocity and wind speed.By replacing this result, it yields: ie a cubic dependence of the maximum power value of the wind speed V.
In cases where the wind speed varies significantly over time, the result that is obtained must be re-analyzed and an equivalent wind speed is required.
For power wind turbines: P TV-MAX = 11 KW the experimental power characteristics and P WT (ω, V) are modeled by: Obtaining the maximum WT power value.

Case 1
The maximum P WT (ω) function is obtained by canceling the derivative with the solution: By replacing this value in the function P WT (ω), maximum velocity is obtained, at the wind speed V: or Case 2 To obtain the maximum of P WT (ω) function, we can also use the no-load MAV value from a low-power auxiliary WT (WT AUX ), operating at MAV, ω MAX .The auxiliary turbine is a model of the main turbine; it is much smaller and models the main turbine power characteristic.The main and auxiliary turbines are in the same location and both are exposed to the same wind.This MAV value takes into account the evolution of wind speed in time.The value of the ω OPTIM /ω MAX ratio is constant for a given WT and it does not depend on the time variation of the wind speed [11][12][13][14] This value is very easy to obtain during the operation of the EES.

The GSMP Mathematical Model, (MM-GSMP)
To analyze the behavior of the system WT-PMSG for the time-varying wind speeds, it uses the orthogonal mathematical model for permanent magnet synchronous generator (PMSG), which is given by the following equations [11][12][13][14]: where U-stator voltage; I d , I q -d-axis and q-axis stator currents; Sustainability 2019, 11, 1249 5 of 16 θ-load angle; R 1 -phase resistance of the generator; L d -synchronous reactance after d axis; L q -synchronous reactance after q axis; Ψ PM -flux permanent magnet; M PMSG -PMSG electromagnetic torque; and, The two functions: P G (R, ω)-electric power provided by the generator and M PMSG (R, ω)-moment at the generator shaft depend on: R-load resistance and ω-MAV.
For R = ct the moment M PMSG depends linearly on ω, and the power depends squarely on ω.
From the nominal values of the PMSG [1], for the nominal power: From the equations of the PMSG, it obtains The analysis of the transient phenomena specific to variable wind speeds is done by simulation.
The simulations are based on the motion equation: where J-equivalent moment of inertia; M WT -Moment on WT; M PMSG -Moment of PMSG.

Simulation Results at the Time Variable Wind Speed
This analyzes the EES operation at a time variation of wind speed of the form: as shown in Figure 2.  (26) and, therefore, power becomes: under these conditions and when considering friction losses as 5% of PECH(571), the motion equation was obtained in the form of:

No-Load Operation of See
For the no-load operation, the motion equation is: The MAV over time is presented in Figure 3. (26) and, therefore, power becomes: under these conditions and when considering friction losses as 5% of PECH(571), the motion equation was obtained in the form of: After two hours, the wind speed drops to 0 and the WT switches to fan mode.The power absorbed by it can be calculated with: where k, the coefficient of proportionality, is determined by knowing that at: ω = 571 [rad/s] the equivalent power of WT is: It yields: and, therefore, power becomes: under these conditions and when considering friction losses as 5% of P ECH (571), the motion equation was obtained in the form of: and, therefore, it can be considered that WT stopped, as it can be seen in Figure 4. ], and, therefore, it can be considered that WT stopped, as it can be seen in Figure 4.The energy that is captured during this time period is significantly dependent on MAV values, which are load-dependent.
For a given time interval, the optimal MAV value, OPTIM, is calculated based on the ratio: OPTIM / MAX = 0.68 where the MAX value is obtained by a no-load operation of a low power auxiliary WTAUX,

No-load operation of WTAUX
The moment of inertia moment, J, of the WTAUX is much smaller than the power WT, for example: For a time variation of wind speed of the form:  The energy that is captured during this time period is significantly dependent on MAV values, which are load-dependent.
For a given time interval, the optimal MAV value, ω OPTIM , is calculated based on the ratio: where the ω MAX value is obtained by a no-load operation of a low power auxiliary WT AUX .

No-Load Operation of WT AUX
The moment of inertia moment, J, of the WT AUX is much smaller than the power WT, for example: For a time variation of wind speed of the form: the motion equation, for WT AUX , becomes: the same as the PMSG, PPMSG(520).PMSG load resistance is calculated from the algebraic system: or: with the solution: For EES to operate in the MPP area, at: OPTIM(33)= 514.64[rad / s]

Load Operation of the EES
Based on these values and measuring, at time t k , the current MAV, ω tk , from PMSG, we can estimate the amount of kinetic energy that is to be taken from PMSG, as follows: This energy is added by the wind energy captured by WT, W WT , during the time ∆t: When considering that, at the beginning, the operation of the EES is stable at the following MAV: , the wind energy captured, W wind , has the value: (33) and therefore, the average WT power is: the same as the PMSG, P PMSG (520).PMSG load resistance is calculated from the algebraic system: with the solution: ω = 520 [rad/s] and R = 216.12 [Ω] For EES to operate in the MPP area, at: must take the kinetic energy ∆W KINETIC corresponding to the both MAV ω(0) and ω OPTIM (33) and the wind energy that was captured by the WT in the time interval ∆t = 33 [s].
The calculation of the wind energy captured by the WT over the analyzed time frame and at wind speed, V(t), can be done through the integration of WT power over time: 3600 − 2 sin 017943t /ω − 0.02 •e −98.06((15•e −t/3600 − 2 sin 017943t)/ω) 15•e − t /3600 − 2 sin 017943t In determining the value of wind energy captured by WT, two problems arise: (1) The estimated value of captured wind energy is based on the use of MM-WT, which is valid only under certain conditions, usually different from the operating conditions.(2) When calculating the integrity of the WT power, it is necessary to know the time variation of MAV, which is not known in advance.This can only be known later by solving the motion equation or by direct measurements.We can determine the maximum power of the TV corresponding to the wind speed at that time using the relationship: The wind speed varies, but between a maximum value: V MAX and a minimum value: V MIN , it is necessary to introduce the equivalent wind speed term, V ECH , the speed at which the WT power has the same value as in the real case over a given time interval.
Determining the equivalent speed value in the graph of the function V(t) that was obtained from the measurement of wind speed over time is complicated because V ECH depends both on the evolution of wind speed and MAV, ω, on which WT works.
Considering the results of [11][12][13][14] the equivalent wind speed is calculated while using the relation [11][12][13][14]: or Based on this value, the maximum WT power can be obtained: While considering the curve F->MPP from Figure 6, the medium WT power during ∆t interval, it yields: where P WT-F is the power in point F. It results: The total win energy captured in this time interval is: The sum of energies: the kinetic energy ∆W KINETIK corresponding to the two MAV ω(0) and ω OPTIM (33) and the wind energy that is captured by the WT in the time interval ∆t = 33 [s] implies the amount of energy that is required to be taken over by the generator on the F-MPP and it has the value The control is performed considering time intervals ∆t = 33 [s]. Step The value of the kinetic energy to be captured by the PMSG, between MAV and ω OPTIM (33) = 514.64[rad/s] is: The PMSG power, to reach the optimum MAV for the period ∆t, is: The load resistance of the PMSG to reach the movement of the F point to MPP can be calculated from: with the solution ω = 517.32 In these conditions, the motion equation for the EES becomes: The MAV over time is presented in Figure 7: The differences are quite insignificant, below 0.025%.
Step 2; t = 66 ÷ 99 [s] The value of the kinetic energy to be captured by the PMSG, between MAV (7) The prescribed power value at the generator to reach the optimal MAV, ω OPTIM-tk , in the time interval ∆t, is calculated from the energy value, W G-REQUIRED , as: Regardless of the evolution of wind speed over time, we can estimate ω OPTIM and thus achieve optimal energy control that is based on the value of the ω OPTIM /ω MAX ratio, which does not change regardless of the wind speed evolution in time.
Based on the connection between ω OPTIM and ω MAX , and calculating the variation of kinetic energies, a simple and economically efficient driving system can be conceived, as seen in Figure 9.
Sustainability 2019, 11 FOR PEER REVIEW 12  − =   + ∆  (66) 7) The prescribed power value at the generator to reach the optimal MAV, OPTIM-tk, in the time interval Δt, is calculated from the energy value, WG-REQUIRED, as: − =  − / .(67) Regardless of the evolution of wind speed over time, we can estimate OPTIM and thus achieve optimal energy control that is based on the value of the OPTIM/MAX ratio, which does not change regardless of the wind speed evolution in time.
Based on the connection between OPTIM and MAX, and calculating the variation of kinetic energies, a simple and economically efficient driving system can be conceived, as seen in Figure 9.

Conclusions
This paper analyzes the operation of a wind power system so as to achieve optimal energy performance.By analyzing several cases, it was possible to obtain the basic parameters that lead to an optimal functioning from the energy point of view and the maintenance of the EES in the MPP area by measuring the speed and the estimation of the wind energy that is captured by the WT.The estimation of the wind energy captured by the WT is based on MM-WT, which requires a more accurate determination of it.The equivalent wind speed is defined and the optimal angular velocity is calculated, as a function of it.By knowing the optimal speed, the system can operate at the Maximal Power Point.The energy that is captured by the WT has maximum values in the MPP area.Through the simulations presented, the PMSG loads could be identified so that the WT+PMSG will reach the optimal area of in terms of energy in the shortest possible time.The method is based on the optimal MAV determination with an auxiliary WT operating in a no-load regime.The method is based on the fact that the optimal MAV value, OPTIM, is directly proportional to MAX.By measuring the maximum MAV, MAX, using an auxiliary WT can determine, at any time, OPTIM, regardless of the time variation of the wind speed.By calculating the variations of the kinetic energies and the measurement of the electric energy that is flowing by the generator, the value of the power to it is determined, a value that is achieved by controlling the switching elements of the power converter between the generator and the network.

Conclusions
This paper analyzes the operation of a wind power system so as to achieve optimal energy performance.By analyzing several cases, it was possible to obtain the basic parameters that lead to an optimal functioning from the energy point of view and the maintenance of the EES in the MPP area by measuring the speed and the estimation of the wind energy that is captured by the WT.The estimation of the wind energy captured by the WT is based on MM-WT, which requires a more accurate determination of it.The equivalent wind speed is defined and the optimal angular velocity is calculated, as a function of it.By knowing the optimal speed, the system can operate at the Maximal Power Point.The energy that is captured by the WT has maximum values in the MPP area.Through the simulations presented, the PMSG loads could be identified so that the WT + PMSG will reach the optimal area of in terms of energy in the shortest possible time.The method is based on the optimal MAV determination with an auxiliary WT operating in a no-load regime.The method is based on the fact that the optimal MAV value, ω OPTIM , is directly proportional to ω MAX .By measuring the maximum MAV, ω MAX , using an auxiliary WT can determine, at any time, ω OPTIM , regardless of the time variation of the wind speed.By calculating the variations of the kinetic energies and the measurement of the electric energy that is flowing by the generator, the value of the power to it is

Figure 2 .
Figure 2. Wind speed variation over time.

Figure 2 .
Figure 2. Wind speed variation over time.
29) the motion equation, for WTAUX, becomes: and the time variations of  and OPTIM are obtained over the interval 0-333 [s], as shown in Figure5.

Figure 6 .Figure 6 .
Figure 6.The movement of the operating point towards MPP.

Figure 9 .
Figure 9. Optimal control of a wind system based on the connection between OPTIM and MAX.

Figure 9 .
Figure 9. Optimal control of a wind system based on the connection between ω OPTIM and ω MAX .