Novel Modified Elman Neural Network Control for PMSG System Based on Wind Turbine Emulator

The novel modified Elman neural network (NN) controlled permanent magnet synchronous generator (PMSG) system, which is directly driven by a permanent magnet synchronousmotor (PMSM) based on wind turbine emulator, is proposed to control output of rectifier (AC/DC power converter) and inverter (DC/AC power converter) in this study. First, a closed loop PMSM drive control based on wind turbine emulator is designed to generate power for the PMSG system according to different wind speeds. Then, the rotor speed of the PMSG, the voltage, and current of the power converter are detected simultaneously to yield better power output of the converter. Because the PMSG system is the nonlinear and time-varying system, two sets online trained modified Elman NN controllers are developed for the tracking controllers of DC bus power and AC power to improve output performance of rectifier and inverter. Finally, experimental results are verified to show the effectiveness of the proposed control scheme.


Introduction
Since the petroleum is gradually exhausting and environmental protection is progressively rising, the usage of the clean energy sources such as wind, photovoltaic, and fuel cells has become very important and quite popular in electric power industries.Clean energy sources such as wind, photovoltaic, and fuel cells can be interfaced to a multilevel converter system for a high power application [1][2][3].
Wind turbine which acted as sources of energy has progressively increased in the whole earth.The various control methods and convert technologies of wind energy conversion systems are fast developed in energy conversion application.The PM synchronous generator system has been used for wind power generating system due to many advantages such as simpler structure, better reliability, lower maintenance, and higher efficiency [4][5][6][7][8].Therefore, the PM synchronous generator generation system stands for a significant trend in progress of wind power applications [4][5][6][7][8].The output power behavior of wind turbine is nonlinear.The provided power of vertical-axis turbines is very sensitive to the load variation due to different structure effect [4][5][6][7][8].Thus, the control of operating point is indispensable for maximum output power.The controllable rectifier is used to convert varied AC voltage generated by PM synchronous generator into DC bus voltage.Then, the controllable inverter is used to convert DC bus voltage into AC at a fixed frequency in order to provide for the stand alone or grid applications of electrical utilizations.The major purposes of utilizing wind turbines are to extract maximum power of turbine and deliver appropriate energy to stand alone power or grid power.According to these purposes, the better structure of the power conversion in wind turbines is the AC to DC to AC power converter [9][10][11][12].Reference [11] proposes the intelligent control of a windturbine emulator and an induction-generator (IG) system with an AC/DC power converter using a radial basis function network (RBFN).An on-line trained RBFN is developed for the tracking controller of DC-link power to improve the control performance.Reference [12] proposes a radial basis function network (RBFN) controlled three-phase IG system using AC/DC and DC/AC power converters.Two online trained RBFNs using backpropagation learning algorithm with improved particle swarm optimization (IPSO) are used as the regulating controllers for both the DC-link voltage and the AC line voltage of the DC/AC power inverter.The IPSO is adopted in this study to adapt the learning rates in the backpropagation process of the RBFNs to improve the learning capability.
The Elman neural network (NN) is a partial recurrent network model that was first proposed by Elman [13].Typical Elman NN has one hidden layer with delayed feedback.The Elman NN is capable of providing the standard state-space representation for dynamic systems.The Elman NN can be considered to be a special type of recurrent neural network with feedback connections from the hidden layer to the context layer.The context layer is an additional layer that is used as an extra memory to memorize previous activations of the hidden neurons and to feed all the hidden neurons after the one-step time delay.Therefore, compared with the general recurrent neural networks, Elman NN has a special explicit memory to store the temporal information.Due to the context neurons, it has certain dynamical advantages over static neural network [14][15][16] and it also has been widely applied in dynamical systems identification and control [17][18][19][20].Generally, Elman NN can be considered to be a special kind of feed-forward neural network with additional memory neurons [13].Furthermore, the Elman NN can approximate high-order systems with high precision, and its converge speed is fast.Reference [20] proposes a fieldprogrammable gate array (FPGA)-based Elman NN control system to control the mover position of a linear ultrasonic motor (LUSM).A piecewise continuous function is adopted to replace the sigmoid function in the hidden layer of the ENN to facilitate implementation of the FPGA chip hardware in order to reduce cost and raise high performance.
The recurrent neural network has received increasing attention due to its structural advantage in nonlinear system modeling and dynamic system control [21][22][23][24][25].The most important characteristic of the recurrent neural network is its self-connection to memorize feedback information of the historical influence in the same neuron.Moreover, in the general recurrent neural networks, the specific selfconnection feedback of the hidden neuron or output neuron is responsible for memorizing the specific previous activation of the hidden neuron or output neuron and feeding itself only.Therefore, the outputs of the other neurons have no ability to affect the specific neuron.However, in the complicated nonlinear dynamic system such as PMSG system directdriven by PMSM, the friction torque, various wind, and external nonlinear interference are always a factor.Hence, if each neuron in the recurrent neural networks is considered as a state in the nonlinear dynamic systems, the self-connection feedback type is unable to approximate the dynamic systems efficiently.On the other hand, the feedbacks in Elman NN not only are self-connecting but they also store in the context neurons and feed all the hidden neurons.Thus, the structure of Elman NN is more powerful than the general recurrent neural networks for dealing with time varying, and nonlinear dynamic systems can be approximated efficiently with the additional context layer.In order to improve the ability of identifying high-order systems, some modified Elman NN [26][27][28] have been proposed recently, which proved to have more advantages than the basic Elman NNs, including a better performance, higher accuracy, dynamic robustness, and a fast transient performance.The modified Elman neural network adopted in this paper has not only the feedback connection from the context layer in the hidden layer but also the delay feedback connection from the output layer in the input layer to raise control and transient performance.
Since the PMSGs have robust construction, lower initial and lower maintenance cost, PMSG are suitable for stand alone or grid power sources in small wind energy application.Therefore a PMSM direct-drive PMSG system using the two sets of the same modified Elman NN controllers is introduced as the adjusting controllers for both the DC bus voltage of the rectifier and the AC 60 Hz line voltage of the inverter in this study.Two online trained modified Elman NNs are introduced as the adjusting controllers for both the DC bus voltage of the controllable rectifier and the AC 60 Hz line voltage of the controllable inverter.Moreover, the training algorithms of two sets of the same online trained modified Elman NNs based on backpropagation are derived to train the recurrent weights, connective weights, translations, and dilations.Additionally, for the comparison of the control performance, the proportional integral (PI) controller can be also executed in the PMSG system.However, the control gains of the PI controller are obtained by trial and error method which is very time-consuming in practical applications.Due to the PMSG system with many uncertainties, adjusted capacity and tracking capacity of output voltage controlled by using the PI controller is less improved.To raise the desired robustness and overcome the above problem, the modified Elman NN controller is proposed to control output DC bus voltage of the rectifier produced by PMSM direct-driven PMSG system and control output voltage of the inverter provided by DC bus power.In the proposed modified Elman NN controller, the recurrent weights, connective weights, translations, and dilations are trained online via learning algorithm.Meanwhile, to demonstrate better dynamic characteristics of the proposed controller, comparative studies with the PI controller and the conventional NN controller are demonstrated by experimental results.Therefore, the control performance of the proposed modified Elman NN control system is much improved and can be verified by some experimental results.
This paper is organized as follows.The configuration of PMSG system is reviewed in Section 2. The novel modified Elman NN control system design is presented in Section 3. Experimental results are illustrated in Section 4. Some conclusions are given in Section 5.

Configuration of PMSG System
The variable speed wind turbine of the PMSG system directdriven by PMSM is a complex electromechanical system, which includes the mechanical components and the PMSG.The description of these components is presented as follows.

Model of Wind Turbine.
The characteristic curve of the wind power versus rotor speed for model of wind turbine at different wind speeds in steady state shown in Figure 1 is very important for PMSG system direct-driven by PMSM.The power specification of the adopted wind turbine is the 1.5 kW in this paper.Its diameter is 2 m.It is the three-blade horizontal axis type.It is capable of obtaining the working point of the wind turbine that used the intersection point of the load characteristic curve and the turbine characteristic curve at a designated wind speed.It is a very important characteristic curve that the shaft power of the wind turbine relates to the wind speed V 1 and rotor speed  1 to the maximum power tracking in Figure 1.For convenient usage and application, the characteristic curve shown in Figure 2 can also be expressed as the characteristic curve of the wind turbine model.It represents relationship curve between coefficient   () of power performance and tip speed ratio .According to aerodynamic principle [1,[4][5][6][7], the tip speed ratio  of the wind turbine can be represented as where  is the tip speed ratio of the wind turbine,   is the rotor radius of the wind turbine in meter,   is the rotor speed of the wind turbine in rad/s, and V  is the wind speed in m/s.The output mechanical power   of the wind turbine can be expressed as [1,[4][5][6][7]] where   is the density of air in kg/m 3 , and   is the undraped area in m 2 .The generated torque of the wind turbine for different wind speeds can be obtained from the   () −  curve in modeling usage.It is a very important case that the aerodynamic efficiency is maximum at the optimum tip speed ratio.The produced torque of the wind turbine can be indicated as follows [1,[4][5][6][7]: where   is the produced torque of the wind turbine in Nm.A block diagram for the wind turbine model is shown in Figure 3.The block diagram shown in Figure 3 is to clearly describe the function between the input variable and output variable of the model.Since the rotor of the wind turbine and the rotor of the PMSG are directly coupled through iron coupler, the mechanical angular speed of the rotor of the wind turbine is the same as the mechanical angular speed of the rotor of the PMSG for neglecting the stiffness of the wind turbine and the PMSG.From [29], the mechanical dynamic equation of torque, which the produced torque   of the wind turbine subtract to the electromagnetic torque  er of the PMSG, can be represented as where   is the total moment of inertia in the direct-decoupled system of the wind turbine and the PMSG;   is the total viscous friction coefficient in the direct-decoupled system of the wind turbine and the PMSG,  er is the electrical angular speed of rotor, and P is the number of poles of the PMSG.

Wind Turbine
Emulator Based on PMSM.The wind turbine emulator, which is proposed in [4][5][6][7][8][9][10], is adopted in this study in order to emulate the wind turbine.Additionally, the adopted field-oriented controlled PMSM can be emulated the power speed characteristic curve of a wind turbine in this paper.In addition, a closed-loop robust speed controller, which can fight the intrinsic nonlinear and time-varying characteristic of the PMSM drive, is adopted to adjust the rotor speed with the relevant wind speed in order to emulate the wind variation.

Field-Oriented
Controlled PMSG System.The voltage equations for the PMSG in the rotating reference frame can be indicated as follows [1,[7][8][9][10]: where V 1 is the d-axis stator voltage, V 1 is the q-axis stator voltage,  1 is the d-axis stator current,  1 is the q-axis stator current,  1 is the d-axis stator inductance,  1 is the q-axis Aerodynamic power and torque calculation Calculation of the voltage and electromagnetic torque of PMSG from equations ( 5) and ( 6)  stator inductance, and  1 is the stator resistance.A fieldoriented control is adopted in [7][8][9][10].By using the fieldoriented control, the d-axis stator current can be set zero, that is,   = 0.Moreover, the electromagnetic torque of the PMSG can be expressed as where  pm is the permanent magnet flux linkage, and   = 3 pm /4 is the torque constant.For convenient analysis, the field-oriented controlled PMSG system is adopted.To emulate the operation of the wind turbine, the primary machine has adopted PM synchronous motor which directly mounted to PMSG.The control principle of the PMSG system is based on field orientation.Due to  1 =  1 and  1 = 0 in PMSG system, the second term of ( 6) is zero.Moreover,  pm is constant for a field orientation control of PMSG system.The electromagnetic torque  er is a function of  1 .The electromagnetic torque  er is linearly proportional to qaxis current  1 .When the d-axis rotor flux is constant, the maximum electromagnetic torque per ampere can be reached for the field-oriented control at the  er proportional to the  1 .

PMSG System.
The control block diagram of the two sets of the same four-layer modified Elman NNs controlled PMSM direct-driven PMSG system is shown in Figure 4.The wind pattern can be acquired by appropriately programming the PMSM speed.The AC power of variable frequency and voltage generated by the PMSG system is rectified to DC power by controllable rectifier.The power converts of the PMSG system direct-driven by a PMSM consist of two field-oriented institutions, two current control loops, two sine pulse-width-modulation (SPWM) control circuits, two interlock and isolated circuits, and two IGBT power modules for rectifier and inverter.The DC bus voltage of the PMSG system directly driven by a PMSM via controllable rectifier can be controlled by using the first of modified Elman NN controller.Then, the inverter, which is controlled by using the second of modified Elman NN controller based on fieldoriented control, can convert the DC bus voltage into the AC 60 Hz line voltage to provide for the stand-alone load.
The specification of PMSG is a three-phase four-pole 1.5 kW 220 V 10 A 2000 rpm type for experimental test in this study.
The electric parameters of the PMSG are  1 = 0.2 Ω,  1 =  1 = 6mH, and   = 6.2 mH.The specification of three-phase PMSM which acted as prime machine is a 1 kW 220 V 7 A 2000 rpm type.In practical applications, the pure differentiator may amplify the high-frequency noise, so the stability of the closed-loop PMSG drive system will be greatly affected.Thus, a filter is implemented as an alternative for the pure differentiators shown in Figure 4.It is designed to behave as a pure differentiator for the main low-frequency dynamic signal and become a low-pass filter for the highfrequency signals.
The output voltages of rectifier and inverter controlled by two sets of the same modified Elman NN controllers were implemented by using two independent sets TMS320C32 DSP control system in Figure 4, where   is the rotor position of the PMSG;  *  is the  axis control current of the rectifier;  *  is the q axis e control current of the rectifier;   bus voltage in output end of the rectifier;  *  is the  axis control current of the inverter;  *  is the  axis control current of the inverter;   is the electric angular angle of the inverter which integrates the command electric angular frequency with respect to t;  *  ,  *  , and  *  are the desired phase currents of the inverter in phases ai, bi, and ci, respectively;   ,   , and   are the actual measured phase currents of the inverter in phases ai, bi, and ci, respectively; V  , V  , and V  are the actual measured phase voltages of the inverter in phases ai, bi, and ci, respectively;   ,   , and   are the SPWM control signals of the inverter in phases ai, bi, and ci, respectively;  rms is the actual root-mean-square magnitude of the AC 60 Hz line voltage in output end of the inverter;  * rms is the desired root-mean-square magnitude of the AC 60 Hz line voltage in output end of the inverter.

Description of Modified Elman NN.
In the proposed two sets of the same four-layer modified Elman NNs with input layer using feedback signals from output layer are taken into account to result in better learning efficiency.The architecture of the two sets of the same four-layer modified Elman NNs, which consists of the first layer (the input layer), the second layer (the hidden layer), the third layer (the context layer), and the forth layer (the output layer), is shown in Figure 5.The exciting functions and signal propagations of nodes in

Hidden layer
Output layer each layer of the modified Elman NN can be described as follows. 3 where  1 , is the input of the th nod in the mth modified Elman NN, and  1 , is the output of the th nod in the mth modified Elman NN.The different inputs of the two sets of modified Elman NNs are  1  1,1 =  1 =  *  −   ,  ,  = 1, . . ., ,  = 1, 2, where  , are the connective weights between the context layer and the hidden layer in the mth modified Elman NN,  , are the connective weights between the input layer and the hidden layer in the mth modified Elman NN, and n is the number of neurons in the hidden layer;  2 , is the activation function in the mth modified Elman NN, which is also a sigmoid function;  1 , () =  2 , () represents the th output node of input layer in the mth modified Elman NN, and  3  , () =  3 , () represents the kth input to the node of context layer in the mth modified Elman NN.

Third Layer 3: Context Layer.
In the context layer, the node input and the node output are represented as nod 3  , () =  2 , ( − 1) +  3 , ( − 1) , where  2 , () =  4 , () represents the th output node of hidden layer in the mth modified Elman NN;  3  , () represents the th output to the node of context layer in the mth modified Elman NN;  is the number of neurons in the context layer; 0 ≤  < 1 is the self-connecting feedback gain of context layer.

Online Learning Algorithm of Modified Elman NN.
To explain the online learning algorithm of the modified Elman NN using supervised gradient decent method, firstly the energy function  , is defined as where  1 equals to  *  −   in rectifier end of the PMSG system;  2 equals to  * rms −  rms in the inverter end of the PMSG system.Then, the learning algorithm is described as follows.

Fourth
where connective weights  , between hidden layer and input layer in the mth modified Elman NN can be updated as Then variation Δ , of connective weights in the mth modified Elman NN by using the chain rule and the gradient descent method can be calculated as where  , ≡  2 , / , can be calculated from (7).The recurrent weights  , between output layer and input layer in mth modified Elman NN can be renewed according to the following equation:  , ( + 1) =  , () + Δ , ,  = 1,2. ( Due to the uncertainty effect of the system dynamics, the accurate computation of the Jacobian  , / 4 , ,  = 1, 2 in the PMSG system cannot be determined.To dispel the difficulty and endure the above matter, using the delta adaptation law [20] can raise the online learning capacity of the connective weights.Therefore, the delta adaptation law can be calculated as [20] Thus       ()     < √ , .

Experimental Results
The two sets of the same modified Elman NNs controlled PMSG system are realized in two sets TMS320C32 DSP control system.A photo of the experimental setup is shown in Figure 6.To implement current controlled PWM rectifier and inverter by field-oriented control, the two sets IGBT power modules are adopted BSM 100 GB-120DLC manufactured by Eupec Co.The switching frequency of both IGBT power modules is all 15 kHz.The two programs of the two sets TMS320C32 DSP control system used for executing the two sets of the same modified Elman NNs and online training of the two sets of the same modified Elman NNs need 2 ms sampling interval.The proposed methodology for the real-time control implementation consists of the two main programs and two interrupt service routines (ISRs) in the two sets of DSP control system as shown in Figure 7.In the main program #1, parameters and input/output (I/O) initialization are processed first.Then, the interrupt interval for the ISR #1 is set.After enabling the interrupt, the main program #1 is used to monitor control data.The ISR #1 with 2 ms sampling interval is used for reading the rotor position of the PM synchronous generator from encoder, reading mechanic torque from torque transducer, reading measured DC bus voltage and current from analog/digital (A/D) converters, calculating maximum DC bus power of the PM synchronous generator and DC bus power, and executing the modified Elman NN control system #1.On the other   magnitude  *  of DC bus voltage, the desired root-meansquare magnitude  * rms of AC 60 Hz line voltage are set as 157 rad/s (1500 rpm), 220 V, and 110 V, respectively.Thirdly, case 3 is the Δ connection three-phase load 18 Ω and the rotor speed   , the desired magnitude  *  of DC bus voltage, the desired root-mean-square magnitude  * rms of AC 60 Hz line voltage are set as 209.3 rad/s (2000 rpm), 220 V, and 110 V, respectively.The Δ connection three-phase loads of 100 Ω, 50 Ω, and 18 Ω dispatched powers as 121 W, 242 W, and 672 W, respectively.The Δ connection three-phase loads of 100 Ω and 50 Ω, and 18 Ω dispatched powers as 121 W, 242 W, and 672 W, respectively.Some experimental results of the PI controlled PMSG system direct-driven by PMSM are demonstrated for the comparison of the control performance.Since the PMSG system is a nonlinear time-varying system, the gains of the PI controllers for both the DC bus voltage adjustment and AC 60 Hz line voltage adjustment are obtained by trial and error to achieve steady state control performance.The control gains are   = 5.2,   = 10.2 for the DC bus voltage adjustment and   = 4.8,   = 10.8 for the AC 60 Hz line voltage adjustment in the two sets PI controllers.The experimental results of the PI controlled PMSG system direct-driven by PMSM for the Δ connection three-phase loads of 100 Ω with   = 78.5 rad/s (  = 750 rpm) at case 1, the Δ connection three-phase loads of 50 Ω with   = 150 rad/s (  = 1500 rpm) at case 2, and the Δ connection three-phase loads of 18 Ω with   = 209.results, few sluggish DC bus voltage and AC 60 Hz line voltage adjusting responses are obtained for the conventional NN controlled PM synchronous motor direct-drive PM synchronous generator system because the conventional NN is static input/output mapping schemes that can approximate a continuous function to an arbitrary degree of accuracy.Some experimental results of the modified Elman NN controlled PMSG system direct-driven by PMSM for the Δ connection three-phase loads of 100 Ω with   = 78.5 rad/s

Conclusions
This study demonstrated the implementation of both the DC bus voltage and AC 60 Hz line voltage adjustment of the PMSG system direct-driven by PMSM based on wind turbine emulator by using the two sets of the same modified Elman NN controllers for stand alone power applications.Firstly, the field-oriented control was implemented for the control of the PMSG system direct-driven by PMSM based on wind turbine emulator.Then, the proposed two sets of the same PMSG system direct-driven by PMSM based on wind turbine emulator with a rectifier and an inverter for stand alone power applications.

Figure 3 :
Figure 3: Block diagram for the wind turbine model.

Figure 4 :
Figure 4: System configuration of the two sets of the same modified Elman NNs controlled PMSG system direct-driven by PMSM with rectifier and inverter.

Figure 5 :
Figure 5: Structure of the two sets of the same four-layer novel modified Elman NNs.

Remark 2 .
The values of the learning-rate parameter  1, is dependent on the selection of the value  1 .

1 Figure 6 :
Figure 6: A photo of the experimental setup.

Figure 7 :
Figure 7: Flowcharts of two sets of the same modified Elman NN control systems controlled PMSG system.

Figure 8 :
Figure 8: Experimental results of PMSM direct-driven PMSG system using the PI controller for the Δ connection three-phase loads of 100 Ω with   = 78.5 rad/s (  = 750 rpm) at case 1: (a) rotor speed   (  ); (b) adjusting response of step desired magnitude  *  of the DC bus voltage and actual measured magnitude   of the DC bus voltage in output end of the rectifier; (c) adjusting response of step desired root-mean-square magnitude  * rms of the AC 60 Hz line voltage and actual measured root-mean-square magnitude  rms of the AC 60 Hz line voltage in output end of the inverter  rms ; (d) tracking response of the desired phase current  *  and actual measured phase current   in phase ai of the inverter.

Figure 9 :
Figure 9: Experimental results of PMSM direct-driven PMSG system using the PI controller for the Δ connection three-phase loads of 50 Ω with   = 150 rad/s (  = 1500 rpm) at case 2: (a) rotor speed   (  ); (b) adjusting response of step desired magnitude  *  of the DC bus voltage and actual measured magnitude   of the DC bus voltage in output end of the rectifier; (c) adjusting response of step desired root-mean-square magnitude  * rms of the AC 60 Hz line voltage and actual measured root-mean-square magnitude  rms of the AC 60 Hz line voltage in output end of the inverter  rms ; (d) tracking response of the desired phase current  *  and actual measured phase current   in phase ai of the inverter.
, and 16, respectively; where rotor speed   (  ) is shown in Figures 14(a), 15(a), and 16(a); adjusting response of step desired magnitude  *  of the DC bus voltage and actual measured magnitude   of the DC bus voltage in output end of the rectifier is shown in Figures 14(b), 15(b), and 16(b); adjusting

Figure 10 :
Figure 10: Experimental results of PMSM direct-driven PMSG system using the PI controller for the Δ connection three-phase loads of 18 Ω with   = 209.3rad/s (  = 2000 rpm) at case 3: (a) rotor speed   (  ); (b) adjusting response of step desired magnitude  *  of the DC bus voltage and actual measured magnitude   of the DC bus voltage in output end of the rectifier; (c) adjusting response of step desired root-mean-square magnitude  * rms of the AC 60 Hz line voltage and actual measured root-mean-square magnitude  rms of the AC 60 Hz line voltage in output end of the inverter  rms ; (d) tracking response of the desired phase current  *  and actual measured phase current   in phase ai of the inverter.

Figure 11 :
Figure 11: Experimental results of PMSM direct-driven PMSG system using the conventional NN controller for the Δ connection three-phase loads of 100 Ω with   = 78.5 rad/s (  = 750 rpm) at case 1: (a) rotor speed   (  ); (b) adjusting response of step desired magnitude  *  of the DC bus voltage and actual measured magnitude   of the DC bus voltage in output end of the rectifier; (c) adjusting response of step desired root-mean-square magnitude  * rms of the AC 60 Hz line voltage and actual measured root-meansquare magnitude  rms of the AC 60 Hz line voltage in output end of the inverter  rms ; (d) tracking response of the desired phase current  *  and actual measured phase current   in phase ai of the inverter.

Figure 12 :
Figure 12: Experimental results of PMSM direct-driven PMSG system using the conventional NN controller for the Δ connection three-phase loads of 50 Ω with   = 150 rad/s (  = 1500 rpm) at case 2: (a) rotor speed   (  ); (b) adjusting response of step desired magnitude  *  of the DC bus voltage and actual measured magnitude   of the DC bus voltage in output end of the rectifier; (c) adjusting response of step desired root-mean-square magnitude  * rms of the AC 60 Hz line voltage and actual measured root-meansquare magnitude  rms of the AC 60 Hz line voltage in output end of the inverter  rms ; (d) tracking response of the desired phase current  *  and actual measured phase current   in phase ai of the inverter.

Figure 13 :Figure 14 :Figure 15 :
Figure 13: Experimental results of PMSM direct-driven PMSG system using the conventional NN controller for the Δ connection three-phase loads of 18 Ω with   = 209.3rad/s (  = 2000 rpm) at case 3: (a) rotor speed   (  ); (b) adjusting response of step desired magnitude  *  of the DC bus voltage and actual measured magnitude   of the DC bus voltage in output end of the rectifier; (c) adjusting response of step desired root-mean-square magnitude  * rms of the AC 60 Hz line voltage and actual measured root-meansquare magnitude  rms of the AC 60 Hz line voltage in output end of the inverter  rms ; (d) tracking response of the desired phase current  *  and actual measured phase current   in phase ai of the inverter.

Figure 16 :
Figure 16: Experimental results of PMSM direct-driven PMSG system using the modified Elman NN controller for the Δ connection three-phase loads of 18 Ω with   = 209.3rad/s (  = 2000 rpm) at case 3: (a) rotor speed   (  ); (b) adjusting response of step desired magnitude  *  of the DC bus voltage and actual measured magnitude   of the DC bus voltage in output end of the rectifier; (c) adjusting response of step desired root-mean-square magnitude  * rms of the AC 60 Hz line voltage and actual measured root-meansquare magnitude  rms of the AC 60 Hz line voltage in output end of the inverter  rms ; (d) tracking response of the desired phase current  *  and actual measured phase current   in phase ai of the inverter.
=   (  /) + The input and the output for each node  in this layer are expressed as nod1, () = Π   1 , () ⋅  , ⋅  4 , ( − 1) , .1.1.First Layer: Input Layer.Each node  in this layer is indicated by using, Π which multiplies by each other between each other for input signals.Then outputs signals are the results of product.
12,1 = ė 1 for the rectifier end of the PMSG system in the first modified Elman NN, and  1 1, 2 =  2 =  * rms −  rms ,  1 2,2 = ė 2 for the inverter end of the PMSG system in the second modified Elman NN, respectively.The  indicates the number of iterations.The connecting weights  , are the recurrent weights between the output layer and the input layer in the mth modified Elman NN.  4 , is the output value of the output layer in the mth modified Elman NN. 3.1.2.Second Layer: Hidden Layer.The single node th in this layer is labeled with Σ.It computes outputs of the input layer and the context layer as the summation of all input signals.The net input and the net output for node th in this layer are expressed as ,  3 , () + ∑   ,  1 , (), 3.1.4.Fourth Layer: Output Layer.The single node oth in this layer is labeled with Σ.It computes the overall output as the summation of all input signals.The net input and the net output for node oth in this layer are expressed as  are the connective weights between the hidden layer and the output layer in the mth modified Elman NN; 4, is the activation function in the mth modified Elman NN, which is set to be unit;  2 , () =  4 , () represents the jth input to the node of output layer in the mth modified Elman NN.The outputs in the mth recurrent wavelet NN can be represented as 4The output values of the two sets of the same fourlayer modified Elman NNs can be rewritten as  1 = ( 1 )   1 =  *  for rectifier and  2 = ( 2 )   2 =  * are to be adjusted parameters between the mother layer and the output layer of the two sets of the same four-layer modified Elman NNs.The   = [41,  4 2, ⋅ ⋅ ⋅  4 , ]  ,  = 1,2 are the inputs vectors in the output layer of the two sets of the same fourlayer modified Elman NNs, in which  4 , are determined by the selected sigmoid function and 0 ≤ 4, ≤ 1.
Layer.The propagated error term in the mth modified Elman NN is  is the learning rate between hidden layer and output layer in the mth modified Elman NN.The connective weights  , between hidden layer and output layer in the mth modified Elman NN can be renewed according to the following equation: where connective weights  , between context layer and hidden layer in the mth modified Elman NN can be renewed according to the following equation:  , ( + 1) =  , () + Δ , ,  = 1,2.(18)Thenvariation Δ , of connective weights in the mth modified Elman NN can be calculated as , is the number of nodes in hidden layer of the mth modified Elman NN.
Theorem 1.Let   be the learning-rate parameter of the mth modified Elman NN and let  max, be defined as  max, ≡ max  ‖  ()‖, where   () =  4 , / , in the mth modified Elman NN and ‖ ⋅ ‖ is the Euclidean norm in R n .The convergence is guaranteed if  1, is chosen as  1, =  1 /( 2 max, ) =  1 / , , in which  1 is a positive constant gain;