energies Article Application of the Feedback Linearization in Maximum Power Point Tracking Control for Hydraulic Wind Turbine Chao Ai 1,2, Wei Gao 1,2

Transcription

1 energies Article Application of Feedback Linearization in Maximum Power Point Tracking Control for Hydraulic Wind Turbine Chao Ai 1,2, Wei Gao 1,2, Qinyu Hu 1,2, Yankang Zhang 3, Lijuan Chen 3, *, Jiawei Guo 1,2 and Zengrui Han 1,2 1 Hebei Heavy Machinery Fluid Power Transmission and Control Laboratory, Yanshan University, Qinhuangdao , China; aichao@ysueducn (CA); gaowei2018@njiteducn (WG); huqingyu@stumailysueducn (QH); @stumailysueducn (JG); hzr@stumailysueducn (ZH) 2 School of Mechanical Engineering, Yanshan University, Qinhuangdao , China 3 Nanjing Institute of Technology, Nanjing , China; zhangyk@njiteducn * Correspondence: chenlj@njiteducn; Tel: Received: 11 February 2020; Accepted: 17 March 2020; Published: 24 March 2020 Abstract: Taking hydraulic wind turbine as research object, method is studied to improve utilization ratio of wind energy for hydraulic wind turbine, when wind speed is lower than rated wind speed The hydraulic fixed displacement pump speed and generating power can be used as control output to realize maximum power point tracking control The characteristics of maximum power point tracking control are analyzed for hydraulic wind turbine, and hydraulic output power is taken as control output based on comprehensive performance requirements Because hydraulic wind turbine is a strong multiplication nonlinear system, system is globally linearized based feedback linearization method, and maximum power point tracking control law is obtained The simulation and experiment results show that system has good dynamic performance with proposed control law The control provides oretical guidance for optimal power tracking control law application for hydraulic wind turbine Keywords: hydraulic wind turbine; maximum power point tracking control; hydraulic fixed displacement pump; hydraulic variable displacement motor; feedback linearization method 1 Introduction Wind energy is a vast renewable energy resource being increasingly tapped by wind turbines which are growing in number and size Conventional wind turbines transmissions consist of a mechanical gearbox and high-speed generator, or sometimes direct drive low-speed generators are used as described in [1] But se two traditional transmission schemes have some disadvantages, such as high cost and high failure rate In order to reduce cost and improve reliability, hydrostatic transmission has been proposed in [2 4] Thus, power captured by turbine rotor will be transmitted to generator through hydrostatic transmission, and generator can be flexibly installed, eir in nacelle of wind turbine or installed on ground driven by hydraulic pipelines, which greatly reduces difficulty of installation and maintenance Benefits of hydraulic wind turbine (HWT) are that hydrostatic transmission is inherently more compliant than mechanical transmission, which makes it more reliable in face of shock loading, and transmission ratio is adjustable, so system does not need additional frequency conversion devices to coordinate wind turbine speed and power grid frequency Meanwhile, with development of digital hydraulic technology, low part-load efficiency of conventional hydraulics is solved [5] Energies 2020, 13, 1529; doi:103390/en wwwmdpicom/journal/energies

2 Energies 2020, 13, of 18 In order to improve efficiency of wind energy utilization, scholars at home and abroad have done a lot of research on maximum power point tracking (MPPT) control for HWT An optimal H ring forming pressure controller was designed for a mechano-hydraulic hybrid wind turbine, which can accurately track optimal load pressure in [6] The small signal linearization and variable gain proportional integral differentiation (PID) were combined to solve problem of inconsistent power response, and speed control loop is used to optimize MPPT strategy in [7] A sensorless double integral sliding mode controller was designed to achieve maximum power extraction under condition of large parameter uncertainty and nonlinearity for hydrostatic tidal turbine in [8] A PID controller and a sliding mode controller were applied for speed control to track a predefined speed in [9] To track maximum power point, a takagi-sugeno (TS) fuzzy model was proposed for HWT with two variable displacement pumps and a fixed displacement motor, and a parameter varying model predictive control (PVMPC) approach was designed in [10] Shengquan Li proposed a combined model predictive control strategy to achieve purpose of MPPT control for HWT in [11] SP Mulders proposed an implementation, model-free, gradient-based and extremum seeking control optimization strategy to achieve an average maximum power increase in [12] M, Deldar designed a decentralized control configuration to collect maximum energy without directly measuring wind speed in [13] A digital hydrostatic drive solution was introduced in HWT to largely utilize energy captured by rotor in [14], and kw 2 control law combined with digital hydrostatic method was proposed to improve HWT energy production Active disturbance rejection controller (ADRC) control method was introduced to achieve HWT MPPT control, and pump speed was taken as control output in [15] However, MPPT control smoothness needed to be furr improved The frequency control and MPPT control rely on mechanical torque adjustment on hydraulic motor that was coupled with generator in [16] A MPPT control strategy based on tip speed ratio (TSR) was adopted for 600 kw hydraulic energy storage wind turbine, and maximum power tracking performance was simulated and analyzed by subjecting to turbulent speed condition in [17] The above-mentioned researches on MPPT control methods verify proposed controller effectiveness from a oretical point view and provide a certain reference However, according to HWT with a fixed displacement pump and a variable displacement motor, it is also necessary to explore a MPPT controller Therefore, based on HWT working principle, a state space model is established Because hydraulic main transmission system has strong nonlinearity, wind has randomness and volatility, and sometimes fluctuation range is large, feedback linearization method can rid system of small signal linearization limitation, and analysis and synsis of system in a large range is realized [18] Thus, feedback linearization method is adopted to conduct MPPT controller The paper is organized as follows: HWT working principle and requirements for MPPT control are introduced in Section 2 The HWT mamatical model is established in Section 3 In Section 4, feedback linearization method is used to globally linearize system, and MPPT controller is proposed with output of power In Section 5, simulation results are carried out under different conditions, and experiment results are also carried out based on 24 kw HWT experiment platform Finally, conclusions are drawn in Section 6

3 Energies 2020, 13, of 18 2 Description of HWT 21 HWT Working Principle HWT is mainly composed of a rotor, a fixed displacement pump, a variable displacement motor Energies 2020, 13, of 21 and a synchronous generator An inverter for frequency conversion is not required The rotor and pump are are coaxially connected, connected, and and pump pump supplies supplies high pressure high pressure oil to oil motor, so motor, fluctuation so of fluctuation wind speed of tends wind to speed affecttends motor affect speed [19,20] motor The speed motor [19,20] and The generator motor and are alsogenerator coaxially connected, are also coaxially and connected, motor and and generator motor speed and generator are controlled speed by are changing controlled by motor changing displacement motor When displacement synchronous When generator synchronous is stable generator within is stable required within speed range required (1500 speed r/min range ± 6 r/min), (1500 r/min it can be ± 6 connected r/min), it can to be grid connected for power to generation grid for power At generation same time, At charge same pump time, provides charge hydraulic pump oil provides for hydraulic low-pressure oil for pipeline, low-pressure control pump pipeline, provides control hydraulic pump oilprovides for control hydraulic pipelineoil of for variable control pipeline displacement of motor, variable anddisplacement safety relief motor, valveand and relief safety valve relief are used valve for and system relief protection valve are Aused schematic for system diagram protection of A HWT schematic is shown diagram in Figure of 1 HWT is shown in Figure 1 wind turbine rotor U E flow control valve v ω p safety relief valve pump p h1 γ p h 2 synchronous generator ω m G power grid Variable displacement motor relief valve charge pump control pump relief valve 22 Control Effects of MPPT Figure 1 Schematic of hydraulic wind turbine (HWT) The maximum wind power captured by wind turbine under demand wind speed [21] is expressed as P rmax = 1 2 ρπr2 v 3 C pmax (1) The rotor speed is related to windprmax speed, = ρπ R v C p max (1) 2 and rotor speed is expressed as The rotor speed is related to wind speed, and rotor speed is expressed as ω = vλ opt (2) R vλopt From Equations (1) and (2), relationship ω = between captured maximum wind power and (2) rotor speed ( pump speed) can be expressed as R From Equations (1) and (2), relationship between captured maximum wind power and rotor speed ( pump speed) P can be expressed as rmax = ρπr5 C pmax ω 3 = K P ω 3 (3) 2λ 3 opt 5 ρπ RCpmax 3 3 rmax ω K 3 Pω 2λopt The principle of MPPT method P for = direct generation = power control is shown in Figure(3) 2, and tracking process is expressed as follows: The principle of MPPT method for direct generation power control is shown in Figure 2, and tracking process is expressed as follows:

4 Energies 2020, 13, of 18 Energies 2020, 13, of 21 Figure 2 Schematic of maximum power point tracking (MPPT) method for direct generation power control Assuming Assuming wind wind speed speed v v1 1 < v v2 2 < v3, v 3, E, E, A, A, and and B are are maximum maximum power power points points at at three three kinds kinds of wind speed, respectively The initial wind speed is v of wind speed, respectively The initial wind speed is 2, and initially wind turbine works stably at v2, and initially wind turbine works stably point A, which is maximum power point at wind speed v at point A, which is maximum power point at wind speed 2 v2 When wind speed increases from v 2 to v 3, since rotor speed cannot change suddenly, input When power wind is speed power increases P C corresponding from v2 to v3, to since point C, rotor andspeed power cannot value change is suddenly, power P 2 corresponding input power is to point power A Since PC corresponding input power to Ppoint C is always C, and greater power than value generating is power PP2 2 before corresponding point B, to point rotora will Since accelerate input Aspower rotor PC is speed always increases, greater than input generating power moves power fromp2 point before Cpoint to B, maximum rotor power will accelerate point B along As power rotor speed characteristic increases, curve of input v 3 ; power actual moves generating from power point C moves to from maximum point A power to point point B along B along demand power valuecharacteristic of generating curve power of v3; ( optimal actual power generating curve) power Themoves rotor input from point power A isto equal point to B along demand demand powervalue at point of B, and generating system power is re-balanced ( optimal power At thiscurve) point, The MPPT rotor input process power is completed is equal to when demand windpower speedat increases point B, from and v 2 to system v 3 is re-balanced At this point, MPPT process is completed when wind speed increases from Inv2 ato similar v3 way, when wind speed decrease from v 2 to v 1, since rotor speed cannot change suddenly; input power is power P D corresponding to point D, and power value is power In Pa similar way, when wind speed decrease from v2 2 corresponding to point A Since input power P D is always to v1, less since than rotor generating speed cannot power Pchange suddenly; input power is power PD 2 before point E, rotor will slow down As rotor corresponding speed decreases, to point D, input and power power moves value fromis point power D to P2 corresponding optimal power to point point E along A Since power input characteristic power PD is curve always of vless 1 The than actual generating power moves P2 before frompoint point E, A torotor pointwill E along slow down demand As value rotor of speed generating decreases, power input ( optimal power power moves curve) from point The input D to power optimal is equal power to point demand E along value of powercharacteristic generation at curve point E, of and v1 The system actual generating is re-balanced power Atmoves this point, from point MPPT A to process point is E completed along demand when value windof speed generating decreases from power v 2 ( to voptimal 1 power curve) The input power is equal to demand value of power generation at point Based E, on and above system principle is re-balanced analysis, At this MPPT point, method MPPT of direct process generation is completed power control when can automatically wind speed decreases track from windv2 turbine to v1 to maximum power point The rapidity and accuracy of MPPT process is taken into account Based on above principle analysis, MPPT method of direct generation power control can 3 automatically The Mamatical track Model wind of turbine Hydraustatic to maximum Transmission power point The rapidity and accuracy of MPPT process is taken into account 31 The Pump Model 3 The Mamatical Model of Hydraustatic Transmission The pump shaft torque is related to pressure, and torque is expressed as 31 The Pump Model T p = D pp h1 (4) The pump shaft torque is related to pressure, η mech,p and torque is expressed as

5 Energies 2020, 13, of 18 The difference between rotor and pump torque accelerates combined inertia and overcomes viscous friction The torque balance equation is expressed as T v ( ωp, v ) T p = dω p dt Flow continuity equation of fixed displacement pump is expressed as + B p ω p (5) Q p = D p ω p C t1 p h1 (6) From Equations (4) (6), state equation of pump speed can be expressed as ω p = 1 ( T r D ) pp h1 B p ω p η mech,p (7) 32 The Motor Model Similarly, motor output torque is related to pressure, and torque is expressed as T m = D m p h2 η mech,m = K m γp h2 η mech,m (8) The difference between motor and generator torque accelerates motor inertia and overcomes viscous friction The torque balance equation is expressed as T m T L = J m dω m dt And variable motor flow continuity equation is expressed as + B m ω m (9) Q m = D m ω m + C t2 p h2 (10) From Equations (8) (10), state equation of motor speed is ω m = 1 ( ) Km γp J h2 η mech,m B m ω m T L m (11) 33 The Flow Control Valve Model The flow through proportional flow control valve is expressed as 34 The Hose Model Q bl = KU E (12) The oil flows between pump and motor inside a hose, and general equation for additional flow caused by oil compressibility and hose compliance is expressed as Q c = V β e dp h dt (13) The proportional flow control valve divides high-pressure line into two parts, one is pump to flow control valve, with volume V 1, and or part is flow control valve to motor, with volume V 2

6 Energies 2020, 13, of The Hydraulic Transmission System State Space Model The compressibility flow between pump and flow control valve is expressed as Q c1 = Q p Q bl = D p ω p C t1 p h1 KU E (14) The compressibility flow between flow control valve and motor is expressed as Q c2 = Q bl Q m = KU E D m ω m C t2 p h2 (15) To establish state space model of main transmission system for HWT, following assumptions need to be made: (1) The pressure in low-pressure line is a constant (2) The leakage coefficient, viscous damping coefficient and bulk modulus of oil are fixed values, which do not change with temperature or or factors (3) The pressure loss in hydraulic lines is ignored (4) The pump mechanical efficiency η mech,p is designed 1 The system state space model is obtained by combining Equations (7), (11) (15) ω p = B p ω p D ( p p h1 + J 1 p T v ωp, v ) p h1 = D pβ V 1 ω p C t1β V 1 p h1 KβU E V 1 p h2 = KβU E V 2 C t2β V 2 p h2 K mβω m V 2 γ ω m = B m Jm ω m + K mp h2 J m γ J 1 m T L (16) Due to MPPT control process, grid-connected speed control has been completed, and motor speed is constant; that is, ω m = 1500r/min Moreover, proportional throttle valve opening is open Thus, system status values are pump speed and pressure between pump and variable displacement motor The state equation during power tracking can be expressed as ω p = B p ω p D ( p p h + J 1 p T r ωp, v ) p h = D pβ V ω p C tβ V p h K mω md β V γ The state variables is define as x 1 = ω p, x 2 = p h, and system control input is u = γ Therefore, system state space model after grid-connection can be expressed as x 1 = B p x 1 D p x 2 + J 1 p T r (x 1, v) x 2 = D pβ V x 1 C tβ V x 2 K mω md β V u The system model is rewritten into affine nonlinear form ẋ = f(x) + g(x)u, system state space model after grid grid-connection can be expressed as f(x) = B p D p β V D p 1 Jp C tβ V 0 g(x) = 0 K mω md β V x 1 x 2 T r (x 1, v) (17) (18) (19) (20)

7 Energies 2020, 13, of 18 4 The MPPT Controller Based on Feedback Linearization Method 41 Control Thoughts In MPPT method for direct power generation control, control system mainly includes three control loops, which are motor displacement reference control loop, motor speed control loop, and power control loop Among three control loops, motor displacement reference control loop is calculated from fixed displacement pump speed based on traffic conservation, and purpose of motor displacement reference control loop is to make variable motor work at excitation synchronous speed The motor speed control loop is motor displacement fine-tuning amount, formed by deviation between demand synchronous speed (1500 r/min) and actual motor speed The function of motor speed control loop is to control variable motor to work precisely at synchronous speed and achieve quasi-synchronous grid connection Power control loop is reference value of maximum power, which is calculated by actual pump speed according to optimum blade tip ratio and maximum wind energy utilization coefficient Compared with actual power, deviation is formed, and displacement compensation value is adjusted by controller The specific control block diagram is shown in Figure 3 Energies 2020, 13, of 21 Figure Figure 3 3 HWT MPPT control block diagram 42 MPPT 42 MPPT Controller Design The The process process of solving of system control law using feedback linearization method is shown is shown in Figure in Figure 4 [18 22] 4 [18 22] The determination of system status, order and control output The system order Judging relationship between relative order r and system order n r<n Second standard type Lie derivative, Lie brackets,and relative order Reconstructed coordinate transformation Zero dynamic design and stability analysis Unstable Method I Method II L ϕ ( x) = 0 Lgϕ( x) 0 g The system is linearized and a new control input v is constructed

8 Figure 3 HWT MPPT control block diagram 42 MPPT Controller Design The process of solving system control law using feedback linearization method is shown in Figure 4 [18 22] Energies 2020, 13, of 18 The determination of system status, order and control output The system order Judging relationship between relative order r and system order n r<n Second standard type Lie derivative, Lie brackets,and relative order Reconstructed coordinate transformation Zero dynamic design and stability analysis Unstable Method I Method II L ϕ ( x) = 0 Lgϕ( x) 0 g The system is linearized and a new control input v is constructed The structure of new control input v is determined based on control objective The new final control law u is obtained 421 Control Output The control output is expressed as as 422 Relative Order Figure 4 Feedback linearization control law flowchart y = h(x) = D p x 1 x 2 = D p ω p p h (21) The definition of relative order is as follows: (1) The Lie derivative value of k-order Lie derivative of output function h(x) to vector field F(x) to vector field F(x) in neighborhood of x = x 0 is zero; that is, L g L k h(x) = 0 (2) The Lie derivative value of (r 1)-order Lie derivative f (k < r 1) of output function h(x) to vector field F(x) to vector field F(x) in neighborhood of x = x 0 is not zero; that is, L g L r 1 h(x) 0 Then relative degree of nonlinear system (18) in x = x f 0 is defined as r [18 22] From equations (17) (20), according relevant definition of Lie derivative, relative order evaluation expression is L g L 0 f h(x) = K mβω md D p x 1 V 0 (22) From expression (22), relative order r = 1 < 2, system cannot be linearized in full state and zero dynamic design is required From above analysis, system cannot be completely linearized Therefore, input output linearization and zero dynamic design method are adopted to determine system controller [22] In process of zero dynamic design, system dynamic behavior can be divided into external dynamics and internal dynamics The external dynamics are not only required to be stable but also have tracking performance and reject disturbance The internal dynamics are only required to be stable

9 Energies 2020, 13, of Zero Dynamic Controller Design The zero dynamics selected in this paper is ϕ 2 (x) = x 1, pump speed, which is considered as internal dynamics First, a coordinate transformation of system gives { z1 = ϕ 1 (x) = h(x) = D p x 1 x 2 z 2 = η 2 (x) = x 1 (23) Thus, Jacobian matrix of vector function ϕ(x) = [z 1 (x) z 2 (x)] T at x = x 0 is expressed as J ϕ = φ(x) (x) [ Dp x x=x0 = 2 D p x This is a non-singular at x = x 0, so coordinate transformation is valid And n inverse mapping of z = ϕ(x) is x = ϕ 1 (z) Therefore, x can be expressed as ] (24) x 1 = z 2 x 2 = z 1 D p z 2 (25) From equations (18) and (25), final expression of system in z coordinates can be expressed as z 1 = D p z 1 ( D p z 2 B p z 2 D p z 1 z 2 = B p z 2 J 1 z 1 p z 2 + J 1 p T r (z 2, v) ) D p z 2 + J 1 p T r (z 2, v) The system output in z coordinate system is expressed as ( Dp β + D p z 2 V z 2 C tβ z 1 V D p z 2 )u (26) y = z 1 (27) The external state, which is pump outlet power, is close to zero and in a stable state, that is, z 1 = 0, ż 1 = 0 At this point, equations (26) can be expressed as z 2 = B p z T r (z 2, v) (28) When external state is close to zero, internal state stability is related to pump external load, and it is irrelevant to u, but all are asymptotically stable Therefore, system has zero dynamic stability and can be solved by controller [23] After coordinate transformation and zero-dynamic design, HWT MPPT controller can be designed based on feedback linearization control method The specific method is to linearize system in coordinate system, and output and control input of system are linearly related First, desired control input is constructed as ż 1 = v At this point, system output and constructed input v are linearly dependent The constructed input is expressed as ( v = D p x 2 B p x 1 D p x ) ( Dp β T r (x 1, v) + D p x 1 V x 1 C ) tβ V x 2 u (29) Then system can be linearized as follows: z 1 = v z 2 = B p z 2 1 z 1 z T v (z 2, v) (30)

10 Energies 2020, 13, of 18 Thus, zero dynamics is asymptotically stable, and whole system is asymptotically stable That is, selected coordinate transformation can be used to solve controller, and system is in a stable state under controller Thus, motor control input can be expressed in terms of v* ( ( V u = v [D p x 2 B p x 1 D p x ) ( Dp T v (x 1, v β ) + D p x 1 K m βω md D p x 1 V x 1 C )]) tβ V x Output Reference Design When power is taken as control output, demand maximum power at each wind speed is expressed as 425 Controller Design (31) y d = K p ω 3 p = ρπr5 C pmax 2λ 3 ω 3 p (32) opt In control process of MPPT, oretically, system is linearized precisely However, re will be tracking errors due to model parameters The closed loop PI control is used to reduce tracking error The tracking error is defined as e = y d y (33) and new control input [24] is expressed as v = ẏ d + k 1e + k 2 edt (34) is: In (34), values of k 1, k 2 are turned according to control requirements The final controller is given by (31) and (34) The controller expressed in terms of physical variables γ = D pω p C t p h K m ω md V K m βω md D p ω p (v D p p h ω p ) (35) According to (35), system states and constructed control inputs are combined linearly to form control signal for motor displacement Through real-time detection of system states, calculated control signals are sent to system to realize MPPT control Comprehensively considering rapidity, accuracy and stability of power tracking of hydraulic wind turbine and combining effects of proportional and integral links on control system, control parameters are shown in Table 1 Table 1 Control parameters Number Parameter Symbol Value 1 k k Simulation and Experiment Research 51 Simulation Platform A mamatical model, as showed in Figure 5, is established in software The nonlinear controller accuracy, which is based on feedback linearization ory, can be verified under different kinds of wind speed The corresponding parameters of system are shown in Table 2

11 51 Simulation Platform A mamatical model, as showed in Figure 5, is established in software The nonlinear controller accuracy, which is based on feedback linearization ory, can be verified Energiesunder 2020, 13, different 1529 kinds of wind speed The corresponding parameters of system are shown in Table 11 of 18 2 ε = d Pr = ρπr v Cp( λ, β ) 2 Pr Tr = ω y ke k edt u = f ( ωp, ωp, ph, ε) ε p Bp Dp 1 ωp = ωp p h+ T r( ωp, v ) Jp Jp Jp Dpβ Ctβ Kmβωmd p h = ωp ph γ V V V ω p Figure Figure 5 5 Simulation model Table 2 Parameters of hydraulic drive system for experimental platform Table 2 Parameters of hydraulic drive system for experimental platform Parameter Symbol Parameter Name Value Unit Parameter Name Value Unit Symbol viscous damping coefficient of B p 04 Nms/rad hydraulic pump B D viscous damping coefficient of hydraulic p p displacement of hydraulic pump Nms/rad m 3 inertia ofpump hydraulic pump 400 kg/m 3 displacement gradient of variable D K m p displacement of hydraulic motor pump m 3 3 /rad /rad viscous damping coefficient of variable J B m p inertia displacement of hydraulic motor pump Nms/rad kg/m 3 J m inertia of hydraulic motor 0462 kg/m 3 β displacement gradient of variable K e bulk modulus of hydraulic oil m 5366 Pa total leakage 10 displacement coefficient motor of hydraulic 6 m 3 /rad C t pump and variable displacement motor m 3 /(spa) viscous volumedamping of oil affected coefficient by pressure of variable effect B V 28 m in hydraulic hose Nms/rad m 3 displacement motor P output power of wind turbine 24 kw J R radius of wind turbine 748 m m inertia of hydraulic motor 0462 kg/m 3 maximum coefficient of wind C pmax β energy utilized e bulk modulus of hydraulic oil Pa λ max optimal tip speed ratio 2277 total leakage coefficient of hydraulic pump C t 52 Simulation Analysis and variable displacement motor m 3 /(spa) In order to verify proposed control method effectiveness, HWT MPPT control are researched based on above simulation platform Under condition of 7 m/s 8 m/s step wind speed at 10 s, response characteristics of system are observed during process of power tracking, and specific curves are shown in Figure 6 Under condition of 8 m/s 7 m/s step wind speed at 10 s, response characteristics of system are observed during process of power tracking, and specific curves are shown in Figure 7 From results of Figure 6; Figure 7, pump speed and high pressure change with step wind speed, and demand power curve coincides with output power curve; that is, output power can track demand power The MPPT controller effectiveness is verified There is a difference between input wind power and output power, and difference is 1 kw; reason is that in pump motor system re is transmission efficiency There are two main types of energy loss in process of energy flow conversion, which are energy lost by hydraulic transmission system and energy lost by generator Among m, energy lost by hydraulic transmission system mainly includes volume loss and oil heating loss, and energy lost by generator mainly includes mechanical loss and electromagnetic loss The wind energy absorbed by wind turbine cannot be completely converted into electrical energy with se unavoidable energy losses

12 52 Simulation Analysis There are two main types of energy loss in process of energy flow conversion, which are In order to verify proposed control method effectiveness, HWT MPPT control are researched based on above simulation platform Under condition of 7 m/s 8 m/s step wind speed at 10 s, response characteristics of system are observed during process of power Energies tracking, 2020, 13, and 1529 specific curves are shown in Figure 6 12 of 18 Wind speed v/(m/s) Pump speed w p /(r/min) (a) (b) (c) (d) Figure Energies , The 13, response 1529 curve under condition of 7 m/s-8 m/s step wind speed at 10 s (a) 14 of wind 21 Figure 6 The response curve under condition of m/s-8 m/s step wind speed at 10 s (a) wind speed; speed; (b) (b) pump pump speed; speed; (c) (c) high high pressure; pressure; (d) (d) output output power power Under condition of 8 m/s 7 m/s step wind speed at 10 s, response characteristics of system are observed during process of power tracking, and specific curves are shown in Figure 7 (a) (b) (c) (d) Figure 7 Figure response 7 response curve curve under under condition of 8 m/s-7m/s m/s step step wind wind speed speed at 10 at s (a) 10 s (a) wind wind speed; (b) speed; (b) pump pump speed; speed; (c) (c) high high pressure; (d) output power From results of Figure 6; Figure 7, pump speed and high pressure change with step wind speed, and demand power curve coincides with output power curve; that is, output power can track demand power The MPPT controller effectiveness is verified There is a difference between input wind power and output power, and difference is 1 kw; reason is that in pump motor system re is transmission efficiency

13 Energies 2020, 13, of 18 Under conditions of 7 m/s ( ± 05 m/s) 8 m/s ( ± 05 m/s) in wind speed, response characteristics of system are observed during process of power tracking, and specific curves Energies 2020, 13, of 21 are shown in Figure 8 (a) (b) Power P/kW hydraulic power input power demand power Time t/s (c) (d) Figure 8 The response curve under 8 ± 05 m/s wind speed (a) wind speed; (b) pump speed; Figure 8 The response curve under 8 ± 05 m/s wind speed (a) wind speed; (b) pump speed; (c) high pressure; (d) output power (c) high pressure; (d) output power From From results results of Figure of Figure 8, 8, pump pump speed and andhigh highpressure also also change change with with fluctuated fluctuated speed, speed, and and demand demand power power curve curve also coincides with output power power curve; curve; that that is, is, output output power power can track can track demand power The motor swing angleis is adjusted to achieve to achieve above above results, results, and and MPPT MPPT controller effectiveness is is verified furr Thereis is a a difference between input input wind wind power power and and output output power; power; difference is is 12 kw, and reasonis is that that in in pump motor system system re re is transmission transmission efficiency efficiency 53 Experiment 53 Experiment Platform Platform The 24 kw HWT experiment simulation platform is shown in Figure 9 The system major The 24 kw HWT experiment simulation platform is shown in Figure 9 The system major hardware hardware models are shown in Table 3 The experimental platform mainly consists of four parts: wind models are shown in Table 3 The experimental platform mainly consists of four parts: wind turbine turbine simulation system, hydraulic transmission system, grid-connected generation system and simulation control system, system Among hydraulic m, transmission frequency system, converter grid-connected is used to control generation frequency system conversion and control system motor Among to simulate m, wind frequency turbine converter is used to control frequency conversion motor to simulate wind turbine

14 Energies 2020, 13, of 18 Energies 2020, 13, of 21 Figure semi-physical simulation simulation platform platform Figure 9 9 The The structure structure diagram of 30 kva semi-physical Table 3 Table kw kw semi semi physical physical simulation simulation experiment experiment platform platform components components SerialNumber Number Serial Experiment Analysis Name Name Model Model variable frequency motor yvp250m-4 variable frequency yvp250m-4 speed torque sensormotor jn338a hydraulic fixed displacement a2f63r2p3 speedpump torque sensor jn338a pressure sensor zq-bz-1/hk/m20 hydraulicpressure fixed displacement pump a2f63r2p3 gauge yn-63-i-315 relief valve db10a-1-b30/315 pressure sensor zq-bz-1/hk/m20 check valve s20p10b accumulator sb330-10a1/112a9-330a pressure gauge yn-63-i-315 variable displacement motor a4vso40ds1/10w-ppb13t013z excitation synchronous generator wf relief valve db10a-1-b30/315 vane pump pfe relief valve agm20-a-20/100y check valve s20p10b relief valve agm20-a-20/315y constant pressure variable pump 63pcy14-1b accumulator sb330-10a1/112a9-330a relief valve dbds10g10b/25/2 cooler ok-el5s/31/m/a/1 variable air displacement motor a4vso40ds1/10w-ppb13t013z excitation synchronous generator wf above experimental platform vane pump pfe MPPT control Based on and pre-determined parameters, effect are researched under step wind speed 8 m/s 9 m/s at 8 s, and wind speed, pump speed, 12 relief of valve agm20-a-20/100y system power, and high pressure are shown in Figure relief valve agm20-a-20/315y 14 constant pressure variable pump 63pcy14-1b 15 relief valve dbds10g10b/25/2 16 air cooler ok-el5s/31/m/a/1 54 Experiment Analysis

15 Energies 2020, 13, of 21 Based on above experimental platform and pre-determined parameters, MPPT control effect are researched under step wind speed of 8 m/s 9 m/s at 8 s, and wind speed, pump speed, system power, and high pressure are shown in Figure 10 Energies 2020, 13, of 18 (a) (b) (c) (d) Figure 10 The system response curve under step wind speed (a) wind speed; (b) pump Figure 10 The system response curve under step wind speed (a) wind speed; (b) pump speed; (c) high pressure; (d) output power speed; (c) high pressure; (d) output power From Figure 10, through adjusting motor displacement, high pressure and pump speed change From with Figure10, step wind through speed, adjusting actual power motor also displacement, changes with high demand pressure power and The pump response speed change characteristics with step are wind basically speed, same actual as power simulation, also changes and with system demand state changes power stably The under response characteristics are basically same as simulation, and system state changes stably under step wind speed However, re are differences between demand power and actual power, and step wind speed However, re are differences between demand power and actual power, and reason is that MPPT controller accuracy depends on parameters reason is that MPPT controller accuracy depends on parameters The experimental parameters can be re-determined by experimental data V/β can be obtained by above experimental data When system is in a stable state, dynamic change The experimental parameters can be re-determined by experimental data V / β is equal to can be zero Therefore, acceleration term in controller is 0, and calculating formula for motor obtained by above experimental data When system is in a stable state, dynamic change is swing angle can be expressed as equal to zero Therefore, acceleration term in controller is 0, and calculating formula for motor swing angle can be expressed as γ = D pω p C t p h V k 1 P (36) K m ω md β K m ω md D p ω p Dpωp Ctph V k1δp γ = where v = d ( ) D (36) p ω p p h /dt Kmωmd β KmωmdDpωp According to Formula (36), controller contains three items, including pump input flow, where leakagevcaused = d( Dby pω pph) pressure dt and oil-hydraulic contraction caused by pressure The first two items are steady-state term, and oil-hydraulic contraction caused by pressure is dynamic adjustment term The HWT MPPT controller essentially converts power deviation into pressure change rate to dynamically adjust motor displacement The final steady-state term has a great influence on system control When system reaches a stable state, motor displacement should be

16 Energies 2020, 13, of 18 corresponding steady-state value related to pump speed and high pressure so that different power can be generated under each stable pump speed Orwise, repeated dynamic adjustments are needed to achieve stability The pump displacement is a constant, so leakage coefficient at different pump speed will ultimately affect system stability If steady value is not accurate, it is easy to cause power generation instability by adjusting dynamic term Instead, system will stabilize in or states Therefore, it is difficult to guarantee control accuracy in practical application, but controller provides oretical direction 6 Conclusions This paper takes HWT as research object; state space model is established for HWT; hydraulic power is taken as control output, and MPPT control strategy for HWT is proposed based on feedback linearization control method The strategy effectively solves power tracking control problem and strong nonlinearity problem, and system smoothly and quickly tracks demand power The efficiency is about 83%, which mainly depends on mechanical efficiency and volumetric efficiency of HWT key hydraulic components Although output power tracks demand power, it fluctuates greatly The main reason is that proposed controller depends on model parameters and includes leakage coefficient and or time-varying parameters, which leads to poor real-time performance of controller However, proposed oretical controller provides a reference for practical applications Therefore, based on engineering requirements, control law is simplified and verified by experiments; that is, output power can track demand power accurately and smoothly The impact of uncertainty and strong time-varying parameters of HWT MPPT control will be furr studied later Author Contributions: Data curation, LC; formal analysis, YZ, JG and ZH; funding acquisition, CA; validation, QH; writing original draft, WG; writing review and editing, HQ All authors have read and agreed to published version of manuscript Funding: This research was funded by National Natural Science Foundation of China grant number [ ] and funded by Excellent Youth Project of Hebei grant number [E ] Conflicts of Interest: The authors declare no conflict of interest Nomenclature Variable Symbol K P P rmax ρ R v C pmax ω r λ opt K P T p D p p h η mech,p ( T v ωp, v ) ω p Variable Specification The optimal power coefficient, K P = ρπr5 C pmax 2λ 3 opt Maximum wind power absorbed by rotor The air density The radius of a rotor blade The wind speed The maximum coefficient of wind energy utilize The wind turbine speed The optimum tip speed ratio The maximum wind power coefficient The pump torque The pump displacement The pressure difference between pump suction and discharge lines The constant pump mechanical efficiency, which is assumed to be unity The rotor pneumatic torque The pump moment of inertia The pump speed

17 Energies 2020, 13, of 18 B p The pump viscous damping coefficient Q p The pump flow rate C t1 The pump leakage coefficient T m The motor torque D m The motor displacement p h2 The pressure difference between motor suction and discharge lines η mech,m The motor mechanical efficiency, which is assumed to be 1 K m The motor maximum displacement γ The motor swing angle, ranging from 0 to 1 T L The motor load torque Q bl The proportional flow valve flow rate K The proportional coefficient U E The voltage signal Q c The flow rate caused by oil compression V The pressure-affected oil volume β e The effective oil bulk modulus including a correction for hose expansion J m The motor moment of inertia ω m The motor speed B m The motor viscous damping coefficient Q m The motor flow rate C t2 The motor leakage coefficient Q c1 The compressibility flow between pump and flow control valve Q c2 The compressibility flow between motor and flow control valve x 1 The state variable 1 x 2 The state variable 2 ω md The demand motor speed z 1 The state variable after coordinate transformation 1 z 2 The state variable after coordinate transformation 2 e The error y d The system control input y The system control output v The system controller k 1 The control parameter 1 k 2 The control parameter 2 References 1 Wei, J; Sun, W; Guo, A; Wang, L Analysis of wind turbine transmission system considering bearing clearance and rmo-mechanical coupling In Proceedings of World Non-Grid-Connected Wind Power Energy Conference, Nanjing, China, September 2009; pp Liu, ZG; Yang, GL; Wei, LJ; Yue, D Variable speed and constant frequency control of hydraulic wind turbine with energy storage system Adv Mech Eng 2017, 9, 1 10 [CrossRef] 3 Stelson, KA Saving world s energy with fluid power In Proceedings of 8th JFPS Int Symp Fluid Power, Okinawa, Japan, October 2011; pp Jiang, Z; Yang, L; Gao, Z; Moan, T Numerical simulation of a wind turbine with a hydraulic transmission system Energy Procedia 2014, 53, [CrossRef] 5 Pedersen, NH; Johansen, P; Andersen, TO Optimal control of a wind turbine with digital fluid power transmission Nonlinear Dyn 2018, 91, [CrossRef] 6 Yin, X; Tong, X; Zhao, X; Karcanias, A Maximum Power Generation Control of a Hybrid Wind Turbine Transmission System Based on H Loop-Shaping Approach IEEE Trans Sustain Energy 2019 [CrossRef] 7 Ai, C; Bai, W; Zhang, T; Kong, X Research on key problems of MPPT strategy based on active power control of hydraulic wind turbines J Renew Sustain Energy 2019, 11, [CrossRef] 8 Yin, X; Zhao, X Sensor-less Maximum Power Extraction Control of a Hydrostatic Tidal Turbine Based on Adaptive Extreme Learning Machine IEEE Trans Sustain Energy 2019, 11, [CrossRef]

18 Energies 2020, 13, of 18 9 Do, HT; Dang, TD; Truong, HVA; Ahn, KK Maximum power point tracking and output power control on pressure coupling wind energy conversion system IEEE Trans Ind Electron 2018, 65, [CrossRef] 10 Farbood, M; Sha-Sadeghi, M; Izadian, A; Niknam, T Advanced Model Predictive MPPT and Frequency Regulation in Interconnected Wind Turbine Drivetrains In Proceedings of 2018 IEEE Energy Conversion Congress and Exposition (ECCE), IEEE, Portland, OR, USA, September 2018; pp Li, S; Shi, Y; Li, J; Cao, W Maximum power point tracking control using combined predictive controller for a wind energy conversion system with permanent magnet synchronous generator In Proceedings of rd Youth Academic Annual Conference of Chinese Association of Automation (YAC), IEEE, Nanjing, China, May 2018; pp Mulders, SP; Diepeveen, NFB; van Wingerden, JW Extremum Seeking Control for optimization of a feed-forward Pelton turbine speed controller in a fixed-displacement hydraulic wind turbine concept J Phys Conf Series IOP Publ 2019, 1222, [CrossRef] 13 Deldar, M; Izadian, A; Anwar, S A decentralized multivariable controller for hydrostatic wind turbine drivetrain Asian JControl 2019 [CrossRef] 14 Wang, F; Chen, J; Xu, B; Stelson, KA Improving reliability and energy production of large wind turbine with a digital hydrostatic drivetrain Appl Energy 2019, 251, [CrossRef] 15 Ai, C; Wu, C; Zhao, F; Kong, X Optimal power tracking control of a hydraulic wind turbine based on active disturbance rejection control Trans Can Soc Mech Eng 2019 [CrossRef] 16 Akbari, R; Izadian, A; Weissbach, R An Approach in Torque Control of Hydraulic Wind Turbine Powertrains In Proceedings of 2019 IEEE Energy Conversion Congress and Exposition (ECCE), IEEE, Baltimore, MD, USA, 29 September 3 October 2019; pp Wei, L; Zhan, P; Liu, Z; Tao, Y; Yue, D Modeling and analysis of maximum power tracking of a 600 kw hydraulic energy storage wind turbine test rig Processes 2019, 7, 706 [CrossRef] 18 Alberto, I Nonlinear Control Systems II; Anaeum Press Ltd: Gateshead, UK, Kong, XD; Ai, C; Wang, J A summary on control system of hydrostatic drive train for wind turbines Chin Hydraul Pneum 2013, 1, Ai, C; Chen, LJ; Kong, XD Characteristics simulation for hydraulic wind turbine China Mach Eng 2015, 26, Jiang, ZH; Yu, XW Modeling and control of an integrated wind power generation and energy storage system In Proceedings of Power Energy Soc General Meeting, Calgary, AB, Canada, July 2009; pp Wang, JH Advanced Nonlinear Control Theory and Its Application; Beijing Science Press: Beijing, China, Sreenath, K; Park, HW; Poulakakis, I; Grizzle, JW A compliant hybrid zero dynamics controller for stable, efficient and fast bipedal walking on MABEL Int J Rob Res 2011, 30, [CrossRef] 24 Bharadwaj, S; Rao, AV; Kenneth, DM Entry trajectory tracking law via feed-back linearization J Guid Control Dyn 1998, 21, [CrossRef] 2020 by authors Licensee MDPI, Basel, Switzerland This article is an open access article distributed under terms and conditions of Creative Commons Attribution (CC BY) license (