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A robust H∞ controller based frequency control approach using the wind-battery coordination strategy in a small power system
Electrical Power and Energy Systems 58 (2014) 190–198
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
A robust H1 controller based frequency control approach using the wind-battery coordination strategy in a small power system Abdul Motin Howlader a,⇑, Yuya Izumi a, Akie Uehara a, Naomitsu Urasaki a, Tomonobu Senjyu a, Ahmed Yousuf Saber b,1 a b
Faculty of Engineering, University of the Ryukyus, 1 Senbaru, Nishihara-cho, Nakagami, Okinawa 903-0213, Japan Principal Power Engineer, Operation Technology, Inc., ETAP, Irvine, CA, USA
a r t i c l e
i n f o
Article history: Received 3 August 2012 Received in revised form 4 January 2014 Accepted 18 January 2014
Keywords: Battery energy storage system Frequency control H1 control Pitch angle control Wind turbine generator
a b s t r a c t In this paper, a coordinated control method for a wind turbine generator (WTG) and a battery energy storage system (BESS) of a small power system have been presented. The coordinated control approach applies to control the system frequency to reduce the size of BESS and to control the pitch angle system to mitigate the wind turbine blades stress. To achieve these objectives simultaneously, the robust H1 control method is applied in this paper. The pitch angle system of the WTG and the output power command system of the BESS are controlled by the H1 controllers. The output power command of the WTG is determined from the wind velocities and it is controlled by the pitch angle control system. Concurrently, the output power command of the BESS is calculated according to the state of charge and the frequency deviations in the small power system. The small power system includes a WTG, a BESS and a diesel generator. Numerical simulations are conducted by the MATLAB/SIMULINKÒ environment to validate the proposed method. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction There are many isolated communities in the world. Diesel generators are the main source of electric power for the isolated communities. In recent years, limited fossil fuels (e.g. coal, oil, gas) and many conflicting objectives such as environmental impact, reliability, imported fuels are serious issues [1]. The diesel-generated electricity is more expensive than large electricity production plants (e.g. gas, hydro, wind), and incurs the transport, fuel and storage costs [2]. Diesel generators are inherently inefficient when operated at a low load factor (below 40–50% of their rated capacity) and diesel engines burn fossil fuels which release harmful emissions into the atmosphere [3,4]. To overcome these problems, renewable energies (e.g. wind, solar, hydro, tide, biomass) have been drawing attention as the clean electricity sources. Wind energy is one of the most rapid growing renewable power sources in the world, and wind power penetration has been increasing to the power grid [5–8]. On the other hand, the wind turbine generator (WTG) can deliver a cheap electricity power as compared with other renewable sources [9]. Therefore, wind–diesel hybrid power ⇑ Corresponding author. Tel.: +81 98 895 8686; fax: +81 98 895 8708. E-mail addresses: [email protected] (A.M. Howlader), aysaber@ ieee.org (A.Y. Saber). 1 Tel.: +1 949 716 5839. http://dx.doi.org/10.1016/j.ijepes.2014.01.024 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.
systems can install to the isolated communities in order to improve their life standard. This hybrid power system also can be a part of a micro-grid system. With the hybrid power system, the annual diesel fuel can be reduced and, at the same time, the level of emissions can be minimized [10]. However, wind velocity is a highly stochastic element which can deviate quickly. The output power is delivered by the WTG which is proportional to the cube of wind speed. Hence, any deviations in wind speed will cause the wind power fluctuations. The power fluctuations may lead the frequency fluctuation and voltage flicker inside the power system [11]. Therefore, various power smoothing methods are applied to overcome these problems. The pitch angle control is utilized for the output power control of the WTG. There are various pitch angle methods to control the output power of WTGs [11–15]. By using these control methods, the output power fluctuations of WTGs can be reduced at wind speed variations. But these methods increase the wind turbine blade stress extensively due to the rapid pitch actions. To reduce the blade stress, the control regions of the pitch angle and the output power of the WTG should be considered in a frequency domain. The pitch angle control system in a high frequency domain increases the mechanical stress to the wind turbine blades. Moreover, wind energy is an uncertain fluctuating resource and requires tight control management. The conventional proportional–integral (PI) controller is not ideal due to the high turbulence wind velocities and
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parameter uncertainties. Therefore, the researchers have promoted to interest in the robust control concepts (e.g. H2 or H1 controllers). In [15], the H1 control method is applied to generate a smooth output power. However, the control region of the pitch angle in a frequency domain is not clarified. The H1 controllers based load mitigation methods for the WTG systems are proposed in [16,17], and there are nothing to mention about the frequency control strategies. Nowadays, the energy storage system (ESS) is integrated with the renewable sources which are connected into the power grid to maintain the safe operation of the power grid, and balance the supply and demand sides [18–20]. Moreover, the integrated ESS and WTG system can control the frequency in small power systems. There are different types of ESSs in the power systems such as batteries, superconducting magnetic energy storage (SMES), electric double layer capacitor (EDLC), and flywheels [18–25]. Since the SMES has to maintain the conducting state, the operation cost is very high. The installation cost of the EDLC is very high and the flywheel creates a high noise at the large capacity range of operation. The battery energy storage system (BESS) is one of the most rapid growing storage technologies. The BESS installation cost and generating noise are relatively lower than the other storage technologies. However, the installation cost of the large BESS is also very high. If a large battery storage installs in a small power system, the system cost will be increased significantly. Therefore, to reduce the BESS capacity, the power system can provide an enormous economic benefit. A short-term operation can reduce the size of a BESS. For an integrated battery storage and wind turbine systems, the BESS should operate in a short term basis to decrease the size of it. For that reason, this paper presents a coordinated control method for wind turbine and battery storage in a small power system. The coordinated control method applies to control the system frequency which can reduce the size of a BESS, and the pitch angle control system which can mitigate the wind turbine blade stress. The proposed coordinated control method is implemented through the robust H1 controller. The H1 controllers are applied to the pitch angle control system of the WTG and the output power command system of the BESS. By using the H1 controllers, the desired frequency characteristics of the control systems can be achieved by the proper weighting functions in a frequency domain. The output power command of the WTG is depended on wind velocities and is controlled by the pitch angle control system. In addition, the wind speed variations are assumed as an uncertain parameter to design the H1 controller for the pitch angle control system. The H1 controller can reduce the low frequency components for the output power errors of the WTG. Therefore, the pitch action decreases which can mitigate the wind turbine blade stress. Concurrently, the output power command of the BESS is determined according to the state of charge and the frequency deviations in the small power system. By using the output power command system, the H1 controller can limit the BESS operation within the high frequency domain. Thus, the charge/discharge operations of the BESS are not required to perform at the long term. This phenomenon can reduce the size of a BESS in a small power system. The proposed method is compared with the conventional PI control method which can reduce the frequency deviation, size of BESS and mechanical stress of WTG. Effectiveness of the proposed method is verified by the numerical simulations in MATLABÒ/SIMULINKÒ environment.
a diesel generator and a load. It is assumed that the small power system can operate independently. The detail model of the small power system model is shown in Fig. 2 [26]. In Figs. 1 and 2, the symbols P d ; P c ; P g , and P b are the output power of the diesel generator, the combined power of the WTG and the BESS, the output power of the WTG and the output power of the BESS, respectively. The symbols P L and P e are the load power and the supply error, respectively. The ‘’ represents the commanded values for different powers. The WTG system is connected to the AC side through a transformer (AC link method). Since the main purpose of this paper is frequency control, the frequency analytical model of the frequency control system is expressed as a simple transfer function for simplified simulations. The power system model does not include the characteristics such as substation facilities, electrical instruments, communication delay. The flat frequency control (FFC) approach is applied to control the frequency of the diesel generator. The FFC method is commonly used in a stand-alone power system. This method includes an integral control loop which can reduce the frequency deviation, Df . The system capacity is 687.5 kW. Pd
Pd + Pc Load
DG Pc
Diesel generator
Pb Pg
Inverter
IG1
WTG
Battery
Fig. 1. Proposed small power system model.
Fig. 2. Small power system model.
2. Small power system model An integrated power system diagram is shown in Fig. 1. The small power system consists of a wind turbine, a battery storage,
Fig. 3. Wind turbine generator system.
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2.1. Wind turbine generator system The wind turbine generator system is shown in Fig. 3. The output power of the WTG, P w , is determined by the following equation [27,28]:
Pw ¼
1 C p ðk; bÞqpR2 V 3w 2
ð1Þ
where V w is the wind speed, q is the air density, R is the rotor radius, and C p is the aerodynamic power coefficient. C p is approximated by the following equations [12,15]:
C p ðk; bÞ ¼ c1 ðbÞk2 þ c2 ðbÞk3 þ c3 ðbÞk4
ð2Þ
9 c1 ðbÞ ¼ c10 þ c11 b þ c12 b2 þ c13 b3 þ c14 b4 > = c2 ðbÞ ¼ c20 þ c21 b þ c22 b2 þ c23 b3 þ c24 b4 2
3
c3 ðbÞ ¼ c30 þ c31 b þ c32 b þ c33 b þ c34 b
4
2.2. Linearization of wind turbine generator system To apply the H1 control theory to the pitch angle control system, the WTG system should be linearized [15]. When the wind speed, V w , is given, the wind turbine torque, T w , and the generator torque, T g , are denoted by the following equations:
T w ðxw ; V w ; bÞ ¼ ð3Þ
> ;
Tg ¼
where c10 c34 are the constants represented by the performance characteristic of the wind turbine, b is the pitch angle, k is the tip speed ratio. The fixed speed squirrel-cage induction generator, that has the advantages of low cost and robustness, is used in this paper. The generator output power, P g , can be expressed by
Pg ¼
a low-frequency component of the wind speed, V w low , is determined through a low-pass filter which is used for calculating the output power command of the WTG, Pg . The time constant of the low-pass filter is 10 s. By using the large time constant, the output power command of the WTG can be smoothed.
3V 2 sð1 þ sÞR2
ð4Þ
ðR2 sR1 Þ2 þ s2 ðX 1 þ X 2 Þ2
where V the phase voltage, the slip of induction generator is as s ¼ ðxo xÞ=xo which comprises the synchronous angular speed of the rotor, xo and the angular speed of rotor, x; R1 and R2 are the generator’s stator and rotor resistances, X 1 and X 2 are the generator’s stator and rotor reactances. The H1 controller based pitch angle control system will be explained in Section 3. In Fig. 3, by subtracting the output power command of the WTG, Pg , from the output power of the WTG, P g , determines the output power error of the WTG, DP g , that evaluates the pitch angle command, bCMD , via the pitch angle control system. Pg is controlled by a hydraulic servo system that drives the blades according to the pitch angle command bCMD . The hydraulic servo system is modeled as a first-order lag system, which has the time constant, T c , of 1s [27,28]. The pitch angle command, bCMD , is limited by a limiter within 10–90°. In the conventional methods of WTG, output power command is constant at the rated power (275 kW). The pitch angle, b, operates in the region between the rated wind speed and the cut-out wind speed, and b is constant at 10 in the region between the cut-in wind speed and the rated wind speed. Therefore, in the region between the cut-in wind speed and the rated wind speed, the torque controller maximizes the power depending on the wind speed. To achieve the output power leveling control of the WTG, the pitch angle control law is extended for all operating regions in this paper [12,15]. The proposed output power command system of the WTG is shown in Fig. 4. The output power command of the wind turbine is approximated by [12]
Pw ¼ d1 þ d2 V 2w
ð5Þ
In (5), d1 and d2 are expressed as a function of the pitch angle, b, and are set at d1 ¼ 46:8623 and d2 ¼ 1:95. In practical systems, since the accurate measuring and forecasting of wind speed is difficult,
1 C p ðk; bÞV 3w qA 2xw
3V 2 sð1 þ sÞR2
xe fðR2 sR1 Þ2 þ s2 ðX 1 þ X 2 Þ2 g
ð6Þ
ð7Þ
here, the wind turbine torque is a high nonlinearity component. For simplification, (6) is substituted by the following equation:
T w ¼ f ðxw ; V w ; bÞ
ð8Þ
By indicating each operating point, xwo ; V wo ; bo , and using the Taylor expansion which ignores higher order terms (over second order), this expression as follows
T w T wo ¼
@f @f @f ðxw xwo Þ þ ðV w V wo Þ þ ðb bo Þ @ xw @V w @b
Rewriting Aslope
ð9Þ
@f @f ¼ @@f xw , Bslope ¼ @V w , C slope ¼ @b, in (8), then
DT w ¼ Aslope Dxw þ Bslope DV w þ C slope Db
ð10Þ
where D indicates a small-signal value around the operating point. Aslope ; Bslope ; C slope can be determined by [15]:
qAV 2w Vw @C p C p þR 2xw xw @k qAV w Vw @C p R 3C p Bslope ¼ 2 xw @k
Aslope ¼
C slope ¼ where
1 V w3 @C qA p 2 xw @b
@C p @k
;
@C p @b
ð11Þ ð12Þ ð13Þ
are expressed as:
@C p ¼ 2c1 k þ 3c2 k2 þ 4c3 k3 @k @C p ¼ c01 k2 þ c02 k3 þ c03 k4 @b c01 ¼ c11 þ 2c12 b þ 3c13 b2 þ 4c14 b3 c02 ¼ c21 þ 2c22 b þ 3c23 b2 þ 4c24 b3 c03 ¼ c31 þ 2c32 b þ 3c33 b2 þ 4c34 b3 The value of Aslope ; Bslope ; C slope are determined by the calculation of the operating point. In second step, linearization of the generator torque is described. The generator torque is also a nonlinear component. The linear expression of the generator torque is as follows:
T g ¼ K g xw
ð14Þ
where K g is a constant value which is around the operating point. The small-signal generator torque, DT g , can be expressed as
Vw
1 10s+1
Vw_low
2
d1+d2Vw_low
LPF Fig. 4. Output power command system of WTG.
Pg*
DT g ¼ K g Dxw
ð15Þ
where Dxw is the difference between xw and xwo . In addition, the small-signal generated power, DPg , can be expressed as
DPg ¼ K e Dxw
ð16Þ
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where K e is a constant value which is around the operating point. In third step, linearization of the rotational speed of wind turbine is described. The rotational speed of the wind turbine can be expressed as
J
dxw ¼ Tw Tg dt
Pb*
ð17Þ
+0.2pu 1 Tbs+1 -0.2pu
where J is the inertia of the wind turbine. The small-signal model around the operating point is expressed as follows
1 _ w ¼ ð DT w DT g Þ Dx J
ð18Þ
where ‘‘’’ is the differential operator. By substituting (10) and (15) in (18), the following equation is obtained:
1 _ w ¼ f Aslope K g Dxw þ Bslope DV w þ C slope Dbg Dx J
ð19Þ
The small-signal rotational speed of the wind turbine around the operating point is derived from (19) as [15]:
Dx w ¼
1 sJ fAslope K g g
fBslope DV w þ C slope Dbg
ð20Þ
where Db can be expressed using the pitch angle command, DbCMD , as
Db ¼
1 Db T c s þ 1 CMD
ð21Þ
Substituting (21) into (20) yields the following expression:
Dx w ¼
C slope Bslope DV w þ DbCMD sJ fAslope K g g Tcs þ 1 1
ð22Þ
From (22), G1 and G2 are derived as
C slope ðT c s þ 1ÞfsJ ðAslope K g Þg Bslope G2 ¼ sJ ðAslope K g Þ
G1 ¼
ð23Þ ð24Þ
These transfer function models of the wind speed, V w , and the generator output power, Pg , are shown in Fig. 5. 2.3. Battery energy storage system model The model of the BESS is shown in Fig. 6. The output power command of the BESS, P b , is limited by the capacity. Although it is assumed that a bidirectional power converter is controlled by the pulse width modulation, the BESS is modeled as a first-order lag system and the time constant, T b , is 0.3 s. The state of charge, n, is calculated by integrating the output power of the BESS, Pb . In this paper, the state of charge is indicated as a dischargeable energy in percentage of the battery rated capacity. The state of charge should always be maintained in the proper range for a stable operation [29]. Therefore, the state of charge is maintained near 50% by the charge/discharge controller in this paper. The rated capacity of the battery is 200 kW h and the rated capacity of the power converter is 137 kW (0.2 pu).
ΔVw
Δ βCMD
Pb 1 Wb Capacity ξ calculation s
Fig. 6. BESS model.
The output power command of the BESS, Pb , is determined by two control inputs; the frequency deviation, Df , and the state of charge error of the BESS, Dn. The output power command system of the BESS will be explained in the next section. 3. Proposed controller design approach 3.1. H1 control theory This section presents the design principles of the H1 controllers. The detail design of the H1 controllers is shown in [30–33]. Since this paper deals with the design of the pitch angle control system and the output power command system of the BESS, the H1 control theory is briefly explained. A closed loop system which includes weighting functions is shown in Fig. 7. In this figure, P is an intended open-loop plant, K is a H1 controller, and W S and W T are weighting functions, respectively. The weighing functions determine the performance of the controller, K. The transfer function be~ is given as tween the command value, w, and the output, zT ¼ ½~e; y follows.
~e W S SðsÞ=c ¼ w ~ W T TðsÞ y
ð25Þ
where c is the boundary of H1 norm, it is a small positive value which is indicated the control efficiency. In addition, SðsÞ ¼ ð1 þ PKÞ1 and TðsÞ ¼ PKSðsÞ are the sensitivity function and the complementary sensitivity function, respectively. Generally, it is possible to decrease the control error for the command value by selecting SðsÞ and it is possible to design a robust controller for disturbance by selecting TðsÞ. The design index as mixed sensitivity problem can be given as follows.
W S SðsÞ W TðsÞ < c T 1
ð26Þ
In this paper, linear matrix inequality (LMI) approach [31,32] is applied to design the H1 controllers. 3.2. Design process and configurations of H1 controllers The H1 controllers based pitch angle control system and output power command system of the BESS are described in this section. These control systems operate as decentralized control systems because the H1 controllers are designed individually. At first, the H1 controller of the pitch angle control system is designed because the rated capacity of the WTG is larger than that of the BESS. Subsequently, the H1 controller of the output power
WS
G2
G1
Δωw
w Ke
Fig. 5. Linearized model of WTG.
e
K(s)
u
P(s)
y
e z
WT
Δ Pg Fig. 7. Closed loop system including weight function.
y
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Singular Values (dB)
command system of the BESS is designed. The H1 controllers based proposed control system are shown in Fig. 8. To design the H1 controller based output power command system of the BESS, the entire power system is shown in Fig. 1 which is considered as the nominal plant. Therefore, the H1 controller design procedures consider the characteristics and performance of the WTG. The H1 controller based proposed pitch angle control system is shown in Fig. 8(a). From this figure, the input of the H1 controller is the output power error of the WTG, DP g . The output of the H1 controller is the pitch angle command, bCMD , to maintain the error DPg is about zero. Fig. 8(b) shows the H1 controller based output power command system for the BESS. From this figure, the inputs of the H1 controller are the frequency deviation, Df , and the state of charge error, Dn. The output of the H1 controller is the output power command of the BESS, Pb , to keep these errors (Df ; Dn) around zero. The singular value plots of the control loops with/without H1 controllers are illustrated in Fig. 9. The open-loop transfer functions, GðsÞDPg !bCMD ; GðsÞDf !Df , and GðsÞDn !Dn , are expressed by the following equations:
100
Without H-infinity controller With H-infinity controller
50 0 -50 -100 10-3
10-2
10-1
235:3 s2 þ 1:446s þ 0:4465 33:33s3 þ 340s2 þ 66:67s GðsÞDf !Df ¼ 5 s þ 14:73s4 þ 52:24s3 þ 72:09s2 þ 100:1s þ 60 0:3183 GðsÞDn !Dn ¼ 2 s þ 3:333s
Singular Values (dB)
Without H-infinity controller With H-infinity controller
50 0 -50 -100 10-4
10-2
102
104
Singular Values (dB)
ð28Þ ð29Þ
100
Without H-infinity controller With H-infinity controller
50 0 -50 -100 10-4
10-2
102
104
Fig. 9. Singular value plots (DP g ! bCMD ; Df ! Df ; Dn ! Dn).
in Fig. 10(a) and (b). To design the H1 controllers, one can choose the weighting functions as a low pass filter or a high pass filter with an arbitrary order. In this paper, the weighting functions are designed by using LMI Control Toolbox of MATLAB/SIMULINKÒ [32]. From Fig. 10(a) and (b), W S1 and W T1 are related to the pitch
Plant
8
P(s)
Pg
WT1
WS12
Plant
−Δ f 8
H
100
Frequency (rad/sec)
βCMD
H
Δξ
100
ð27Þ
WS11
ξ *= 50%
103
100
WS1
Δ f *= 0
102
Frequency (rad/sec)
The singular value plots of the pitch angle control loop (DPg to bCMD ) are shown in Fig. 9(a). From this figure and GðsÞDPg !bCMD , it is seen that the numerator of the transfer function is large and there is a high-gain at the low frequency domain (below 1 rad/s). Due to the high gain, the pitch action for the output power fluctuations of the WTG may increase at the low frequency domain. In the frequency control loop, it is seen that the singular value has a resonance point as shown in Fig. 9(b). The resonance frequency can be calculated by GðsÞDf !Df and the value is 1.4007 rad/s. The resonance point might affect the frequency control in the small power system. On the other hand, the state of charge control loop is shown in Fig. 9(c). From this figure and GðsÞDn !Dn , the singular value has an integral characteristic at the low frequency domain (below 1 rad/s). Due to the integral characteristic, the state of charge keeps to increase so that the characteristic can be eliminated. According to achieve these design goals, the weighting functions, W S1 , W T1 ; W S11 ; W S12 ; W T11 , and W T12 are selected as shown
Pg*
101
Frequency (rad/sec)
GðsÞDPg !bCMD ¼
ΔPg
100
Pb*
P(s)
Fig. 8. Proposed control systems.
Δf
WT11 ξ
WT12
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method, the PI controllers are applied instead of the H1 controllers. The parameters of the PI controllers are listed in Table 2. These parameters are selected by the trial and error to obtain a good control performance.
Singular Values (dB)
150
WS1
100 50
WS12
0
4.1. Evaluation of the pitch action in a short-term simulation
WS11
-50 -100 10-4
10-2
100
The simulation results are shown in Fig. 11. Fig. 11(a) and (b) shows the wind speed and the load power, which are modeled by the random functions. The pitch angles with the different time ranges are shown in Fig. 11(c) and (d), respectively. From these figures, it is confirmed that the pitch action with the proposed method is moderated as compared with the conventional method. For the more quantitative measurement of the pitch action, a performance index, bact , is derived as the following equation:
102
Frequency (rad/sec)
Singular Values (dB)
0 -20
WT1
-40 -60
WT11 WT12
bact ¼
-80 -100 10-4
10-2
100
102
Frequency (rad/sec)
Fig. 10. Singular value plot of weighting functions.
angle control loop. W S1 is selected as a second-order low pass filter, which contains a high-gain at the region below the 1 rad/s. W S11 and W T11 are related to the frequency control loop, while W S12 and W T12 are related to the state of charge control loop, respectively. W S11 is selected as a second-order low pass filter, which has the cutoff frequency at 0.01 rad/s. As a results, the band width of the frequency control loop is improved within the high frequency domain (about 0.01–100 rad/s). Similarly, W S12 is selected as a second-order low pass filter, which has the cutoff frequency at 1 rad/s. This weighting function eliminates the integral characteristic at the region below the 0.01 rad/s. Therefore, the of charge/discharge operations of the BESS is performed within a short-term basis which can reduce the size of the BESS. The weighting functions of the complementary sensitivity function, W T1 ; W T11 and W T12 , decide the robustness of the H1 controllers against the disturbances. Hence, the weighting functions are set to a low-gain as illustrated in Fig. 10(b). The weighting functions of the proposed system are expressed as:
W S1 ¼
0:05023s þ 0:5467
s2 þ 0:002s þ 106 0:00571 W S11 ¼ 2 s þ 0:04398s þ 0:001265 2:22 1016 s þ 49:6 W S12 ¼ 2 s þ 0:959s þ 0:9681 1:635s2 þ 13:79s þ 11:39 W T1 ¼ s2 þ 3:246 104 þ 1:671 106 0:001134s2 þ 0:001277s þ 0:0001125 W T11 ¼ s2 þ 7:168s þ 12:75 2 0:00845s þ 0:05187s þ 0:143 W T12 ¼ s2 þ 468:1s þ 1691
ð30Þ ð31Þ ð32Þ ð33Þ ð34Þ ð35Þ
4. Simulation results The simulation parameters of the small power system, WTG system, and the operating points for linearization are listed in Table 1. The proposed H1 controller methods are compared with the conventional PI controller methods. In the conventional
Z t dbðtÞ dt dt 0
ð36Þ
If bact is large, it is means that the pitch action of the wind turbine also is large, and v ice v ersa. bact is shown in Fig. 11(e). It is evident that the mechanical stress of the wind turbine blades can be reduced by the proposed method. The output powers of the WTG system for the conventional and proposed methods are shown in Fig. 11(f) and (g). Reduction of the mechanical stress of the wind turbine blades and the output power command of the WTG are related to the performance trade-off. Therefore, the output power fluctuation of the WTG with the proposed method is larger than that of the conventional method. However, the charge/discharge of the BESS is shown in Fig. 11(h). The charge/discharge in the long-term is reduced by the proposed H1 controller. In addition, the variation of the state of charge with the proposed method is smaller as compared with the conventional method, as shown in Fig. 11(i). Fig. 11(j)–(l) shows the output power of the diesel generator, the supply error, and the frequency deviation, respectively. As can be seen from these figures, the output power of the diesel generator is almost constant with both methods. Also, the frequency deviation is nearer to zero. The frequency control method to the small power system, the control performance of the proposed H1 controller is same as the conventional PI controller. However, from the simulation results, it is
Table 1 Simulation parameters. Parameters of small power system Inertia constant, M Damping constant, D Governor time constant, T g Diesel generator time constant, T d Parameters of WTG system Blade radius, R Inertia coefficient, J Air density, q Rated output, P g Phase voltage, V Stator resistance, R1 Stator reactance, X 1 Rotor resistance, R2 Rotor reactance, X 2
0.15 puMW s/Hz 0.008 puMW/Hz 0.1 s 1.0 s 14 m 62,993 kg m2 1.225 kg/m3 275 kW pffiffiffi 400 3 V 0.00397 X 0.0376 X 0.00443 X 0.0534 X
Operating point Wind speed, V wo Pitch angle, bo Rotational speed, xwo Aslope
12.5 m/s 12° 4.5326 rad/s
Bslope
9:6695 103
C slope
1:8862 103 24,337 7857
Kg Ke
3:7882 103
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4.2. Evaluation of BESS operation in a long-term simulation
Pitch angle control system Frequency deviation State of charge
P gain K p
I gain K I
2 2 0.1
0.1 4 0.01
Wind speed Vw [m/s]
evident that the proposed method can reduce the wind turbine blade stress as compared with the conventional method. The proposed method also can reduce the size of the BESS.
18 16 14 12 10 8 100
200
300
400
500
600
700
In the long-term simulation, a cost analysis approach is given through the BESS operation. The simulation time is extended to 30 min. The simulation results are shown in Fig. 12. The wind speed and the load are shown in Fig. 12(a) and (b), which are modeled by the random functions. The pitch angle and the output power of the WTG for the both methods are shown in Fig. 12(c)– (e), respectively. From Fig. 12(c), it is confirmed that the pitch action is mitigated by the proposed H1 controller method. However, as mentioned in previous section, the power fluctuation of Output power of WTG Pg [pu]
Table 2 PI controller parameters.
0.8 Output power command Pg*
0.6
Output power Pg
0.4 0.2 0 100
200
300
Time t [s]
0.7 0.6 200
300
400
500
600
700
Output power of BESS Pb [pu]
Load PL [pu]
0.8
0.5 100
15
0 -0.1 -0.2 100
Conventional method Proposed method
200
300
400
500
600
700
Conventional method Proposed method
200
250
300
Output power of diesel generator Pd [pu]
Pitch angle β [deg]
15
50 49 48 100
State of charge command ξ∗ Conventional method Proposed method
200
300
500
600
700
Conventional method Proposed method
0.6 0.4 0.2 0 100
200
300
400
500
600
700
0.10
Supply error Pe [pu]
Performance index βact
400
0.8
60 40 Conventional method Proposed method
20 200
300
400
500
600
0.05 0 -0.05 -0.10 100
700
Conventional method Proposed method
200
300
Time t [s]
0.8 Output power command Pg
0.6
*
Output power Pg
0.4 0.2 200
300
400
400
500
600
700
Time t [s]
500
600
700
Frequency deviation Δ f [Hz]
Output power of WTG Pg [pu]
700
Time t [s]
80
0 100
600
51
Time t [s]
0 100
500
Time t [s]
20
150
400
52
Time t [s]
10 100
700
0.1
State of charge ξ [%]
Pitch angle β [deg]
Conventional method Proposed method
300
600
Time t [s]
20
200
500
0.2
Time t [s]
10 100
400
Time t [s]
0.4 0.2 0 Conventional method Proposed method
-0.2 -0.4 100
200
Time t [s]
Fig. 11. Simulation results in a short-term.
300
400
Time t [s]
500
600
700
18 16 14 12 10 8
200
400
600
800 1000 1200 1400 1600 1800
Output power of BESS Pb [pu]
Wind speed Vw [m/s]
A.M. Howlader et al. / Electrical Power and Energy Systems 58 (2014) 190–198
0.3 0.2 0.1 0 -0.1 -0.2 -0.3
Conventional method Proposed method
200
400 600
Time t [s]
State of charge ξ [%]
Load PL [pu]
0.9 0.8 0.7 200
400
600
800 1000 1200 1400 1600 1800
70 60 50 State of charge command ξ∗ Conventional method Proposed method
40 30
200
400
600
Time t [s]
200
400
600
800 1000 1200 1400 1600 1800
Output power of diesel generator Pd [pu]
Pitch angle β [Hz]
Conventional method Proposed method
15 10
0.8 Conventional method Proposed method
0.6 0.4 0.2 0 200
400
600
0.10 *
Supply error Pe [pu]
Output power of WTG Pg [pu]
0.8 Output power command Pg Output power Pg
0.6 0.4 0.2 200
400
600
800 1000 1200 1400 1600 1800
0.05
Conventional method Proposed method
0 -0.05 -0.10
200 400
600 800 1000 1200 1400 1600 1800
Time t [s]
0.8 *
Output power command Pg Output power Pg
0.6 0.4 0.2 200
400
600
800 1000 1200 1400 1600 1800
Frequency deviation Δ f [Hz]
Output power of WTG Pg [pu]
Time t [s]
0
800 1000 1200 1400 1600 1800
Time t [s]
Time t [s]
0
800 1000 1200 1400 1600 1800
Time t [s]
25 20
800 1000 1200 1400 1600 1800
Time t [s]
1.0
0.6
197
0.4 0.2 0 -0.2 -0.4
Conventional method Proposed method
200 400
Time t [s]
600 800 1000 1200 1400 1600 1800
Time t [s]
Fig. 12. Simulation results in a long term.
the WTG with the proposed method is larger than the conventional method (in Fig. 12(d) and (e)). The output power of the BESS, state of charge, output power of the diesel generator, supply error, and frequency deviation with the both methods are shown in Fig. 12(f)–(j), respectively. In this paper, the kW and kW h capacities of the BESS are set to 200 kW h and 137 kW (0.2 pu), respectively. From Fig. 12(f), it is confirmed that the BESS with the conventional method cannot discharge due to the capacity limit at around t ¼ 1200 s. As a result, the large supply error and frequency deviation can occur which is shown in Fig. 12(i) and (j). If a NaS battery uses for the BESS, the cost per unit energy is about 700 $/kW h and the cost per unit power is about 1500 $/kW, respectively [34]. When the proposed method is employed, it is evident that the required kW capacity is less than 137 kW and the total cost is about 205,500 $. However, if the conventional method is used, more large kW capacity will be required. Then the cost of the BESS is about 412,500 $. In addition, from
Fig. 12(g), the state of charge of the conventional method fluctuates nearer to 40%. The state of charge should be kept within proper limits (i.e. between 30% and 70%) [35], more large kW h capacity is required for the long period of operation. On the other hand, the state of charge with the proposed method fluctuates within ±5% deviations. Therefore, the value of kW h capacity is used in this simulation which is suitable for the frequency control. The output power of the diesel generator is little bit more fluctuated than the conventional method (in Fig. 12(h)). Although this section focuses on the operation performance of the BESS, but an optimum capacity of the BESS considering the diesel generator operation is investigated for a long period of time. 5. Conclusion This paper presents a coordinated control method for an integrated BESS and WTG in a small power system. The coordinated
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control focuses on the frequency control in a small power system. It can reduce the charge/discharge of the BESS in a long period of operation, and can mitigate the wind turbine blades stress extensively. The proposed method is implemented through the robust H1 controllers. The H1 control method is applied to the pitch angle control system of the WTG, and the output power command system of the BESS. The proposed method is compared with the conventional PI control method. From the simulations results, the proposed method can mitigate the wind turbine blades stress and can decrease the frequency deviation. It can also decrease the size of a BESS in a small power system which can reduce the system cost significantly.
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[15] Sakamoto R, Senjyu T, Kaneko T, Urasaki N, Takagi T, Sugimoto S. Output power leveling of wind turbine generator by pitch angle control using H1 control. Electr Eng Jpn 2008;126(4):17–24. [16] Corcuera AD, Arrese AP, Ezquerra JM, Segurola E, Landaluze J. H1 based control for load mitigation in wind turbines. Energies 2012;5(4):938–67. [17] Muhando BE, Wies RW. Nonlinear H1 constrained feedback control for gridinteractive WECS under high stochasticity. IEEE Trans Energy Convers 2011;26(4):1000–9. [18] Han X, Chen F, Cui X, Li Y, Li X. A power smoothing control strategy and optimized allocation of battery capacity based on hybrid storage energy technology. Energies 2012;5(5):1593–612. [19] Mousavi SMG. An autonomous hybrid energy system of wind/tidal/ microturbine/battery storage. Int J Electr Power Energy Syst 2012;43(1):1144–54. [20] Sebastian R. Modelling and simulation of a high penetration wind diesel system with battery energy storage. Int J Electr Power Energy Syst 2011;33(3):767–74. [21] Du W, Wang HF, Cheng S, Wen JY, Dunn R. Robustness of damping control implemented by energy storage systems installed in power systems. Int J Electr Power Energy Syst 2011;33(1):35–42. [22] Hajizadeh A, Golkar MA. Control of hybrid fuel cell/energy storage distributed generation system against voltage sag. Int J Electr Power Energy Syst 2010;32(5):488–97. [23] Qudaih YS, Elbaset AA, Hiyama T. Simulation studies on ECS application in a clean power distribution system. Int J Electr Power Energy Syst 2011;33(1):43–54. [24] Chatterjee A, Ghoshal SP, Mukherjee V. Transient performance improvement of grid connected hydro system using distributed generation and capacitive energy storage unit. Int J Electr Power Energy Syst 2012;43(1):210–21. [25] Kinjo T, Senjyu T, Urasaki N, Fujita H. Output leveling of renewable energy by electric double-layer capacitor applied for energy storage system. IEEE Trans Energy Convers 2006;21(1):221–7. [26] Howlader AM, Urasaki N, Pratap A, Senjyu T, Saber AY. A fuzzy control strategy for power smoothing and grid dynamic response enrichment of a gridconnected wind energy conversion system. Wind Energy. doi: 10.1002/ we.1637. [27] Howlader AM, Urasaki N, Yona A, Senjyu T, Saber AY. Design and implement a digital H1 robust controller for a MW-class PMSG-based grid-interactive wind energy conversion system. Energies 2013;6(4):2084–109. [28] Senjyu T, Kaneko T, Uehara A, Yona A, Sekine H, Kim CH. Output power control for large wind power penetration in small power system. Renew Energy 2009;34(11):2334–43. [29] Yoshimoto K, Nanahara T, Koshimizu G, Uchida Y. New control method for regulating state-of-charge of a battery in hybrid wind power/battery energy storage system. IEEE Power Syst Conf Expos 2006:1244–51. [30] Apkarian P, Gahinet P, Becker G. Self-scheduled H1 control of linear parameter-varying systems: a design example. Automatica 1995;31(9):1251–61. [31] Scherer C, Gahinet P, Chilali M. Multi objective output–feedback control via LMI optimization. IEEE Trans Autom Control 1997;42(7):896–911. [32] Gahinet P, Nemirovski A, Laub AJ, Chilali M. LMI control toolbox for use with MATLAB. The Mathworks Inc.; 2000. [33] Tanaka K, Goya T, Senjyu T, Yona A, Urasaki N, Funabashi T, et al. Balancing control method by dispersed generators based on H1 control theory in DC power feeding system. Renew Energy 2011;36(1):163–8. [34] Electricity Storage Association.
Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
A robust H1 controller based frequency control approach using the wind-battery coordination strategy in a small power system Abdul Motin Howlader a,⇑, Yuya Izumi a, Akie Uehara a, Naomitsu Urasaki a, Tomonobu Senjyu a, Ahmed Yousuf Saber b,1 a b
Faculty of Engineering, University of the Ryukyus, 1 Senbaru, Nishihara-cho, Nakagami, Okinawa 903-0213, Japan Principal Power Engineer, Operation Technology, Inc., ETAP, Irvine, CA, USA
a r t i c l e
i n f o
Article history: Received 3 August 2012 Received in revised form 4 January 2014 Accepted 18 January 2014
Keywords: Battery energy storage system Frequency control H1 control Pitch angle control Wind turbine generator
a b s t r a c t In this paper, a coordinated control method for a wind turbine generator (WTG) and a battery energy storage system (BESS) of a small power system have been presented. The coordinated control approach applies to control the system frequency to reduce the size of BESS and to control the pitch angle system to mitigate the wind turbine blades stress. To achieve these objectives simultaneously, the robust H1 control method is applied in this paper. The pitch angle system of the WTG and the output power command system of the BESS are controlled by the H1 controllers. The output power command of the WTG is determined from the wind velocities and it is controlled by the pitch angle control system. Concurrently, the output power command of the BESS is calculated according to the state of charge and the frequency deviations in the small power system. The small power system includes a WTG, a BESS and a diesel generator. Numerical simulations are conducted by the MATLAB/SIMULINKÒ environment to validate the proposed method. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction There are many isolated communities in the world. Diesel generators are the main source of electric power for the isolated communities. In recent years, limited fossil fuels (e.g. coal, oil, gas) and many conflicting objectives such as environmental impact, reliability, imported fuels are serious issues [1]. The diesel-generated electricity is more expensive than large electricity production plants (e.g. gas, hydro, wind), and incurs the transport, fuel and storage costs [2]. Diesel generators are inherently inefficient when operated at a low load factor (below 40–50% of their rated capacity) and diesel engines burn fossil fuels which release harmful emissions into the atmosphere [3,4]. To overcome these problems, renewable energies (e.g. wind, solar, hydro, tide, biomass) have been drawing attention as the clean electricity sources. Wind energy is one of the most rapid growing renewable power sources in the world, and wind power penetration has been increasing to the power grid [5–8]. On the other hand, the wind turbine generator (WTG) can deliver a cheap electricity power as compared with other renewable sources [9]. Therefore, wind–diesel hybrid power ⇑ Corresponding author. Tel.: +81 98 895 8686; fax: +81 98 895 8708. E-mail addresses: [email protected] (A.M. Howlader), aysaber@ ieee.org (A.Y. Saber). 1 Tel.: +1 949 716 5839. http://dx.doi.org/10.1016/j.ijepes.2014.01.024 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.
systems can install to the isolated communities in order to improve their life standard. This hybrid power system also can be a part of a micro-grid system. With the hybrid power system, the annual diesel fuel can be reduced and, at the same time, the level of emissions can be minimized [10]. However, wind velocity is a highly stochastic element which can deviate quickly. The output power is delivered by the WTG which is proportional to the cube of wind speed. Hence, any deviations in wind speed will cause the wind power fluctuations. The power fluctuations may lead the frequency fluctuation and voltage flicker inside the power system [11]. Therefore, various power smoothing methods are applied to overcome these problems. The pitch angle control is utilized for the output power control of the WTG. There are various pitch angle methods to control the output power of WTGs [11–15]. By using these control methods, the output power fluctuations of WTGs can be reduced at wind speed variations. But these methods increase the wind turbine blade stress extensively due to the rapid pitch actions. To reduce the blade stress, the control regions of the pitch angle and the output power of the WTG should be considered in a frequency domain. The pitch angle control system in a high frequency domain increases the mechanical stress to the wind turbine blades. Moreover, wind energy is an uncertain fluctuating resource and requires tight control management. The conventional proportional–integral (PI) controller is not ideal due to the high turbulence wind velocities and
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parameter uncertainties. Therefore, the researchers have promoted to interest in the robust control concepts (e.g. H2 or H1 controllers). In [15], the H1 control method is applied to generate a smooth output power. However, the control region of the pitch angle in a frequency domain is not clarified. The H1 controllers based load mitigation methods for the WTG systems are proposed in [16,17], and there are nothing to mention about the frequency control strategies. Nowadays, the energy storage system (ESS) is integrated with the renewable sources which are connected into the power grid to maintain the safe operation of the power grid, and balance the supply and demand sides [18–20]. Moreover, the integrated ESS and WTG system can control the frequency in small power systems. There are different types of ESSs in the power systems such as batteries, superconducting magnetic energy storage (SMES), electric double layer capacitor (EDLC), and flywheels [18–25]. Since the SMES has to maintain the conducting state, the operation cost is very high. The installation cost of the EDLC is very high and the flywheel creates a high noise at the large capacity range of operation. The battery energy storage system (BESS) is one of the most rapid growing storage technologies. The BESS installation cost and generating noise are relatively lower than the other storage technologies. However, the installation cost of the large BESS is also very high. If a large battery storage installs in a small power system, the system cost will be increased significantly. Therefore, to reduce the BESS capacity, the power system can provide an enormous economic benefit. A short-term operation can reduce the size of a BESS. For an integrated battery storage and wind turbine systems, the BESS should operate in a short term basis to decrease the size of it. For that reason, this paper presents a coordinated control method for wind turbine and battery storage in a small power system. The coordinated control method applies to control the system frequency which can reduce the size of a BESS, and the pitch angle control system which can mitigate the wind turbine blade stress. The proposed coordinated control method is implemented through the robust H1 controller. The H1 controllers are applied to the pitch angle control system of the WTG and the output power command system of the BESS. By using the H1 controllers, the desired frequency characteristics of the control systems can be achieved by the proper weighting functions in a frequency domain. The output power command of the WTG is depended on wind velocities and is controlled by the pitch angle control system. In addition, the wind speed variations are assumed as an uncertain parameter to design the H1 controller for the pitch angle control system. The H1 controller can reduce the low frequency components for the output power errors of the WTG. Therefore, the pitch action decreases which can mitigate the wind turbine blade stress. Concurrently, the output power command of the BESS is determined according to the state of charge and the frequency deviations in the small power system. By using the output power command system, the H1 controller can limit the BESS operation within the high frequency domain. Thus, the charge/discharge operations of the BESS are not required to perform at the long term. This phenomenon can reduce the size of a BESS in a small power system. The proposed method is compared with the conventional PI control method which can reduce the frequency deviation, size of BESS and mechanical stress of WTG. Effectiveness of the proposed method is verified by the numerical simulations in MATLABÒ/SIMULINKÒ environment.
a diesel generator and a load. It is assumed that the small power system can operate independently. The detail model of the small power system model is shown in Fig. 2 [26]. In Figs. 1 and 2, the symbols P d ; P c ; P g , and P b are the output power of the diesel generator, the combined power of the WTG and the BESS, the output power of the WTG and the output power of the BESS, respectively. The symbols P L and P e are the load power and the supply error, respectively. The ‘’ represents the commanded values for different powers. The WTG system is connected to the AC side through a transformer (AC link method). Since the main purpose of this paper is frequency control, the frequency analytical model of the frequency control system is expressed as a simple transfer function for simplified simulations. The power system model does not include the characteristics such as substation facilities, electrical instruments, communication delay. The flat frequency control (FFC) approach is applied to control the frequency of the diesel generator. The FFC method is commonly used in a stand-alone power system. This method includes an integral control loop which can reduce the frequency deviation, Df . The system capacity is 687.5 kW. Pd
Pd + Pc Load
DG Pc
Diesel generator
Pb Pg
Inverter
IG1
WTG
Battery
Fig. 1. Proposed small power system model.
Fig. 2. Small power system model.
2. Small power system model An integrated power system diagram is shown in Fig. 1. The small power system consists of a wind turbine, a battery storage,
Fig. 3. Wind turbine generator system.
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2.1. Wind turbine generator system The wind turbine generator system is shown in Fig. 3. The output power of the WTG, P w , is determined by the following equation [27,28]:
Pw ¼
1 C p ðk; bÞqpR2 V 3w 2
ð1Þ
where V w is the wind speed, q is the air density, R is the rotor radius, and C p is the aerodynamic power coefficient. C p is approximated by the following equations [12,15]:
C p ðk; bÞ ¼ c1 ðbÞk2 þ c2 ðbÞk3 þ c3 ðbÞk4
ð2Þ
9 c1 ðbÞ ¼ c10 þ c11 b þ c12 b2 þ c13 b3 þ c14 b4 > = c2 ðbÞ ¼ c20 þ c21 b þ c22 b2 þ c23 b3 þ c24 b4 2
3
c3 ðbÞ ¼ c30 þ c31 b þ c32 b þ c33 b þ c34 b
4
2.2. Linearization of wind turbine generator system To apply the H1 control theory to the pitch angle control system, the WTG system should be linearized [15]. When the wind speed, V w , is given, the wind turbine torque, T w , and the generator torque, T g , are denoted by the following equations:
T w ðxw ; V w ; bÞ ¼ ð3Þ
> ;
Tg ¼
where c10 c34 are the constants represented by the performance characteristic of the wind turbine, b is the pitch angle, k is the tip speed ratio. The fixed speed squirrel-cage induction generator, that has the advantages of low cost and robustness, is used in this paper. The generator output power, P g , can be expressed by
Pg ¼
a low-frequency component of the wind speed, V w low , is determined through a low-pass filter which is used for calculating the output power command of the WTG, Pg . The time constant of the low-pass filter is 10 s. By using the large time constant, the output power command of the WTG can be smoothed.
3V 2 sð1 þ sÞR2
ð4Þ
ðR2 sR1 Þ2 þ s2 ðX 1 þ X 2 Þ2
where V the phase voltage, the slip of induction generator is as s ¼ ðxo xÞ=xo which comprises the synchronous angular speed of the rotor, xo and the angular speed of rotor, x; R1 and R2 are the generator’s stator and rotor resistances, X 1 and X 2 are the generator’s stator and rotor reactances. The H1 controller based pitch angle control system will be explained in Section 3. In Fig. 3, by subtracting the output power command of the WTG, Pg , from the output power of the WTG, P g , determines the output power error of the WTG, DP g , that evaluates the pitch angle command, bCMD , via the pitch angle control system. Pg is controlled by a hydraulic servo system that drives the blades according to the pitch angle command bCMD . The hydraulic servo system is modeled as a first-order lag system, which has the time constant, T c , of 1s [27,28]. The pitch angle command, bCMD , is limited by a limiter within 10–90°. In the conventional methods of WTG, output power command is constant at the rated power (275 kW). The pitch angle, b, operates in the region between the rated wind speed and the cut-out wind speed, and b is constant at 10 in the region between the cut-in wind speed and the rated wind speed. Therefore, in the region between the cut-in wind speed and the rated wind speed, the torque controller maximizes the power depending on the wind speed. To achieve the output power leveling control of the WTG, the pitch angle control law is extended for all operating regions in this paper [12,15]. The proposed output power command system of the WTG is shown in Fig. 4. The output power command of the wind turbine is approximated by [12]
Pw ¼ d1 þ d2 V 2w
ð5Þ
In (5), d1 and d2 are expressed as a function of the pitch angle, b, and are set at d1 ¼ 46:8623 and d2 ¼ 1:95. In practical systems, since the accurate measuring and forecasting of wind speed is difficult,
1 C p ðk; bÞV 3w qA 2xw
3V 2 sð1 þ sÞR2
xe fðR2 sR1 Þ2 þ s2 ðX 1 þ X 2 Þ2 g
ð6Þ
ð7Þ
here, the wind turbine torque is a high nonlinearity component. For simplification, (6) is substituted by the following equation:
T w ¼ f ðxw ; V w ; bÞ
ð8Þ
By indicating each operating point, xwo ; V wo ; bo , and using the Taylor expansion which ignores higher order terms (over second order), this expression as follows
T w T wo ¼
@f @f @f ðxw xwo Þ þ ðV w V wo Þ þ ðb bo Þ @ xw @V w @b
Rewriting Aslope
ð9Þ
@f @f ¼ @@f xw , Bslope ¼ @V w , C slope ¼ @b, in (8), then
DT w ¼ Aslope Dxw þ Bslope DV w þ C slope Db
ð10Þ
where D indicates a small-signal value around the operating point. Aslope ; Bslope ; C slope can be determined by [15]:
qAV 2w Vw @C p C p þR 2xw xw @k qAV w Vw @C p R 3C p Bslope ¼ 2 xw @k
Aslope ¼
C slope ¼ where
1 V w3 @C qA p 2 xw @b
@C p @k
;
@C p @b
ð11Þ ð12Þ ð13Þ
are expressed as:
@C p ¼ 2c1 k þ 3c2 k2 þ 4c3 k3 @k @C p ¼ c01 k2 þ c02 k3 þ c03 k4 @b c01 ¼ c11 þ 2c12 b þ 3c13 b2 þ 4c14 b3 c02 ¼ c21 þ 2c22 b þ 3c23 b2 þ 4c24 b3 c03 ¼ c31 þ 2c32 b þ 3c33 b2 þ 4c34 b3 The value of Aslope ; Bslope ; C slope are determined by the calculation of the operating point. In second step, linearization of the generator torque is described. The generator torque is also a nonlinear component. The linear expression of the generator torque is as follows:
T g ¼ K g xw
ð14Þ
where K g is a constant value which is around the operating point. The small-signal generator torque, DT g , can be expressed as
Vw
1 10s+1
Vw_low
2
d1+d2Vw_low
LPF Fig. 4. Output power command system of WTG.
Pg*
DT g ¼ K g Dxw
ð15Þ
where Dxw is the difference between xw and xwo . In addition, the small-signal generated power, DPg , can be expressed as
DPg ¼ K e Dxw
ð16Þ
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where K e is a constant value which is around the operating point. In third step, linearization of the rotational speed of wind turbine is described. The rotational speed of the wind turbine can be expressed as
J
dxw ¼ Tw Tg dt
Pb*
ð17Þ
+0.2pu 1 Tbs+1 -0.2pu
where J is the inertia of the wind turbine. The small-signal model around the operating point is expressed as follows
1 _ w ¼ ð DT w DT g Þ Dx J
ð18Þ
where ‘‘’’ is the differential operator. By substituting (10) and (15) in (18), the following equation is obtained:
1 _ w ¼ f Aslope K g Dxw þ Bslope DV w þ C slope Dbg Dx J
ð19Þ
The small-signal rotational speed of the wind turbine around the operating point is derived from (19) as [15]:
Dx w ¼
1 sJ fAslope K g g
fBslope DV w þ C slope Dbg
ð20Þ
where Db can be expressed using the pitch angle command, DbCMD , as
Db ¼
1 Db T c s þ 1 CMD
ð21Þ
Substituting (21) into (20) yields the following expression:
Dx w ¼
C slope Bslope DV w þ DbCMD sJ fAslope K g g Tcs þ 1 1
ð22Þ
From (22), G1 and G2 are derived as
C slope ðT c s þ 1ÞfsJ ðAslope K g Þg Bslope G2 ¼ sJ ðAslope K g Þ
G1 ¼
ð23Þ ð24Þ
These transfer function models of the wind speed, V w , and the generator output power, Pg , are shown in Fig. 5. 2.3. Battery energy storage system model The model of the BESS is shown in Fig. 6. The output power command of the BESS, P b , is limited by the capacity. Although it is assumed that a bidirectional power converter is controlled by the pulse width modulation, the BESS is modeled as a first-order lag system and the time constant, T b , is 0.3 s. The state of charge, n, is calculated by integrating the output power of the BESS, Pb . In this paper, the state of charge is indicated as a dischargeable energy in percentage of the battery rated capacity. The state of charge should always be maintained in the proper range for a stable operation [29]. Therefore, the state of charge is maintained near 50% by the charge/discharge controller in this paper. The rated capacity of the battery is 200 kW h and the rated capacity of the power converter is 137 kW (0.2 pu).
ΔVw
Δ βCMD
Pb 1 Wb Capacity ξ calculation s
Fig. 6. BESS model.
The output power command of the BESS, Pb , is determined by two control inputs; the frequency deviation, Df , and the state of charge error of the BESS, Dn. The output power command system of the BESS will be explained in the next section. 3. Proposed controller design approach 3.1. H1 control theory This section presents the design principles of the H1 controllers. The detail design of the H1 controllers is shown in [30–33]. Since this paper deals with the design of the pitch angle control system and the output power command system of the BESS, the H1 control theory is briefly explained. A closed loop system which includes weighting functions is shown in Fig. 7. In this figure, P is an intended open-loop plant, K is a H1 controller, and W S and W T are weighting functions, respectively. The weighing functions determine the performance of the controller, K. The transfer function be~ is given as tween the command value, w, and the output, zT ¼ ½~e; y follows.
~e W S SðsÞ=c ¼ w ~ W T TðsÞ y
ð25Þ
where c is the boundary of H1 norm, it is a small positive value which is indicated the control efficiency. In addition, SðsÞ ¼ ð1 þ PKÞ1 and TðsÞ ¼ PKSðsÞ are the sensitivity function and the complementary sensitivity function, respectively. Generally, it is possible to decrease the control error for the command value by selecting SðsÞ and it is possible to design a robust controller for disturbance by selecting TðsÞ. The design index as mixed sensitivity problem can be given as follows.
W S SðsÞ W TðsÞ < c T 1
ð26Þ
In this paper, linear matrix inequality (LMI) approach [31,32] is applied to design the H1 controllers. 3.2. Design process and configurations of H1 controllers The H1 controllers based pitch angle control system and output power command system of the BESS are described in this section. These control systems operate as decentralized control systems because the H1 controllers are designed individually. At first, the H1 controller of the pitch angle control system is designed because the rated capacity of the WTG is larger than that of the BESS. Subsequently, the H1 controller of the output power
WS
G2
G1
Δωw
w Ke
Fig. 5. Linearized model of WTG.
e
K(s)
u
P(s)
y
e z
WT
Δ Pg Fig. 7. Closed loop system including weight function.
y
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Singular Values (dB)
command system of the BESS is designed. The H1 controllers based proposed control system are shown in Fig. 8. To design the H1 controller based output power command system of the BESS, the entire power system is shown in Fig. 1 which is considered as the nominal plant. Therefore, the H1 controller design procedures consider the characteristics and performance of the WTG. The H1 controller based proposed pitch angle control system is shown in Fig. 8(a). From this figure, the input of the H1 controller is the output power error of the WTG, DP g . The output of the H1 controller is the pitch angle command, bCMD , to maintain the error DPg is about zero. Fig. 8(b) shows the H1 controller based output power command system for the BESS. From this figure, the inputs of the H1 controller are the frequency deviation, Df , and the state of charge error, Dn. The output of the H1 controller is the output power command of the BESS, Pb , to keep these errors (Df ; Dn) around zero. The singular value plots of the control loops with/without H1 controllers are illustrated in Fig. 9. The open-loop transfer functions, GðsÞDPg !bCMD ; GðsÞDf !Df , and GðsÞDn !Dn , are expressed by the following equations:
100
Without H-infinity controller With H-infinity controller
50 0 -50 -100 10-3
10-2
10-1
235:3 s2 þ 1:446s þ 0:4465 33:33s3 þ 340s2 þ 66:67s GðsÞDf !Df ¼ 5 s þ 14:73s4 þ 52:24s3 þ 72:09s2 þ 100:1s þ 60 0:3183 GðsÞDn !Dn ¼ 2 s þ 3:333s
Singular Values (dB)
Without H-infinity controller With H-infinity controller
50 0 -50 -100 10-4
10-2
102
104
Singular Values (dB)
ð28Þ ð29Þ
100
Without H-infinity controller With H-infinity controller
50 0 -50 -100 10-4
10-2
102
104
Fig. 9. Singular value plots (DP g ! bCMD ; Df ! Df ; Dn ! Dn).
in Fig. 10(a) and (b). To design the H1 controllers, one can choose the weighting functions as a low pass filter or a high pass filter with an arbitrary order. In this paper, the weighting functions are designed by using LMI Control Toolbox of MATLAB/SIMULINKÒ [32]. From Fig. 10(a) and (b), W S1 and W T1 are related to the pitch
Plant
8
P(s)
Pg
WT1
WS12
Plant
−Δ f 8
H
100
Frequency (rad/sec)
βCMD
H
Δξ
100
ð27Þ
WS11
ξ *= 50%
103
100
WS1
Δ f *= 0
102
Frequency (rad/sec)
The singular value plots of the pitch angle control loop (DPg to bCMD ) are shown in Fig. 9(a). From this figure and GðsÞDPg !bCMD , it is seen that the numerator of the transfer function is large and there is a high-gain at the low frequency domain (below 1 rad/s). Due to the high gain, the pitch action for the output power fluctuations of the WTG may increase at the low frequency domain. In the frequency control loop, it is seen that the singular value has a resonance point as shown in Fig. 9(b). The resonance frequency can be calculated by GðsÞDf !Df and the value is 1.4007 rad/s. The resonance point might affect the frequency control in the small power system. On the other hand, the state of charge control loop is shown in Fig. 9(c). From this figure and GðsÞDn !Dn , the singular value has an integral characteristic at the low frequency domain (below 1 rad/s). Due to the integral characteristic, the state of charge keeps to increase so that the characteristic can be eliminated. According to achieve these design goals, the weighting functions, W S1 , W T1 ; W S11 ; W S12 ; W T11 , and W T12 are selected as shown
Pg*
101
Frequency (rad/sec)
GðsÞDPg !bCMD ¼
ΔPg
100
Pb*
P(s)
Fig. 8. Proposed control systems.
Δf
WT11 ξ
WT12
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A.M. Howlader et al. / Electrical Power and Energy Systems 58 (2014) 190–198
method, the PI controllers are applied instead of the H1 controllers. The parameters of the PI controllers are listed in Table 2. These parameters are selected by the trial and error to obtain a good control performance.
Singular Values (dB)
150
WS1
100 50
WS12
0
4.1. Evaluation of the pitch action in a short-term simulation
WS11
-50 -100 10-4
10-2
100
The simulation results are shown in Fig. 11. Fig. 11(a) and (b) shows the wind speed and the load power, which are modeled by the random functions. The pitch angles with the different time ranges are shown in Fig. 11(c) and (d), respectively. From these figures, it is confirmed that the pitch action with the proposed method is moderated as compared with the conventional method. For the more quantitative measurement of the pitch action, a performance index, bact , is derived as the following equation:
102
Frequency (rad/sec)
Singular Values (dB)
0 -20
WT1
-40 -60
WT11 WT12
bact ¼
-80 -100 10-4
10-2
100
102
Frequency (rad/sec)
Fig. 10. Singular value plot of weighting functions.
angle control loop. W S1 is selected as a second-order low pass filter, which contains a high-gain at the region below the 1 rad/s. W S11 and W T11 are related to the frequency control loop, while W S12 and W T12 are related to the state of charge control loop, respectively. W S11 is selected as a second-order low pass filter, which has the cutoff frequency at 0.01 rad/s. As a results, the band width of the frequency control loop is improved within the high frequency domain (about 0.01–100 rad/s). Similarly, W S12 is selected as a second-order low pass filter, which has the cutoff frequency at 1 rad/s. This weighting function eliminates the integral characteristic at the region below the 0.01 rad/s. Therefore, the of charge/discharge operations of the BESS is performed within a short-term basis which can reduce the size of the BESS. The weighting functions of the complementary sensitivity function, W T1 ; W T11 and W T12 , decide the robustness of the H1 controllers against the disturbances. Hence, the weighting functions are set to a low-gain as illustrated in Fig. 10(b). The weighting functions of the proposed system are expressed as:
W S1 ¼
0:05023s þ 0:5467
s2 þ 0:002s þ 106 0:00571 W S11 ¼ 2 s þ 0:04398s þ 0:001265 2:22 1016 s þ 49:6 W S12 ¼ 2 s þ 0:959s þ 0:9681 1:635s2 þ 13:79s þ 11:39 W T1 ¼ s2 þ 3:246 104 þ 1:671 106 0:001134s2 þ 0:001277s þ 0:0001125 W T11 ¼ s2 þ 7:168s þ 12:75 2 0:00845s þ 0:05187s þ 0:143 W T12 ¼ s2 þ 468:1s þ 1691
ð30Þ ð31Þ ð32Þ ð33Þ ð34Þ ð35Þ
4. Simulation results The simulation parameters of the small power system, WTG system, and the operating points for linearization are listed in Table 1. The proposed H1 controller methods are compared with the conventional PI controller methods. In the conventional
Z t dbðtÞ dt dt 0
ð36Þ
If bact is large, it is means that the pitch action of the wind turbine also is large, and v ice v ersa. bact is shown in Fig. 11(e). It is evident that the mechanical stress of the wind turbine blades can be reduced by the proposed method. The output powers of the WTG system for the conventional and proposed methods are shown in Fig. 11(f) and (g). Reduction of the mechanical stress of the wind turbine blades and the output power command of the WTG are related to the performance trade-off. Therefore, the output power fluctuation of the WTG with the proposed method is larger than that of the conventional method. However, the charge/discharge of the BESS is shown in Fig. 11(h). The charge/discharge in the long-term is reduced by the proposed H1 controller. In addition, the variation of the state of charge with the proposed method is smaller as compared with the conventional method, as shown in Fig. 11(i). Fig. 11(j)–(l) shows the output power of the diesel generator, the supply error, and the frequency deviation, respectively. As can be seen from these figures, the output power of the diesel generator is almost constant with both methods. Also, the frequency deviation is nearer to zero. The frequency control method to the small power system, the control performance of the proposed H1 controller is same as the conventional PI controller. However, from the simulation results, it is
Table 1 Simulation parameters. Parameters of small power system Inertia constant, M Damping constant, D Governor time constant, T g Diesel generator time constant, T d Parameters of WTG system Blade radius, R Inertia coefficient, J Air density, q Rated output, P g Phase voltage, V Stator resistance, R1 Stator reactance, X 1 Rotor resistance, R2 Rotor reactance, X 2
0.15 puMW s/Hz 0.008 puMW/Hz 0.1 s 1.0 s 14 m 62,993 kg m2 1.225 kg/m3 275 kW pffiffiffi 400 3 V 0.00397 X 0.0376 X 0.00443 X 0.0534 X
Operating point Wind speed, V wo Pitch angle, bo Rotational speed, xwo Aslope
12.5 m/s 12° 4.5326 rad/s
Bslope
9:6695 103
C slope
1:8862 103 24,337 7857
Kg Ke
3:7882 103
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4.2. Evaluation of BESS operation in a long-term simulation
Pitch angle control system Frequency deviation State of charge
P gain K p
I gain K I
2 2 0.1
0.1 4 0.01
Wind speed Vw [m/s]
evident that the proposed method can reduce the wind turbine blade stress as compared with the conventional method. The proposed method also can reduce the size of the BESS.
18 16 14 12 10 8 100
200
300
400
500
600
700
In the long-term simulation, a cost analysis approach is given through the BESS operation. The simulation time is extended to 30 min. The simulation results are shown in Fig. 12. The wind speed and the load are shown in Fig. 12(a) and (b), which are modeled by the random functions. The pitch angle and the output power of the WTG for the both methods are shown in Fig. 12(c)– (e), respectively. From Fig. 12(c), it is confirmed that the pitch action is mitigated by the proposed H1 controller method. However, as mentioned in previous section, the power fluctuation of Output power of WTG Pg [pu]
Table 2 PI controller parameters.
0.8 Output power command Pg*
0.6
Output power Pg
0.4 0.2 0 100
200
300
Time t [s]
0.7 0.6 200
300
400
500
600
700
Output power of BESS Pb [pu]
Load PL [pu]
0.8
0.5 100
15
0 -0.1 -0.2 100
Conventional method Proposed method
200
300
400
500
600
700
Conventional method Proposed method
200
250
300
Output power of diesel generator Pd [pu]
Pitch angle β [deg]
15
50 49 48 100
State of charge command ξ∗ Conventional method Proposed method
200
300
500
600
700
Conventional method Proposed method
0.6 0.4 0.2 0 100
200
300
400
500
600
700
0.10
Supply error Pe [pu]
Performance index βact
400
0.8
60 40 Conventional method Proposed method
20 200
300
400
500
600
0.05 0 -0.05 -0.10 100
700
Conventional method Proposed method
200
300
Time t [s]
0.8 Output power command Pg
0.6
*
Output power Pg
0.4 0.2 200
300
400
400
500
600
700
Time t [s]
500
600
700
Frequency deviation Δ f [Hz]
Output power of WTG Pg [pu]
700
Time t [s]
80
0 100
600
51
Time t [s]
0 100
500
Time t [s]
20
150
400
52
Time t [s]
10 100
700
0.1
State of charge ξ [%]
Pitch angle β [deg]
Conventional method Proposed method
300
600
Time t [s]
20
200
500
0.2
Time t [s]
10 100
400
Time t [s]
0.4 0.2 0 Conventional method Proposed method
-0.2 -0.4 100
200
Time t [s]
Fig. 11. Simulation results in a short-term.
300
400
Time t [s]
500
600
700
18 16 14 12 10 8
200
400
600
800 1000 1200 1400 1600 1800
Output power of BESS Pb [pu]
Wind speed Vw [m/s]
A.M. Howlader et al. / Electrical Power and Energy Systems 58 (2014) 190–198
0.3 0.2 0.1 0 -0.1 -0.2 -0.3
Conventional method Proposed method
200
400 600
Time t [s]
State of charge ξ [%]
Load PL [pu]
0.9 0.8 0.7 200
400
600
800 1000 1200 1400 1600 1800
70 60 50 State of charge command ξ∗ Conventional method Proposed method
40 30
200
400
600
Time t [s]
200
400
600
800 1000 1200 1400 1600 1800
Output power of diesel generator Pd [pu]
Pitch angle β [Hz]
Conventional method Proposed method
15 10
0.8 Conventional method Proposed method
0.6 0.4 0.2 0 200
400
600
0.10 *
Supply error Pe [pu]
Output power of WTG Pg [pu]
0.8 Output power command Pg Output power Pg
0.6 0.4 0.2 200
400
600
800 1000 1200 1400 1600 1800
0.05
Conventional method Proposed method
0 -0.05 -0.10
200 400
600 800 1000 1200 1400 1600 1800
Time t [s]
0.8 *
Output power command Pg Output power Pg
0.6 0.4 0.2 200
400
600
800 1000 1200 1400 1600 1800
Frequency deviation Δ f [Hz]
Output power of WTG Pg [pu]
Time t [s]
0
800 1000 1200 1400 1600 1800
Time t [s]
Time t [s]
0
800 1000 1200 1400 1600 1800
Time t [s]
25 20
800 1000 1200 1400 1600 1800
Time t [s]
1.0
0.6
197
0.4 0.2 0 -0.2 -0.4
Conventional method Proposed method
200 400
Time t [s]
600 800 1000 1200 1400 1600 1800
Time t [s]
Fig. 12. Simulation results in a long term.
the WTG with the proposed method is larger than the conventional method (in Fig. 12(d) and (e)). The output power of the BESS, state of charge, output power of the diesel generator, supply error, and frequency deviation with the both methods are shown in Fig. 12(f)–(j), respectively. In this paper, the kW and kW h capacities of the BESS are set to 200 kW h and 137 kW (0.2 pu), respectively. From Fig. 12(f), it is confirmed that the BESS with the conventional method cannot discharge due to the capacity limit at around t ¼ 1200 s. As a result, the large supply error and frequency deviation can occur which is shown in Fig. 12(i) and (j). If a NaS battery uses for the BESS, the cost per unit energy is about 700 $/kW h and the cost per unit power is about 1500 $/kW, respectively [34]. When the proposed method is employed, it is evident that the required kW capacity is less than 137 kW and the total cost is about 205,500 $. However, if the conventional method is used, more large kW capacity will be required. Then the cost of the BESS is about 412,500 $. In addition, from
Fig. 12(g), the state of charge of the conventional method fluctuates nearer to 40%. The state of charge should be kept within proper limits (i.e. between 30% and 70%) [35], more large kW h capacity is required for the long period of operation. On the other hand, the state of charge with the proposed method fluctuates within ±5% deviations. Therefore, the value of kW h capacity is used in this simulation which is suitable for the frequency control. The output power of the diesel generator is little bit more fluctuated than the conventional method (in Fig. 12(h)). Although this section focuses on the operation performance of the BESS, but an optimum capacity of the BESS considering the diesel generator operation is investigated for a long period of time. 5. Conclusion This paper presents a coordinated control method for an integrated BESS and WTG in a small power system. The coordinated
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control focuses on the frequency control in a small power system. It can reduce the charge/discharge of the BESS in a long period of operation, and can mitigate the wind turbine blades stress extensively. The proposed method is implemented through the robust H1 controllers. The H1 control method is applied to the pitch angle control system of the WTG, and the output power command system of the BESS. The proposed method is compared with the conventional PI control method. From the simulations results, the proposed method can mitigate the wind turbine blades stress and can decrease the frequency deviation. It can also decrease the size of a BESS in a small power system which can reduce the system cost significantly.
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