energies Article
A Digital Hysteresis Current Control for Half-Bridge Inverters with Constrained Switching Frequency Triet Nguyen-Van *
ID
, Rikiya Abe and Kenji Tanaka
Internet of Energy Laboratory, Department of Technology Management for Innovation, The University of Tokyo, Tokyo 113-8656, Japan;
[email protected] (R.A.);
[email protected] (K.T.) * Correspondence:
[email protected] Received: 7 September 2017; Accepted: 12 October 2017; Published: 14 October 2017
Abstract: This paper proposes a new robustly adaptive hysteresis current digital control algorithm for half-bridge inverters, which plays an important role in electric power, and in various applications in electronic systems. The proposed control algorithm is assumed to be implemented on a high-speed Field Programmable Gate Array (FPGA) circuit, using measured data with high sampling frequency. The hysteresis current band is computed in each switching modulation period based on both the current error and the negative half switching period during the previous modulation period, in addition to the conventionally used voltages measured at computation instants. The proposed control algorithm is derived by solving the optimization problem—where the switching frequency is always constrained at below the desired constant frequency—which is not guaranteed by the conventional method. The optimization problem also keeps the output current stable around the reference, and minimizes power loss. Simulation results show good performances of the proposed algorithm compared with the conventional one. Keywords: half-bridge inverters; digital control; hysteresis current control; switching frequency; optimization
1. Introduction Nowadays, inverters and their controllers play an important role in the enabling of a wider proliferation of renewable energy generation [1,2], the realization of new grid concepts with high efficiency [3,4], and various applications in electronic systems [5,6]. Among various current control techniques developed over last few decades, there are three major classes of control scheme: sine-triangle Pulse Width Modulation (PWM), predictive dead-beat, and hysteresis current control. While the asynchronous sine-triangle PWM is a popular technique to modulate current error, it requires a PI (Proportional-Integral) regulator and often results in unavoidable delays. The predictive dead-beat control technique tends to give good performance in terms of response and accuracy. However, this technique is highly dependent on the accuracy of the predictive model and is complicated to implement [7]. Hysteresis current control, on the other hand, is simpler to implement and does not have such drawbacks. It has a fast dynamic response, and does not require any information about the system parameters, which enhances its robustness [5,8]. For this reason, it finds applications in a large variety of switching inverters. The basic implementation of hysteresis current control is based on the switching signal that is derived by comparing the actual current and the reference current so that the current error is kept within the tolerance current band. In classical hysteresis current controllers, the hysteresis current band is normally fixed to a certain value, which makes the switching frequency vary in order to contain the current ripple within the band. This leads to unwanted heavy interference among the phases in the three-phase system [9]. In order to overcome this problem, an adaptive hysteresis current control
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technique has been developed and used in many applications [10–12]. In this technique, the hysteresis current band is not fixed, but controlled adaptively in each switching modulation period, and based on the measured output voltage values and the desired constant switching frequency [13]. Digital hysteresis current control usually requires a sufficiently high sampling frequency to control the switch devices with accurate switching time [14,15]. The ripple current varies during each switching modulation at a very fast rate, which is inversely proportional to the output inductance and proportional to the difference between the dc and output voltages. Because, during the sampling interval, the hysteresis current band could not be observed until the arrival of the next data sample, the low sampling frequency may lead to a large ripple current overshoot from the hysteresis current band. A high sampling frequency at the MHz level seems to be quite high for conventional Digital Signal Processors (DSP) and microcontrollers, which are employed in most power electronic applications at present. Additionally, such a high sampling frequency is beyond the scope of the Field Programmable Gate Array (FPGA) circuit, which has a high-speed clock and is becoming more and more popular in inverter control [16–18]. In many applications, the high switching frequency has several advantages including lower current ripple, faster transient response, and effective size reduction for the inverter circuit [19–21]. In some cases, the switching devices are operated near the highest possible switching frequency, for example when the controller is redesigned for an existing device, or in order to reduce the price of the inverters. For instance, when a motor is accelerated from a standstill to rated speed, an inverter is usually designed to operate near its highest possible switching frequency in order to achieve a low current distortion [22]. In these cases, if the switching devices are forced to change their on-off states at a frequency beyond their limit, especially under the influence of noise, short-circuit fault can occur and may lead to breakage of the switching device [23]. Therefore, it is important to properly control the hysteresis current band so that the frequency of the switching device does not exceed its highest possible switching frequency. However, the switching frequency fluctuation controlled by the conventional method is not guaranteed to never exceed this limit, especially with noise. The conventional method of computing a hysteresis current band uses only the measured voltage values at the instant when the switches start a new modulation period [13,24]. The present study proposes a new digital hysteresis current control algorithm that takes the feedback information from the preceding modulation period into account, keeping the switching frequency constant and always below the highest possible switching frequency of the switching device, even when noise is present. The controller is assumed to be using an FPGA while the sampling interval is at the MHz level. This paper is organized as follows. Section 2 presents a conventional method of using the fixed band hysteresis current. In Section 3, a conventional adaptive hysteresis current control is described and its disadvantages are explained. Then, a new robustly-adaptive hysteresis current control algorithm is proposed in Section 4. In Section 5, the simulation results are shown to demonstrate that the proposed method has better performance than the conventional method. Conclusions are given in Section 6. 2. Classical Fixed Band Hysteresis Current Control Consider a single-phase half-bridge inverter circuit, where the output is connected to a grid ac voltage v g as in Figure 1. The dc voltage is supplied by two constant and balanced dc sources, each of which has a value of Vdc . Parameters L, C, and R present the output inductance, current ripple filter, and load respectively. The inverter output current i L is controlled by the switching devices S1 and S2 to track a given reference current ire f . The ripple component of the output current i L is filtered by the current ripple filter, and the grid current io is derived without a ripple component.
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Figure Figure 1. 1. Single-phase Single-phase half-bridge half-bridge inverter inverter circuit. circuit. Figure 1. Single-phase half-bridge inverter circuit.
Figure Figure 222 shows shows the the structure structure of of the the digital digital control control system. system. The The measured measured output output voltages, voltages, and and Figure structure of the digital control system. The measured output voltages, and Figure 2 shows showsthe the structure of the digital control system. The measured output voltages, currents are sampled by analog/digital converters, and used for computing on/off pulses currents are are sampled sampled by by analog/digital analog/digital converters, converters, and and used used for for computing on/off on/off pulses of of currents and currents are sampled by analog/digital converters, and used forcomputing computing on/off pulses pulses of of S S switches and . The states of the switches and the corresponding inputs and outputs of the S S 1 2 switches SS 1and and states ofswitches the switches switches and the corresponding corresponding inputs and outputs outputs of the the 2 . The switches states of the and the and of S2 . SThe states of the and the corresponding inputs inputs and outputs of the inverter 1 1 and 2 . The inverter are shown in Table 1. inverter areinshown shown in Table Table 1. 1. are shown Table 1. inverter are in
Figure Structure Figure 2. 2. Structure Structure of of the the digital digital control control system. system. Figure 2. Structure of the digital control system. Table Table Table 1. 1. States States of of switches switches and and corresponding corresponding outputs. outputs. Table 1. States of switches and corresponding outputs. Half Period S1 SS22 Input Voltage vvdcdc Output Current iL HalfHalf Period Input Voltage Output Current Half Period S2 S2 Input vdc Output Current Period SS11 S1 Input Voltage Voltage Output Current iL iiLL dc VVdc Positive On Off Increase Positive On Off Increase Vdcdc Positive On Increase Positive On Off Off V Increase V dc Negative Off On Decrease dc V Negative Off On Decrease dc Negative Decrease Negative Off Off OnOn −VVdcdc Decrease
In In the the classical classical fixed fixed band band hysteresis hysteresis current current control, control, the the tolerance tolerance current current band band is is fixed fixed to to aaa In the classical fixed band hysteresis current control, the tolerance current band is fixed to In the classical fixed band hysteresis current control, the tolerance current band is fixed to certain iiibb [25]. certain value value [25]. While While the the measured measured output output current current iiiLL is is between between the the upper upper and and lower lower certain value [25]. While the measured output current is between the upper and lower b L a certain value ∆ib [25]. While the measured output current i L is betweenabove the upperupper and lower limits, limits, limits, no no switching switching occurs. occurs. When When the the measured measured output output current current crosses crosses above above the the upper upper limit limit of of the the limits, no switching occurs. When the measured output current crosses the of the no switching occurs. When the measured output current crosses above the upper limit of thelimit hysteresis iib ,, the hysteresis S11 is S2 is hysteresis band band iiiref the switch switch S is turned turned off, off, the the switch switch S is turned turned on on and and the the ref ibb , the hysteresis band switch is turned off, switch is turned on and band ire f + S1 is turnedS1off, the switch S2the is turned onS 22and the current startsthe to ∆ib , the ref switch current starts to decay. In contrast, when the measured current crosses below the lower limit of the current starts to decay. In contrast, when the measured current crosses below the lower limit of the decay. In contrast, when measured current crosses below the lowerbelow limit of the hysteresis band current starts Inthe contrast, when the measured current crosses the lower limit of the to decay. iiref iib ,, the SS1 is SS2 is hysteresis band switch turned on, the switch turned off and the current hysteresis band the switch is turned on, the switch is turned off and the current ref b 1 2 i − ∆i , the switch S is turned on, the switch S is turned off and the current starts to increase i i hysteresis on, the switch S 2 is turned off and the current re f bband ref b 1 , the switch S1 is turned 2 starts to increase (Figure 3). starts to increase (Figure 3). (Figure 3). starts to increase (Figure 3).
Figure Figure 3. 3. Fixed Fixed band band hysteresis hysteresis current current control. control. Figure 3. Fixed band hysteresis current control. Figure 3. Fixed band hysteresis current control.
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The hysteresis current band value is directly proportional to the current ripple and inversely of 13 switching frequency. Thus, increasing the value of the hysteresis current 4band will increase the current ripple while a decrease in the band will increase the switching losses. In analog controllers, the current ripple is always kept exactly within the hysteresis band. However, in The hysteresis current band value is directly proportional to the current ripple and inversely digital controllers, the hysteresis control is shown to be effective only if the current band is chosen proportional to the switching frequency. Thus, increasing the value of the hysteresis current band will to satisfy the following condition [5]: increase the current ripple while a decrease in the band will increase the switching losses. In analog controllers, the current ripple is always kept exactly diwithin ref 1 the hysteresis band. However, in digital i max , (1) controllers, the hysteresis control is shown bto be effective f sp if the current band is chosen to satisfy dt only the following condition [5]: where f sp is the sampling frequency. The maximum dire f 1switching frequency f sw _ max should be ∆ib > max , (1) dt f sp smaller than half the sampling frequency f sp , i.e. : Energies 2017, 10, to 1610 proportional the
where f sp is the sampling frequency. The maximum switching frequency f sw_max should be smaller 1 (2) than half the sampling frequency f sp , i.e.,: f sw _ max f sp . 2 1 f sw_max ≤ f sp . 2 3. Conventional Adaptive Hysteresis Current Control
(2)
3. Conventional Adaptive Hysteresis Current Control 3.1. Algorithm of the Convention Adaptive Hysteresis Current Control 3.1. Algorithm of mentioned the Convention Adaptive Hysteresiscontrol Currenthas Control The above fixed band hysteresis many advantages: its simplicity, its fast and stable response, and its independence from system parameters. However, a its disadvantage is that The above mentioned fixed band hysteresis control has many advantages: simplicity, its fast the switching frequency needs to vary in order to keep the peak-to-peak current ripple controlled at and stable response, and its independence from system parameters. However, a disadvantage is that all points on the fundamental frequency wave [10]. In order to solve this problem, the adaptive the switching frequency needs to vary in order to keep the peak-to-peak current ripple controlled at all hysteresis current control technique been presented in solve the literature [7, 26].the Inadaptive this technique, the points on the fundamental frequency has wave [10]. In order to this problem, hysteresis hysteresis current band is has not been fixed,presented but controlled in eachInswitching modulation period, current control technique in theadaptively literature [7,26]. this technique, the hysteresis based on the measured output voltage values and the desired constant switching frequency. current band is not fixed, but controlled adaptively in each switching modulation period, based onThe the adaptive hysteresis current control is employed as shown below. measured output voltage values and the desired constant switching frequency. The adaptive hysteresis i t as:below. Define the iscurrent errorasshown current control employed Define the current error ∆i (t) as:
i t iL t iref t ,
(3)
∆i (t) = i (t) − ire f (t), (3) where iL t , and iref t are the inverter outputL and the reference currents, respectively, at instant
t . Consider t0 , when iL tends the output current to cross the lower hysteresisatband, andt. where i L (t), the andinstant ire f (t) are the inverter output and the reference currents, respectively, instant Consider t0 , when current i L tends the lower band, the icross t0 is to the switchtheS1instant is switched on.the Theoutput current error at 4).hysteresis Assume that theand switch t0 (Figure switch S1 is switched on. The error at t0 is ∆i (t0 ) (Figure 4). Assume that the switch S1 is S1 is switched on during t0 , tcurrent 1 , and is switched off during t1 , t2 intervals. These intervals are switched on during [t0 , t1 ), and is switched off during [t1 , t2 ) intervals. These intervals are called called positive and negative half switching periods respectively. positive and negative half switching periods respectively.
Figure Conventional adaptive adaptive hysteresis Figure 4. 4. Conventional hysteresis current current control. control.
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The output current dynamic equation can be written as: vdc (t) − v g (t) di L (t) = dt L
(4)
for t0 ≤ t ≤ t2 , where v g is the instantaneous grid voltage, L is the output inductance, and vdc (t) is the inverter dc voltage, and can be elaborated as: ( vdc (t) =
Vdc −Vdc
i f S1 is On . i f S1 is O f f .
(5) .
Define the output current slopes in the on and off periods by I on , and I o f f respectively. Assuming that the output voltage is slowly varying during the switching modulation period [t0 , t2 ], the output current slopes (4) can be expressed as: .
I on = for t ∈ [t0 , t1 ), and: .
Io f f ,
Vdc − v g (t0 ) di L (t) = dt L
(6)
−Vdc − v g (t0 ) di L (t) = dt L
(7)
for t ∈ [t1 , t2 ). The current errors in the positive and negative half switching periods are given as: ∆i (t1 )
∆i (t2 )
= i L (t1 ) − ire f (t1 ) . = ire f (t0 ) + ∆i (t0 ) + I on Ton − ire f (t1 ) ,
= i L (t2 ) − ire f (t2 )
. . = ire f (t0 ) + ∆i (t0 ) + I on Ton + I o f f To f f − ire f (t2 ).
(8)
(9)
where Ton and To f f are the on and off periods of switch S1 , given as: Ton = t1 − t0
(10)
To f f = t2 − t1
(11)
The reference current ire f (t) is slowly varying during the modulation period, such that it can be approximated as: . ire f (t1 ) = ire f (t0 ) + I re f (t0 ) Ton , (12) . ire f (t2 ) = ire f (t0 ) + I re f (t0 ) Ton + To f f , (13) where:
dire f (t) I re f (t0 ) = dt t=t0 .
(14)
By substituting Equations (12) and (13) into Equations (8) and (9), one can write the current errors ∆i (t1 ), and ∆i (t2 ) as: .0
∆i (t1 ) = ∆i (t0 ) + I on Ton ∆i (t2 ) = ∆i (t0 ) + .0
.0
.0 I on Ton
+
.0 I o f f To f f
= ∆i (t1 ) +
(15) .0 I o f f To f f
(16)
where I on and I o f f are the current error slopes in positive and negative half switching periods, given as:
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.0
.
.
I on = I on − I re f (t0 ) =
Vdc − v g (t0 ) . − I re f (t0 ), L
(17)
−Vdc − v g (t0 ) . − I re f (t0 ) (18) L Let f sw be the desired constant switching frequency. In the conventional adaptive hysteresis current control method, the hysteresis current band ∆ib (t0 ) is derived by using the following conditions: ∆i (t1 ) − ∆i (t0 ) = 2∆ib (t0 ) (19) .0
.
.
I o f f = I o f f − I re f (t0 ) =
∆i (t2 ) − ∆i (t1 ) = −2∆ib (t0 ),
(20)
Ton + To f f = Tsw ,
(21)
where Tsw = 1/ f sw is the desired constant switching period. Substituting Equations (19)–(21) into Equations (15) and (16), we can derive the hysteresis current band as: .
.
1 I on I o f f ∆ib (t0 ) = . . Tsw 2 I o f f − I on
(22)
By substituting Equations (17) and (18) into Equation (22), the hysteresis band in Equation (22) can also be written in the form of [26] : Vdc Tsw 1 − m2 ( t0 ) , 4L
(23)
. 1 v g (t) + L I re f (t) Vdc
(24)
∆ib (t0 ) = where: m ( t0 ) =
3.2. Disadvantages of Conventional Adaptive Hysteresis Current Control Consider the case where the inverter needs to be controlled so that the operating switching frequency is always strictly equal to or lower than a constant frequency, which may be the highest possible switching frequency of the switching devices. The conventional adaptive hysteresis current control described in Section 3.1 has been shown to be able to improve on the disadvantage of the fixed band hysteresis current control of the varying switching frequency [8]. However, when the inverter is operated under the effect of noise or disturbances, this conventional method does not guarantee a stable response and constant switching frequency as, for example, in the following problems. Problem 1. If the negative half switching period of switch S1 in the previous modulation period is To f f _pre , as shown in Figure 5, then: To f f _pre = t0 − t−1
(25)
While the operating switching frequency can be defined from both the time periods Ton + To f f and To f f _pre + Ton , the calculation of the conventional hysteresis current band only guarantees that modulation period Ton + To f f is equal to the desired constant switching period Tsw , as in Equation (21). If the noise or disturbances to voltage and current make the negative half switching period To f f _pre in the previous modulation shorter than the calculated value, then the operating switching frequency of the conventional method may exceed the highest possible switching frequency of the switching devices.
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T in the previous modulation shorter than the calculated value, then the operating switching Toff off __ pre pre in the previous modulation shorter than the calculated value, then the operating switching frequency frequency of of the the conventional conventional method method may may exceed exceed the the highest highest possible possible switching switching frequency frequency of of the the Energies 2017, 10, 1610 7 of 13 switching switching devices. devices.
Figure 5. Conventional Conventional adaptive hysteresis hysteresis current control control does not guarantee that the operating Figure Figure 5. 5. Conventional adaptive adaptive hysteresis current current control does does not not guarantee guarantee that that the the operating operating switching frequency will be lower than the desired constant frequency. switching frequency will be lower than the desired constant frequency. switching frequency will be lower than the desired constant frequency.
Problem 2. 2. Problem 2. In order to keep the average value of the output current equal to to the reference current in each switching modulation, the hysteresis current band should be computed so that: switching modulation, the hysteresis hysteresis current current band band should should be be computed computed so so that: that:
ii tt1 i t −iibb∆ib − ∆i ∆ii(tt222 ) = (t1 ) 1=
(26) (26) (26)
However, the hysteresis current band computed by the conventional method, as in However,the thehysteresis hysteresiscurrent currentband band computed conventional method, in Equations Equations However, computed byby thethe conventional method, as inasEquations (19) (19) and (20), in general, does not satisfy the Condition (26), except in the case of the ideal condition (19) and (20), in general, does not satisfy the Condition (26), except in the case of the ideal condition and (20), in general, does not satisfy the Condition (26), except in the case of the ideal condition shown shown by by the the dashed dashed line line in in Figure Figure 6, 6, where where the the current current error error at at instant instant tt00 is is identical identical to to the the shown by the dashed line in Figure 6, where the current error at instant t0 is identical to the computed computed hysteresis current band , i.e. : i computedcurrent hysteresis current band ibb , i.e. : hysteresis band ∆ib , i.e.,:
(27) ii tt0 iib (27) ∆i (t0 )0= −∆ib b (27) When the Condition (27) is not satisfied, the average value of the output current during a switching When the Condition (27)(27) is not satisfied, thethe average value ofofthe When the Condition is not satisfied, average value theoutput outputcurrent currentduring during aa switching switching modulation period deviates from the reference current, which may lead to an unstable response as modulation period deviates from the reference current, which may lead to an unstable response as as modulation period deviates from the reference current, which may lead to an unstable response shown in Figure 6 by the solid line. shown in in Figure Figure 66 by by the the solid solid line. line. shown
Figure 6. 6. stability of of the the Figure Conventional adaptive adaptive hysteresis hysteresis current not guarantee guarantee the the stability 6. Conventional adaptive hysteresis current control control does does not response current. response current. response current.
4. 4. Proposed Proposed Robustly Robustly Adaptive Adaptive Hysteresis Hysteresis Current Current Control Control
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4. Proposed Robustly Adaptive Hysteresis Current Control
Consider Consider the the instant t0t0,,when whenthe theoutput output current current crosses crosses to to pass pass the the lower lower limit limit of of the the hysteresis hysteresis band i + ∆i t and the switch S starts to be switched on as shown in Figure 7. The negative ( ) band reireff i t00 and the switch S11 starts to be switched on as shown in Figure 7. The negative half half switching period of switch S1 in the previous modulation period To f f _pre is: switching period of switch S1 in the previous modulation period Toff _ pre is: To fTf _pre =tt0 − t −1 . off _ pre 0 t1 .
(28) (28)
Figure Figure 7. 7. Proposed Proposed adaptive adaptive hysteresis hysteresis current current control. control.
In order order to to solve solve the the problem problem of of unstable unstable switching switching frequency frequency of of the the conventional conventional adaptive adaptive In hysteresis current current control control mentioned mentioned in in Section Section 3, 3, this this study study proposes proposes aa new new hysteresis hysteresis current current band band hysteresis computation method method by by setting setting an an optimization optimization problem problemas: as: computation
i T pre Ton Tsw , (i ) To f f −off pre + Ton ≥ Tsw , (ii ) iiTon T+on ToTf offf ≥ TTswsw, , −∆i (iii)iii∆i b= (29) ib ∆i (it1 t)1 = i t(2t2,), (29) (iv) ∆ib ≥ ∆iconv , iv i i ∆i,(t ))2 + (∆i(t ))2 , minimizeb J = (conv 2 1 2 2 ∆i minimize J i t1 ? i t2 , i where ∆iconv is the hysteresis current band computed by the conventional method as in Equation (22), where is the hysteresis iconv objective and J is the function. current band computed by the conventional method as in Equation conditions and (ii) in Equation (29) maintain the operating switching frequency to (22), Constraint and J is the objective(i)function. always be equalconditions to or smaller than desired instant switching Condition (iii) keeps Constraint (i) and (ii)the in Equation (29) maintain the frequency. operating switching frequency to the average value of smaller the output identical toswitching the reference currentCondition during the always be equal to or thancurrent the desired instant frequency. (iii)switching keeps the modulation period. Condition keeps the output to current from deviating from the reference current. average value of the output(iv) current identical the reference current during the switching The objective function J represents the power loss from the inverter in each switching modulation modulation period. Condition (iv) keeps the output current from deviating from the reference period. band is computed such that the loss inverter J is minimized. J represents current.The Thehysteresis objectivecurrent function the power losspower from the in each switching Equations (15) and (16) can be written as: modulation period. The hysteresis current band is computed such that the power loss J is minimized. 1 Ton = .as: (30) Equations (15) and (16) can be written 0 ( ∆i ( t1 ) − ∆i ( t0 )), I on 1 Ton i t1 i t0 , (30) I 1 To f f = . 0on (∆i (t2 ) − ∆i (t1 )) (31) I o1f f Toff i t2 i t1 (31) I of f Equation (29), the optimization problem (29) can Substituting Equations (30) and (31) into be written as: Substituting Equations (30) and (31) into Equation (29), the optimization problem (29) can be written as:
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(i)
i t1 I on Tsw Toff pre i t0 ,
I I (ii ) 1 on i t1 on i t2 I onTsw i t0 , I I off . off 0 Tsw T f fi− pre + ∆i (t0 ), (i ) ∆i (t)1 )i≥ t1Ion (iii ? i t− 2 o . . . I on ( iv ) i iconv (ii ) 1 − . ∆i (, t1 ) + .I on ∆i (t2 ) ≥ I on Tsw + ∆i (t0 ), Io f f I2o f f 2 i t2 (it t)1 = ?∆i, (iii )minimize ∆i (t ) =J−∆i
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2
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(32)
(32)
(iv) ∆i ≥ ∆iconv , The constraints in the optimization problem given in Equation (32) are linear. They can be minimize J = (∆i (t1 ))2 + (∆i (t2 ))2 . solved by using the graphical ∆i method for minimization problems [27], as below. The feasible region representing constraint conditions (i–iv) can be represented by the dark on the They phase-plane The constraints in the optimization problem given in Equation (32) area are linear. can be
J is represented Figure 8.method The objective function problems by theThe distance from the t2 as i t1 ,byiusing solved theingraphical for minimization [27], as below. feasible region representing constraint conditions by the solution dark area the phase-plane i t1 , can i t2be original phase-plane to the point (i–iv) Thus, the optimal foron Problem (32) is the .represented (∆i (t1 ), ∆i (t2 )) as in Figure 8. The objective function J is represented by the distance from the original , i t s on point i t1 to the region, which is closestsolution to the original point. (32) The solution can phase-plane . Thus, the optimal for Problem is the point (∆i (t1feasible ), ∆i (t2 )) s the 2point ∆iderived t2 )s ) on the feasible region, which is closest to the original point. The solution can be (be (t1 )s , ∆i (as: derived as: i max iconv , iA , iB , (33) ∆iopt opt = max(∆iconv , ∆i A , ∆i B ), (33)
where ∆i , conv arethe thepoints pointswhich whichsatisfy satisfypair pairconstraint constraintconditions conditions(i, (i, iii), iii), (ii, (ii, iii), iii), (iv, (iv, iii), iii), iA, ,∆ii,B ∆i iconvare where B A respectively,and andare aregiven givenby: by: respectively,
i . I T T i t0 , ∆i A = AI on onTswsw− Toofff f−prepre + ∆i ( t0 ),
(34) (34)
. I T i t iB I onT sw + ∆i (0t ), (35) on sw 0 (35) ∆i B = 1 2 .I on I.off , 1 − 2 I on / I o f f 1 .I on I. off iconv 1 I I T (36) on o f f sw ∆iconv = 2. I off I on (36) . Tsw 2 I o f f − I on When the optimal solution is at iA , it means that, for example under the effect of noise, the When the optimal solution is at ∆i A , it means that, for example under the effect of noise, negative half switching period Toff _ pre in the previous switching modulation was smaller than the the negative half switching period To f f _pre in the previous switching modulation was smaller than the regular value value(Problem (Problem1 in 1 in Section 3) and hysteresis current to be broadened to regular Section 3) and the the hysteresis current bandband needsneeds to be broadened to satisfy satisfy condition (i) in Equation (29). When the optimal solution is at , it means that the average i B condition (i) in Equation (29). When the optimal solution is at ∆i B , it means that the average output output current deviated from the reference current (Problem 2 in Section 3) and the hysteresis current deviated from the reference current (Problem 2 in Section 3) and the hysteresis current band current needs totobesatisfy broadened to satisfy condition (ii)When in Equation (29). solution When the needs to band be broadened condition (ii) in Equation (29). the optimal is atoptimal ∆iconv , iconv solution is atthe , it means theare conditions and are satisfied and the should hysteresis current it means that conditions (i) that and (ii) satisfied (i) and the(ii) hysteresis current band be brought band be brought to conventional the steady state as in the conventional method. to the should steady state as in the method.
Figure 8. 8. Domain Domain and and optimal optimal solution solution for for the the hysteresis hysteresis current current band. band. Figure
Remark 1. In an ideal state, where the effect of noise is sufficiently small, the proposed method gives the same hysteresis current band as the conventional adaptive hysteresis current control method.
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Remark 1. In an ideal state, where the effect of noise is sufficiently small, the proposed method gives the same hysteresis current band as the conventional adaptive hysteresis current control method. Energies 2017, 10, 1610
5. Simulation Results
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Simulations have been carried out to assess the performance of the proposed method as 5. Simulation Results compared with the conventional method [13] using the Matlab-Simscape Power System. A single-phase half-bridge was employed with the of following parameters: output Simulations have been inverter carried out to assess the performance the proposed method asan compared C 10 F. The Power inductance of L 1 mHmethod and a filter capacitor switching transistors in the inverter with the conventional [13] using the of Matlab-Simscape System. A single-phase halfbridge was employed withInsulated the following parameters: an output inductance of L =in1 mH circuit inverter were modeled by the Gate Bipolar Transistor (IGBT) block the and a filter capacitor of C MathWorks, = 10 µF. TheNatick, switching in the inverter Matlab-Simscape(R2016a, MA,transistors USA) Power System. Thecircuit IGBTswere havemodeled internal by Insulated Gate Bipolar block in the The Matlab-Simscape(R2016a, MathWorks, andthe snubber resistances of 1 Transistor mΩ and 1(IGBT) kΩ, respectively. inverter was connected to the dc Natick, USA) System. The internal and snubber 1 kΩ, Vdc Power 175 V, voltage MA, source and the ac IGBTs voltagehave of the grid was The and sampling vg 100 resistances 2 sin 100 tof V.1 mΩ respectively. The inverter was connected to the dc voltage source V = 175 V, and the ac voltage frequency of the analog/digital converters was f sp 2 MHz. The dcreference current was set at √ of the grid was v g = 100 2 sin(100πt) V. The sampling frequency of the analog/digital converters iref f10sin 100 t A. The noise in the measured current, which may have come from the current was sp = 2 MHz. The reference current was set at ire f = 10 sin(100πt ) A. The noise in the measured sensor or the analog/digital converter, assumed noise with a variance 0.01 A. current, which may have come from the was current sensor to or be thewhite analog/digital converter, wasofassumed Figures 9–11 show the operating switching frequencies and the output currents of the proposed to be white noise with a variance of 0.01 A. Figures 9–11 show the operating switching frequencies and conventional methods compared the referencemethods current compared for the desired constant switching the output currents of the proposedwith and conventional with the reference current frequency at 40 kHz, 20 kHz,frequency and 10 kHz upper sub-figures show a part of the f sw constant for the desired switching f swrespectively. at 40 kHz, 20The kHz, and 10 kHz respectively. The upper sub-figures show a response. part of theRegarding current hysteresis response. Regarding the frequencies, desired constant current hysteresis the desired constant switching bothswitching methods frequencies, both methods yielded hysteresis that were similarHowever, to the reference current. yielded hysteresis output currents that were output similar currents to the reference current. the operating However, the operating switching frequency of the conventional method oscillated and went over the switching frequency of the conventional method oscillated and went over the desired constant desired constant switching frequency. On the other hand, the proposed yielded a switching switching frequency. On the other hand, proposed method yieldedmethod a switching frequency that frequency was more stable than that of the conventional and maintained was always maintained at was more that stable than that of the conventional method and method was always at equal to or equal or lower than the desired constantfrequency. switching frequency. All the Figures that the lower to than the desired constant switching All the Figures also show also that show the hysteresis hysteresis current band when increased when the desiredswitching constant switching was increased. current band increased the desired constant frequencyfrequency was increased. The power efficiency of the inverter has been calculated, taking into account hysteresis current loss. Table power efficiencies of the and conventional methods with the tested Table22shows showsthe the power efficiencies of proposed the proposed and conventional methods with the switching frequencies. The efficiencies of both methods decreased the switching was tested switching frequencies. The efficiencies of both methodswhen decreased when frequency the switching decreased. candecreased. be seen thatItthe performed than performed the conventional in frequency It was canproposed be seenmethod that the proposedbetter method bettermethod than the terms of power efficiency in allof cases. conventional method in terms power efficiency in all cases.
Figure 9. Current and switching period of the proposed method and conventional method for the desired switching frequency at 40 kHz.
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Figure Figure 9. 9. Current Current and and switching switching period period of of the the proposed proposed method method and and conventional conventional method method for for the the 11 of 13 desired switching frequency at 40 kHz. desired switching frequency at 40 kHz.
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Figure 10. Current switching proposed method conventional Figure 10. 10. Current Current and and switching switching period period of of the the proposed proposed method method and and conventional conventional method method for for the the Figure and period of the and method for the desired switching frequency at 20 kHz. desired switching switching frequency frequencyat at20 20kHz. kHz. desired
method Figure Figure 11. 11. Current Current and and switching switching period period of of the the proposed proposed method method and and conventional conventional method method for for the the desired switching frequency at 10 kHz. desired desired switching switching frequency frequency at at 10 10 kHz. kHz.
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Table 2. Power efficiency with various switching frequencies (%). Switching frequency
f sw = 10 kHz
f sw = 20 kHz
f sw = 40 kHz
Proposed method Conventional method
97.1 95.2
98.4 97.2
99.3 98.5
6. Conclusions A new digital hysteresis current control algorithm for single-phase half-bridge inverters has been proposed. By taking advantage of digital controls, the hysteresis current band is computed in each switching modulation period based on both the negative half switching period and the current error during the previous switching modulation period, in addition to the usual measured voltage value at the instant of computing. The hysteresis current control, with its simplicity and its fast and stable response, may be implemented on high-speed FPGA boards with a high sampling frequency. The proposed algorithm for computing the hysteresis current band is derived by solving the optimization problem where (i) the switching frequency is stable and always maintained at equal to or lower than the desired constant frequency; (ii) the output current is kept stable around the reference current; and (iii) the power loss is minimized. Although the proposed control method has not been evaluated in experiments, the theoretical numerical and simulation results show good performance of the proposed algorithm compared with those of the conventional method. The proposed method is expected to contribute to the control theory of high-speed FPGA-based hysteresis current control. Author Contributions: Triet Nguyen-Van, Rikiya Abe, Kenji Tanaka performed and discussed the research; Triet Nguyen-Van carried out the simulations, analyzed the data, and wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.
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