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Computer-aided analysis of stepped sinewave inverters
Solar Cells, 31 (1991) 559-579
559
Computer-aided analysis of stepped sinewave inverters Abdelhalim Zekry Electrical Engineering Department, College of Engineering, King Saud University, Riyadh (Saudi Arabia) (Received July 25, 1990)
Abstract A stepped sinewave full-bridge inverter was analysed theoretically to determine its optimum working conditions, its minimum number of components, the characteristic of its constituent components and control circuit of its power switches. A computer program, written in FORTRAN-77, was developed to analyse this inverter circuit. The program prompts for input data such as the characteristics of the power and control circuit elements, the output fundamental frequency, and the number of available batteries and their voltage. The output of the program is designed to give the optimum stair widths, the loss, the harmonic and the total efficiency, the harmonic distortion, the sensitivity of the efficiency and fundamental voltage to the variation in stair width, and the energy withdrawal from each battery for the power switches controlling the circuit. Interesting design data for this circuit were also obtained.
1. I n t r o d u c t i o n One o f the main units in a p h o t o v o l t a i c p o w e r s y s t e m is the d . c . - a . c . c o n v e r t e r . Different inverter circuits suitable for p h o t o v o l t a i c p o w e r s y s t e m s have b e e n d e v e l o p e d [ 1, 2]. The m o s t efficient a n d at the s a m e time the c h e a p e s t inverter is t h a t p r o p o s e d b y S c h m i t t and Schaetzle [2 ]. A s y s t e m a t i c analysis o f t h e s e inverters t o d e t e r m i n e their o p t i m u m p e r f o r m a n c e a n d their o p t i m u m d e s i g n h a s n o t yet been carried out. In this work, a general inverter circuit is i n t r o d u c e d a n d a n a l y s e d systematically. A design e x a m p l e is also given. The p h o t o v o l t a i c g e n e r a t o r s u p p l y i n g the inverter has the following features: (i) it is a divided s o u r c e , n o r m a l l y in the f o r m o f a large n u m b e r o f batteries; (ii) it is e x p e n s i v e (this applies to b o t h the p h o t o v o l t a i c a r r a y a n d the c h e m i c a l b a t t e r y storage). T h e s e t w o i m p o r t a n t f a c t o r s will b e t a k e n into c o n s i d e r a t i o n in designing the inverter.
2. A n a l y s i s
of the inverter
circuit
A g e n e r a l s t e p p e d sinewave full-bridge inverter circuit is s h o w n schematically in Fig. 1. It c o n s i s t s o f (2Ns + 4) p o w e r switches with Ns the n u m b e r o f b a t t e r y stages. E a c h b a t t e r y s t a g e m a y c o n t a i n m o r e than one b a t t e r y
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© 1991 -- Elsevier Sequoia, Lausanne
560 ! NsVs ~ SNs_i" SNs+~ T SNs-I /
\ Spl S2Ns-~
~Sn I
Si j
RL I
S2Ns-~
, |
$
2
!
~
SI_/
Sp2
Sn2
~V S
SNs+~
Fig. 1. Stepped sinewave full-bridge inverter circuit.
, V0
NsVs
VS
Ai
-i;;i ~11
-Vs
~
i
i sta i r
i b', i , ~pulse
I
l
angle O _
!:ii
-i Vs -Ns Vs Fig. 2. Output with stacked k-pulses having different pulse widths (k = 2 N ~ - 1 ) .
and has an e.m.f. Vs. With p r o p e r control signals for the power switches, this circuit can generate the periodic output voltage Vo(O) shown in Fig. 2. It is in the form of a stepped sinewave. For the positive half-wave Spl and Sp2 are on while Sn, and S,2 conduct in the negative half-cycle only. Each half is c o m p o s e d of Ns stairs stacked on top of each other. The width of the ith stair is Li where i is a running index varying from 1 to AT,. The ith stair is delayed by a phase angle Ai as shown in Fig. 2. The positive stairs are symmetrical around 0= 7r/2 and the negative stairs around O= 3~r/2. The positive half-wave can also be obtained by stacking k-pulses having different pulse widths side by side where k = ( 2 N s - 1 ) . The width of the ith pulse is X~ in radiance. The negative half-wave is identical with the positive half-wave. In generating the ith and ( 2 N s - i ) t h pulses, only the switch St must be on, t oget her with Sp, and Sp2. The group o f switches from SN.+, . . . t o S2Ns pr oduces the pulses of the negative halfcycle. The switching circuit operates as follows. In the positive half-cycle Spl and Sp2 are on and all other switches are off. The pulse X, is p r o d u c e d by putting S~ on during the duration of this pulse as shown in Fig. 2. Directly before putting $2 on to produce the pulse X2, one has to put off S,. To
561 produce the ith pulse the switch S~ is put on directly after making S~_ ~ off. The ith pulse in the falling part of the positive half-cycle is produced by again putting Si on after putting Si+~ off for a duration of X~. It should be noticed that battery 1 in the string delivers the largest current in the positive half-cycle. In the negative half-cycle S,I and S,2 are on. The first pulse i = - 1 in the falling part of the negative half-cycle is produced by making SN.+, on, the second pulse i = - 2 by switching SN~+~ on after switching SN~., off. The ith sample is produced by the switch SN~+i. In the rising part of the negative cycle the - i t h pulse is also produced by putting SN~., on while all other switches are off except Snl and Sn2. It must be mentioned that in the negative half-cycle the largest charge will be delivered by the uppermost battery stage in the string Ns, in contrast with the positive half-cycle. In this way charge equilization among all batteries results. If switching spikes or noise are present, they will be of high frequency and can easily be filtered by connecting a relatively small capacitor across the load. If any short occurs across any battery stage it will be eliminated by melting because of the capabilities of chemical batteries to supply very large surge currents. This means that the batteries will probably not be damaged. In the output waveform only two pulses will not be produced. If the number of stages is large the effect of shorting one battery stage is not serious. A switch failure, which means a switch is always off, would lead to the disappearance of two pulses in a half-cycle only, i.e. the same effect as in the previous case. In the practical circuit one has to add protection as well as monitoring elements. The inverter described here can be used directly in stand-alone photovoltaic systems. This inverter is not regulated and its output voltage varies according to the state of the battery charges. Consequently, loads accepting these voltage variations can be utilized with this inverter type. However, by synchronizing the timing control circuit with the utility voltage waveform, and providing the inverter with a means of voltage control and safety circuits, it can be coupled with the utility. The voltage control can be carried out by (i) controlling the input voltage to the inverter, (ii) controlling the voltage within the inverter or (iii) controlling the voltage delivered by the inverter. Methods (i) and (ii) could be uneconomical, especially when the output voltage varies over a wide range because one has to make full use of the batteries. For limited load voltage variation, however, they could be utilized. Method (iii) would be preferable for large variations in the input or load voltage. The basic concept used to control the output voltage within the inverter is modulation of the pulse width where the durations of both the positive and negative pulses are varied from zero to half the period of one complete cycle. Here, the height and the width of each stair can be controlled separately. To avoid the need of a high voltage photovoltaic array to charge the batteries as the input to the inverter, one can feed each battery separately through a photovoltaic battery charge controller. -
562 The circuit must satisfy the following r e q u i r e m e n t s : (1) high efficiency, (2) m i n i m u m h a r m o n i c c o n t e n t in the o u t p u t voltage, (3) equal e n e r g y withdrawal from e a c h battery, (4) simplicity of the inverter circuit, (5) low cost, (6) ease of control o f the p o w e r switching circuit, (7) reliability o f operation. R e q u i r e m e n t s ( 4 ) - ( 7 ) can be fulfilled by reducing the n u m b e r of switches and by avoiding e x p e n s i v e c o m p o n e n t s such as c h o k e s and t r a n s f o r m e r s . R e q u i r e m e n t s (1) and (2), however, can be only realized by increasing the n u m b e r o f switches. T h e r e f o r e , all r e q u i r e m e n t s m a y not be satisfied simultaneously. B e c a u s e o f the relative i m p o r t a n c e o f high efficiency in c o m p a r i s o n with all the o t h e r r e q u i r e m e n t s , this must be given priority. This will be clear if we think a b o u t the capital cost p e r watt for the solar array and the battery storage. In fact, o n e has to avoid the loss of any joule g e n e r a t e d in the solar p o w e r system, while satisfying all o t h e r r e q u i r e m e n t s by using high quality c o m p o n e n t s and a d v a n c e d t e c h n o l o g y to a s s e m b l e t h e s e devices. In this way, a reliable inverter can be obtained. In summary, the inverter cost per watt m u s t be k e p t less than that of the array and storage.
2.1. Inverter e ~ c i e n c y The efficiency ~? is defined by the ratio o f the f u n d a m e n t a l o u t p u t p o w e r P0: to the p o w e r input to the inverter Pi P0~ . = -Pi
(1)
Multiplying the right-hand side by (Po/Po) where Po is the total o u t p u t power, ~? can be rewritten in the form
(e) with 77L:
T~h
Po
(2a)
Pi
Po~
(2b)
Po
w h e r e ~h is the h a r m o n i c efficiency and ~?L is the efficiency due to p o w e r loss in the inverter circuit. It will be t e r m e d simply the loss efficiency. The allowable h a r m o n i c c o n t e n t in the o u t p u t voltage is d e t e r m i n e d b y the n a t u r e o f the load. T h e r e f o r e , we split the efficiency into two efficiencies ~?L and ~Th t o reflect this.
2.1.1. The harmonic e ~ c i e n v y In this section we shall s t u d y the h a r m o n i c efficiency, the h a r m o n i c c o n t e n t in the o u t p u t voltage and the f u n d a m e n t a l voltage, assuming resistive load.
563
Denoting the effective value of the output voltage by Vrms and the magnitude of its fundamental c o m p o n e n t by V,m, the harmonic efficiency can be expressed by V2m
~?h-- 2V2r~
(3)
For the stepped sinewave shown in Fig. 2 one obtains, according to the Fourier expansion, the nt h component, Vn of the output voltage [3].
N. . [nL,~
4Vs •
Vm~ =
n=
sml--/-I
n~T i=l
\z/
1, 3, 5, ...
(4)
The mean square output voltage is given by ~r
V~m~= -7T
v 2(0) dO
(5)
0
2V~ Ns
~: i ~ x~
V~m~ =
(6)
77" i = 1
Equation (6) can be rewritten in the form V~m~=
77
(2i
)
- 1)L~ - (N~(LNJ2)}
]
(7)
Substituting eqn. (4) for n = 1 (the fundamental) and eqn. (7) in eqn. (3) we obtain
sin(LJ2) 8
\i=1
nh =
(8)
~r [ (,~ (2i--1)Li)--{N2,(LNJ2)) ] Equation (8) relates the harmonic efficiency to the n u m b e r of stairs Ns and their widths L1, L2, . . . , Li . . . . . L~¢,. It has be en implicitly assumed that the heights of the stairs are equal. Consequently, for a given n u m b e r of stairs Ns one can vary the stair widths L~ to make the stepped wave a p p r o a c h a pure sinusoidal wave. Under this condition the harmonic will be minimum and the efficiency b e c o m e s maximum. The harmonic content (HC) of the output is then given by HC = [1 - ~h] °'5
(8a)
Before obtaining the m a x i m u m efficiency one can decide different constraints concerning the relationships bet w e e n the different stair widths. The maximum efficiency will be calculated f or three different regimes.
Control mode 1, CM1. There is no constraint on the variation of the stair width e x c e p t that it m us t be smaller than Pi. This case will be t e r m e d free control mode.
564
Control m o d e 2, CM2. This is the regular t r a p e z i u m with two variables s h o w n in Fig. 3. F o r any given n u m b e r o f stairs t h e r e will be only two optimization p a r a m e t e r s . Control m o d e 3, CM3. This is the regular t r a p e z i u m with one variable base s h o w n in Fig. 4. F o r a n y given n u m b e r of stairs t h e r e will be only one optimizing p a r a m e t e r . Optimization m o d e s 2 and 3 have the a d v a n t a g e that the control circuit o f the p o w e r switches is greatly simplified and can be e x e c u t e d using far fewer c o m p o n e n t s b e c a u s e of the equal on-times of m o s t switches. Moreover, these m o d e s c o u l d be beneficial for e n e r g y equalization a m o n g the batteries supplying the inverter. Their d r a w b a c k is the r e d u c t i o n in h a r m o n i c efficiency for the same n u m b e r of stairs. H o w g r e a t this effect is will be seen in the n e x t section w h e r e the results are presented. 2.1.2. C i r c u i t loss e ~ c i e n c y The circuit loss efficiency is limited by the c o n v e r s i o n of electrical e n e r g y into heat in all circuit c o m p o n e n t s of the inverter including the p o w e r switches,
Vo(O)
,1,'1 Xl
X1
;7/,, x:lXl[
/x 1 x, ix, X0
\\
I
Xo
:~}2
Fig. 3. Control mode 2: each half-wave is in the form of a regular trapezium with two variables X0 and X~ for Ns = 6.
IVo(O) //1 //1 ×I //1 xl //1 Xl Xl (./,.,1/'1 Xl
xl
L~
Xl I\\\ Xl \ ~ X \ l
X1 oXl ~/2I Fig. 4. Control mode 3: each half-wave is in the form of a regular trapezium with one variable XI, for Ns = 6.
565 the control circuit and the internal resistances of the storage batteries supplying the inverter. Our p ur pos e is to reduce the power dissipated in each c o m p o n e n t to obtain as high an efficiency as possible. The power dissipated in the power switches is responsible for the greatest proportion of power loss and hence nearly limits the circuit loss efficiency. Assuming that the current in the switch Si is isw(t) and the voltage across it is vsw(t), the power dissipated in the switch is given by T
Psw =
1 r
Ji w(t)
dt
(9)
0
According to the typical switching waveform in Fig. 5, the power dissipated in a switch Psw can be divided into three parts according to the state of the switch, the off, the on and the dynamic pow e r Pon, Po, and Pdy respectively,
i.e.
Psw=Po,f-i-Pdy-i-Pon
(10)
Since the off-voltage Vo~ and the off-current Ion are independent of time, the off-power can be expr e s s e d by Po~ =
oN orf~ T ]
(11)
where ton is the off-time in one cycle with period T. The on-power during the on-time ton is given by
where Ion is the on-current and Von is the on-voltage. Both are independent o f time. In the transition from the on-state to the off-state and v i c e v e r s a the voltage and current can be considered to vary approximately linearly with time as indicated in Fig. 5. Accordingly, one can write the dynamic p o w e r
-V
Pay-
I
(t~+tf~
off on~ 6 T }
(13)
where tr and tf are the rise and fall times respectively. A c o m m o n p r o p e r t y of all electronic p o w e r switches is that they behave nearly ideally in the offstate. Therefore, their pow er consumption in the off state can be neglected.
Vsw(t)
isw(t) ........
' k
l ~ ..........
!Jr !~ ton _! tf_!_ toff
] 0
.!
Fig. 5. T y p i c a l v o l t a g e a n d c u r r e n t w a v e f o r m s o f a p o w e r s w i t c h .
566
Consequently,
P~w = P,,, +Pay
(14)
After i n t r o d u c i n g the basis for calculating the p o w e r c o n s u m e d in a p o w e r switch we are n o w able to d e t e r m i n e the p o w e r c o n s u m p t i o n in the inverter circuit of c o n c e r n ill Fig. 1. A c c o r d i n g to eqns. (12), (13) a n d (14), the voltage a n d c u r r e n t w a v e f o r m s for e a c h switch of the circuit m u s t be d e t e r m i n e d o v e r o n e c o m p l e t e cycle. F o r distinct division of the v o l t a g e o v e r p o w e r s w i t c h e s c o n n e c t e d in series, we only a s s i g n e d p o l a r i t y r e v e r s a l f u n c t i o n s for the s w i t c h e s Sp~, Sp2, Sn~ and S,2 in the bridge. In the p o s i t i v e half-cycle Sp~ and Sp~ will always b e on while Sn, and S,e will be on in the n e g a t i v e cycle. The r e m a i n i n g s w i t c h e s in the circuit c o n t r o l the c u r r e n t flow to the load f r o m the batteries. B e c a u s e of s y m m e t r y , the p o w e r c o n s u m e d in the positive half-cycle will be equal to t h a t c o n s u m e d in the n e g a t i v e half-cycle, p r o v i d e d the s w i t c h e s a r e identical. H e n c e , it is only sufficient to d e t e r m i n e the p o w e r c o n s u m p t i o n in the p o s i t i v e half-cycle. A c c o r d i n g to the s e c o n d Kirchoff law, the v o l t a g e a c r o s s the ith switch Si is given by:
(15)
v~i(t) = i ys - Vo(t)
w h e r e vo(t) is the load v o l t a g e o f the w a v e f o r m s h o w n in Fig. 2. T h e v o l t a g e w a v e f o r m o f the i t h switch is s h o w n in Fig. 6. It c a n b e s e e n f r o m this figure t h a t in the a s c e n d i n g p a r t o f t h e load voltage, the off-voltage of e a c h switch is equal to Vs. In the d e s c e n d i n g p a r t of the positive half-cycle the off-voltage is - V s . This m a y not b e allowed for real e l e c t r o n i c switches. In fact the s w i t c h e s r e m a i n off until t h e v o l t a g e a c r o s s t h e m h a s p r o p e r polarity. This m e a n s S~ ÷ ~ m u s t be t u r n e d off b e f o r e Si is t u r n e d on. One m u s t wait at least until the off-voltage c r o s s e s zero value to p o s i t i v e values. C o n s e q u e n t l y , the off-voltage in the d e s c e n d i n g p a r t c a n t a k e v a l u e s f r o m zero to i Vs. P o w e r saving could b e a c h i e v e d if S~ w e r e t u r n e d on at zero off-voltage, b u t this condition could be v e r y difficult to fulfil in p r a c t i c e . F o r the calculations, we shall a s s u m e t h a t the off-voltage h a s the s a m e v a l u e as in the a s c e n d i n g case, i.e. Volt= Vs. ~J'sw iV s
x1 V_V-'j
Li
K
-I
i-Ns)V,
Fig. 6. V~, waveform for switch i in the positive half-cycle.
567
The voltage ratings of the p o w e r switches could be d e t e r m i n e d f r o m eqn. (15), i.e. the m a x i m u m positive voltage Vsim=i ITs since Vo=O and the m a x i m u m value o f the negative voltage is given by
V~m = [ ( i V s - N s V s ) l Now it is required to d e t e r m i n e the o n - c u r r e n t in the switches for the positive half-cycle as was d o n e for the voltage. In the on-state, the switch will be m o d e l l e d by the circuit s h o w n in Fig. 7, where, Rsw is the on-resistance o f the switch and Vswo is the cut-in voltage o f the switch. Both are c o n s i d e r e d i n d e p e n d e n t o f the c u r r e n t in the switch. Let us a s s u m e that the load r e s i s t a n c e is RL and the internal r e s i s t a n c e o f the chemical b a t t e r y is Rs. The o n - c u r r e n t Ioni in the ith switch can b e e x p r e s s e d by
iVs-3Y~o
(16)
Ioni = iRs + 3R~,, +RL with
Rs =RBNB/Ns
(17)
w h e r e NB is the total n u m b e r o f the b a t t e r i e s and Rs is the resistance of a b a t t e r y stage. This e q u a t i o n is valid only for the s t e p p e d sinewave full wave inverter. F o r the tri-state inverter, i.e. Ns = 1 the o n - c u r r e n t is given by: Zon =
V s - 2V~wo
(18)
Rs + 2 R ~ + R L
The voltage a c r o s s switch S~ in the on-state follows directly f r o m the equivalent circuit o f Fig. 7, i.e.
Voni = V~o +IoniRo,
(19)
Now the v o l t a g e s and c u r r e n t s can be substituted in eqn. (14) to find the p o w e r c o n s u m e d in switch i, and b y s u m m i n g o v e r all the switches we obtain the total p o w e r loss Pst. F o r the tri-state inverter it follows t h a t Psi = 4 Von/on ~ - -
+ Von/on - ~
(20)
with /on =
V - 2V~o Rs + 2 R ~ + R L !
•
AAA~
Ion
Ron~Rsw
II !
:
Von~Vswo
Fig. 7. Equivalent circuit o f t h e p o w e r s w i t c h in the o n state.
568
Von= Vswo+IonRo, V Yof f --
2
ton~2 =X1 is the half-pulse width The power lost in the stepped sinewave inverter is consequently given by [3]:
Pst=
-~- V o n i l o n i - ~
2
--"
--
with Vo,= V~ and toni/T=Xi/vr. Using eqns. (2a), (6) and (20) or (21) we can calculate the loss efficiency.
2.2. Amp~re hours from each battery Since the voltage of the batteries is constant, the ampere hour supplied by each battery in one complete cycle is the integration of its discharge current over its operating time. The battery stage designated i operates in the positive half-cycle for a period corresponding to N~
iLl+ ~ L i i+l
and in the negative half-cycle for the period corresponding to Ns
(Ns-i+l)Lgs-i+l+
~
Li
Ns-i+2
It follows that the total operating time of the ith battery stage corresponds to Ns
LT(i)=iL,+ ~ L i + ( g s - i + 1) Ln~-i+l i+I Ns
+
~
L~
(22)
Ns--i+2
We are interested in the ratio DSR1 (i) of the discharge period of the battery stage i to the stage i = 1, i.e.
Lr(i)
DSRI(i) = - LT(1)
(23)
3. C o m p u t e r - a i d e d a n a l y s i s o f t h e i n v e r t e r In this section we present the results of the analysis introduced in the previous section for the stepped sinewave full-bridge inverter. The harmonic
569 efficiency ~h will be maximized for a given num ber of stairs and total battery voltage V by finding the optimum widths of the stairs. Under optimum conditions, the fundamental c o m p o n e n t of the output, the energy delivered by each battery and the loss efficiency will be determined. In addition, the sensitivity of the efficiency to variation in stair widths will be defined. This is especially important for the design of the control circuit of the power switches since the accur a c y of the timing circuit could largely affect the stair widths and consequently the efficiency. We also need to calculate the harmonic c o m p o n e n t s and their overall effective value.
3.1. Solution m e t h o d It is clear from eqn. (8) that the harmonic efficiency ~/l, is a non-linear function o f the widths LI, Lz, ... Li . . . . L N s . Therefore, to determine the maximum efficiency, an optimization technique is needed for such a multidimensional non-linear optimization problem. A numerical method based on a QUASI-NEWTON algorithm [4] was utilized. It is based on nulling the Jacobian matrix, followed by inspection of the second derivative matrix for an extremum. A FORTRAN subroutine ZXMIN based on this algorithm as a part of the IMSL Library is available. We developed a FORTtLAN program to calculate the following quantities: the optimum widths of the stairs Li, the widths of the pulses Xi, the harmonic efficiency ~?h, the d e p e n d e n c e of the harmonic efficiency on the stair widths, i.e. the sensitivity of the efficiency to variations in the stair widths, the relative amplitudes of the fundamental RI=Vo~/VsN~, the total harmonic content, the conduction period of each pow e r switch, the maximum forward and reverse voltages on each power switch, the pulse widths of the control pulses for the p o w er switches, the loss efficiency, the overall efficiency and the energy supplied by each battery. The following data must be available: (i) the n u m b e r of batteries NB, (ii) the voltage of each battery VB, (iii) the internal resistance of the battery Rm (iv) the num ber of stages Ns, (v) the frequency of the output voltage, (vi) the load resistance RL, (vii) the onresistance o f the switch Rsw and its cut-in voltage V~o, (viii) the rise and fall-times of the pow e r switch, tr, tr, (ix) the control m ode CM, (x) the nominal discharge current of the battery Im (xi) the maximum on-current of the p o wer switch which must be equal to or greater than IB. The flow chart of the program developed for computer-aided analysis o f the inverter with the above-mentioned inputs and outputs is shown in Fig. 8. 3.2. Results In this section we present the most interesting results of the computeraided analysis in or de r to work out the optimum operating conditions of an inverter havingNB = 14, VB= 12 V, RB= 0.01 12, the nominal discharge current o f the battery I B - - 10 A and the f r equency of the output f = 5 0 Hz. At first the minimum load resistance is calculated from RL=NBVB/IB. The m a x i m u m on-current of the power switch is IB and its maximum off-voltage in either
570
I ~.... I l output fund. f r e q . , the load r e s i s t a n c e , number of a v a i l a b l e b a t t e r i e s and t h e i r characteristics
(a)
l
>
L
Loop: Control mode # i to 3 [ i
t
(b)
Loop: (Ns " I to NB)
I I
Calculate: B a t t e r y s t a g e v o l t a g e I and r e s i s t a n c e , V$ and Rs
L
f t Calculate the initial values of the parameters ( X t . t ' I , 2 , . . , , N s}
.I Form the function of the' harmonic efficiency
E
i Call "ZXMIN" To: find the "mini~" of the previous funetlon
Calculate: i. The optimum 2. The optimum 3. The optimum i
stair width L i delay A i pulse width X I - 1,2,3,...,N s
(in rad.) (in tad.) (in tad.)
Calculate The pulse width t i (~n msec.) I - 1,2,3,...,N B
(c)
I I C a l c u l a t e the optimum p a r a ~ t e r s
[
(~L' ~W ~' Rl' aC, krl)
! I
i . Calculate the s e n s i t i v i t y
of q to the l v a r i a t i o n of Lt, 1 - 1 , 2 , 3 , . . . . , Ns 2. C a l c u l a t e the s e n s i t i v i t y of r e l a t i v e fundamental v o l t a g e R1 t o the v a r i a t i o n of L i , i " 1,2,3,..., Ns
(b) ~
(a) •
b
[
L~***.~..... ~*'*~*'~"1 1,
l ~,o, 1
Fig. 8. n o w chart of computer-aided analysis of the inverter.
571
polarity is V o , = N B V B. T h e r e are m a n y p o w e r e l e c t r o n i c devices which can be u s e d as switches in the p o w e r switching circuit, e.g. bipolar p o w e r transistors, gate-turn-off thyristors and VMOS transistors. T h e i r switching times are m u c h less t h a n their on-time. A typical 10 A bipolar t r a n s i s t o r has Vs~o=0.1 V, an o n - r e s i s t a n c e Rsw=O.07 12 and t r = t r = 5 0 ~s. 3.2.1. T h e i n v e r t e r ejTiciency Figure 9 shows the m a x i m u m h a r m o n i c efficiency as a f u n c t i o n o f the n u m b e r of stairs Ns k e e p i n g the overall b a t t e r y voltage c o n s t a n t for the t h r e e control m o d e s t e r m e d l , 2 and 3. It is clear f r o m this figure that as N~ increases, the h a r m o n i c efficiency b e c o m e s larger, a p p r o a c h i n g unity for Ns > 6. C o n s e q u e n t l y it is n e c e s s a r y to m a k e N~ h i g h e r for l o w e r costs. This is valid for the t h r e e control m o d e s . This can b e easily u n d e r s t o o d b e c a u s e as the step size d e c r e a s e s , the synthetic sinewave a p p r o a c h e s a p u r e sinewave with f u n d a m e n t a l f r e q u e n c y . As h a r m o n i c s are generally related to step size, t h e y are related to the individual b a t t e r y voltage and the p e a k voltage. The h a r m o n i c efficiency ~?h also d e p e n d s on the control m o d e . It has the highest values for the free c o n t r o l m o d e as the n u m b e r of w a v e f o r m adjusting p a r a m e t e r s is the largest. It is interesting to n o t e that as N~ i n c r e a s e s the difference in the efficiency decreases, a p p r o a c h i n g a l m o s t the s a m e value for N~>_6. The loss efficiency is also s h o w n in Fig. 9 as a function of the n u m b e r o f stairs N~ for the different control m o d e s . This result is v e r y interesting. W e see from the figure that the loss efficiency d e c r e a s e s at first slightly with N~, then b e c o m e s constant. It is m o r e o r less i n d e p e n d e n t o f the n u m b e r
I
96
.96
"G
~-.94
.92
.9
0
8
lo
1'2
Number of Battery Stages (NS) Fig. 9. ~?h, ~?L a n d 7; v s . N s for different
control modes
CM1, CM2 a n d CM3.
14
572 o f stages. This is attributed to the negligibly small d y n a m i c loss which c a u s e s the d e p e n d e n c e o f ~?L on/Vs. W e can also see f r o m the figure, that in c o n t r a s t to the h a r m o n i c efficiency, which i n c r e a s e s f r o m m o d e 3 to m o d e 1, the loss efficiency d e c r e a s e s f r o m m o d e 3 to m o d e 1. As a result the overall efficiency s h o w n in the same figure is nearly i n d e p e n d e n t of c o n t r o l mode. It i n c r e a s e s f r o m 0.9 for the bridge inverter w h e r e N~ = 1 to a plateau value of 0 . 9 6 5 at N~>_6. T h e s e values are in g o o d a g r e e m e n t with the m e a s u r e d efficiencies o f 0.88 for N~= 1 b y Anis [ l l and 0.94 for N~-= 16 by Schmitt and Schaetzle [2]. The slightly lower m e a s u r e d efficiencies are a result of losses in their p o w e r switches being larger t h a n t h o s e c o n s i d e r e d here. A c c o r d i n g to these results, one has to k e e p N~ a r o u n d 6 and o p e r a t e the i n v e r t e r in control m o d e 3.
3.2.2. D e p e n d e n c e o f the total eJficiency on the d e v i a t i o n s o f s t a i r w i d t h s f r o m their o p t i m u m values This is an interesting p a r a m e t e r for the design o f a c o n t r o l circuit for the p o w e r switches. It d e t e r m i n e s h o w a c c u r a t e the pulse widths o f the c o n t r o l signals m u s t be. T h e r e f o r e , it could affect the cost o f p r o d u c i n g t h e s e pulses if circuit c o m p o n e n t s with larger t o l e r a n c e s can be utilized. Figure 10 shows the total efficiency ~? as a f u n c t i o n of the relative deviation in Li f r o m their o p t i m u m values for different n u m b e r s of stairs for c o n t r o l m o d e s 1, 2 and 3. Fortunately, it s e e m s that ~ has a b r o a d m a x i m u m . This is m o r e p r o n o u n c e d for the efficiency c u r v e s o f c o n t r o l m o d e 3 (Fig. 10(c)). If one allows a 0.2% d e c r e a s e f r o m the m a x i m u m efficiency one can d e t e r m i n e f r o m t h e s e c u r v e s the permissible p e r c e n t a g e deviation in stair widths. H o w e v e r , it is clear f r o m t h e s e figures that a deviation o f a b o u t 10% in L, c o u l d b e allowed without any significant loss in efficiency. T h e r e f o r e , high p r e c i s i o n timing c o m p o n e n t s in the control signal g e n e r a t o r are n o t needed. 3.2.3. The F o u r i e r c o m p o n e n t s of the output, the f u n d a m e n t a l a n d the h a r m o n i c content T h e relative m a g n i t u d e of the f u n d a m e n t a l and the total h a r m o n i c c o n t e n t are s h o w n in Fig. 11 for different values of N~ u n d e r different c o n t r o l m o d e s . F r o m Fig. l l ( a ) we can o b s e r v e t h a t as Ns increases, the f u n d a m e n t a l c o m p o n e n t d e c r e a s e s slowly. T h e r e f o r e , one n e e d s a slightly larger overall b a t t e r y voltage for the s a m e o u t p u t voltage w h e n increasing Ns. T h e situation is e v e n w o r s e for c o n t r o l m o d e s 1 and 2 since their f u n d a m e n t a l c o m p o n e n t s are smaller t h a n t h a t of m o d e 3. It is clear f r o m Fig. 11 (b) that the h a r m o n i c c o n t e n t d e c r e a s e s as e x p e c t e d b y increasing the n u m b e r o f stairs. Control m o d e 1 results in the lowest h a r m o n i c content. F o r N s = 6 the h a r m o n i c c o n t e n t s are 6.1%, 6.8% and 9.1% for control m o d e s 1, 2 and 3 respectively, w h i c h c o u l d be a c c e p t a b l e for m a n y loads. B e c a u s e o f the slow d e c r e a s e in h a r m o n i c c o n t e n t for Ns > 6, one would have to i n c r e a s e N~ excessively to r e d u c e the h a r m o n i c s further. In fact, the h a r m o n i c c o n t e n t is inversely p r o p o r t i o n a l to /Vs. If h a r m o n i c c o n t e n t less t h a n t h o s e c o r r e s p o n d i n g to
573 n% I00,
%
100.
Total 98. efficiency
9aI 961
Ns=6 Ns=4
941
Ns=2
~ N s = 2
92"
92 NS:I
-0.2
-0.I
0
o.1
,
o.z
-olz
Fr= ALi/L i
-o:1
90 011
O.J2
Relative deviation of the step width Fr: ALi/L i
(a)
(b) i00 ,n % (Total efficiency) 98 _
~Ns=6 Ns:4 Ns:2
92
9O
-o:1 (c)
o
o11
o:2
Fr = ALi/L i
Fig. 10. Total efficiency as a function of control mode 2, and (c) control mode 3.
(ALJL,) at different N~: (a) control mode 1, Co)
Ns= 6 are required, it w o u l d b e m o r e e c o n o m i c a l to u s e filters. F o r filter design, the relative a m p l i t u d e s of the h a r m o n i c s are given in T a b l e 1 for t h e different control m o d e s . W e c a n see t h a t the a m p l i t u d e s o f the h a r m o n i c s a r e relatively l a r g e a n d d e c a y slowly with the o r d e r o f the h a r m o n i c n. In addition, the h a r m o n i c s a l t e r n a t e in sign.
3.2.4. Energy delivered by each battery B e c a u s e w e are i n t e r e s t e d in t h e relative d i s c h a r g e e n e r g y a n d n o t in a b s o l u t e values, t h e d i s c h a r g e t i m e s o f all b a t t e r i e s are r e l a t e d to the o p e r a t i n g t i m e of t h e b a t t e r i e s in s t a g e one. This ratio is t e r m e d DSRi. It is c a l c u l a t e d f r o m eqns. (22) a n d (23) a n d p l o t t e d in Fig. 12 f o r the different c o n t r o l m o d e s t a k i n g the n u m b e r o f s t a g e s Ns as 6. F o r all c o n t r o l m o d e s , t h e b a t t e r i e s in t h e m i d d l e o f the b a t t e r y string deliver m o r e e n e r g y t h a n t h o s e a t t h e e n d s o f t h e string. F o r c o n t r o l m o d e 2 t h e distribution o f t h e d i s c h a r g e e n e r g y is b e t t e r t h a n f o r m o d e 1, a n d is the b e s t for c o n t r o l m o d e 3. H e n c e c o n t r o l m o d e 3 is a d v a n t a g e o u s f r o m this r e s p e c t o v e r the o t h e r p r o p o s e d control modes.
3.2.5. Pulses f o r the control circuit T h e f u n c t i o n o f the c o n t r o l circuit is to switch the p o w e r s w i t c h e s in t h e i n v e r t e r on a n d off in the p r o p e r t i m i n g s e q u e n c e to g e n e r a t e the s t e p p e d
574 1.1702
1.1315-
Y:
1.0930 \
\
\
~ CM3
1.0544
5
CMI
\
1.015B
CM2
.97721 o
g~
.93860
.9
5
0 (a)
10
15
Number of Battery Stages (NS)
.3
.25 -
.2 ,3 .15 ,.?o c
cM3___Z___ c.~,K
.05
0 0 (b)
J
I
I
I
i
i
2
4
6
8
10
12
4
Number of Battery Stages (NS)
Fig. 11. (a) R a t i o o f f u n d a m e n t a l c o m p o n e n t to t h e total b a t t e r y v o l t a g e R1 a s a f u n c t i o n o f t h e n u m b e r o f t h e b a t t e r y s t a g e s Ns. (b) T h e overall h a r m o n i c c o n t e n t vs. N . for different control modes.
N,=I
1.17007 -0.14744 -0.11211 0.17354 -0.12371
1.17023 -0.14783 -0.11173 0.17341 -0.12392
1.17023 -0.14783 -0.11173 0.17341 -0.12392
n
CM1 1 3 5 7 9
CM2 1 3 5 7 9
CM3 1 3 5 7 9
1.11256 -0.04488 -0.13217 0.07553 -0.00594
0.95898 0.04557 0.09665 -0.06323 -0.09836
1.09490 -0.04255 -0.05591 - 0.03537 0.03706
N,=2
1.08732 -0.01419 -0.11153 0.03175 0.00188
0.93840 0.09114 0.04224 -0.06933 0.01411
1.06497 - 0.01958 -0.02745 - 0.02955 -0.01342
Ns=3
1.07408 -0.00192 -0.09570 0.01301 0.00044
0.94322 0.09762 0.01030 -0.04594 0.01439
1.04931 -0.01139 -0.01579 - 0.01949 -0.01668
Ns---4
1.06595 0.00431 -0.08474 0.00324 -0.00119
0.94816 0.09605 - 0.00311 -0.03117 0.00466
1.03931 -0.00729 -0.01002 - 0.01290 -0.01381
Ns=5
1.06113 0.00716 -0.07709 -0.00234 - 0.00214
0.95519 0.09138 -0.01089 -0.02348 -0.00330
1.03287 -0.00508 -0.00702 -0.00927 - 0.01087
Ns=6
1.05688 0.00988 -0.07119 - 0.00629 -0.00308
0.95868 0.08794 - 0.01372 - 0.01908 -0.00794
1.03053 - 0.00490 -0.00641 - 0.00761 - 0.00870
N~=7
1.05421 0.01115 -0.06683 - 0.00889 -0.00357
0.96950 0.07904 -0.01889 -0.01604 -0.01343
1.02485 -0.00275 -0.00430 - 0.00553 - 0.00645
Ns=8
1.05212 0.01208 -0.06337 -0.01082 - 0.00392
0.97166 0.07654 -0.01913 - 0.01537 - 0.01444
1.02271 -0.00261 -0.00326 - 0.00429 - 0.00564
N~=9
The r a t i o Rn of th e h a r m o n i c a m p l i t u d e s r e l a t i v e to t h e t o t a l b a t t e r y v o l t a g e N , Vs for di ffe re nt c o n t r o l m o d e s CM1, CM2 an d CM3
TABLE 1
to
1.2902 0.7145 0.4937 0.3754 0.3051 0.2572 0.2214 0.1956 0.1675
1.2892 0.7189 0.4576 0.3191 0.2284 0.1692 0.1159 0.0826
1.2892 1.0264 0.8115 0.6642 0.5604 0.4833 0.4254 0.3793 0.3421
Ns
CM1 1 2 3 4 5 6 7 8 9
CM2 1 2 3 4 5 6 7 9
CM3 1 2 3 4 5 6 7 8 9
14.843 1.0264 0.8115 0.6642 0.5604 0.4833 0.4254 0.3793 0.3421
14.8433 1.6077 1.1473 0.8830 0.7186 0.6031 0.5213 0.4060
7.4195 1.6102 1.0396 0.7782 0.6221 0.5203 0.4445 0.3892 0.3497
tl
11.789 0.8115 0.6642 0.5604 0.4833 0.4254 0.3793 0.3421
7.1387 1.1473 0.8830 0.7186 0.6031 0.5213 0.4060
5.3506 1.2765 0.8577 0.6602 0.5355 0.4555 0.3994 0.3574
t2
10.2615 0.6642 0.5604 0.4833 0.4254 0.3793 0.3421
6.2320 0.8830 0.7186 0.6031 0.5213 0.4060
4.3805 1.0907 0.7485 0.5903 0.4832 0.4173 0.3654
t3
9.3732 0.5604 0.4833 0.4254 0.3793 0.3421
0.6031 0.5213 0.4060
0.7186
5.8719
3.7957 0.9705 0.6711 0.5304 0.4445 0.3880
t4
8.7921 0.4833 0.4254 0.3793 0.3421
5.6274 0.6031 0.5213 0.4060
3.3870 0.8785 0.6085 0.4900 0.3969
t5
8.4016 0.4254 0.3793 0.3421
5.5267 0.5213 0.4060
3.0944 0.8015 0.5653 0.4611
t6
8.0890 0.3793 0.3421
5.4039 0.4060
2.9099 0.7566 0.5475
t7
7.8634 0.3421
0.4060
2.6840 0.6903
t8
7.6836
5.3846
2.5525
t9
The o p t i m u m pulse widths t~ in milliseconds as a function of the n u m b e r of battery s t a g e s Ns for different control m o d c s CM1, CM2 and CM3
TABLE 2
577 1.16
CMll
1.14 .o
I . 12
1.10 1.08
~-
1.06
g
1.04
=~
i.o2
1. 001 2
3
4
6
Number of battery stages
Fig. 1 2. Discharge amp~re-hour ratio DSRi of any battery stage I relative to battery stage 1
for a number of battery stages Ns= 6 at different control modes.
sinewave. O n c e t h e f r e q u e n c y o f t h e a.c. o u t p u t v o l t a g e a n d t h e p o w e r switch a r e d e t e r m i n e d o n e c a n easily d e d u c e the driving p u l s e s for e a c h switch in t h e inverter. T h e d e v e l o p e d c o m p u t e r p r o g r a m s u p p l i e s t h e d a t a r e q u i r e d to design the c o n t r o l circuit. T a b l e 2 gives c o m p l e t e t i m i n g i n f o r m a t i o n at different v a l u e s o f Ns f o r c o n t r o l m o d e s 1, 2 a n d 3 a s s u m i n g f = 5 0 Hz. As a d e m o n s t r a t i o n , t h e c o n t r o l p u l s e s b a s e d on t h e s e t a b l e s are s h o w n in Fig. 13 f o r Ns = 6. Idp is the driving p u l s e f o r Sp~ a n d Sp2 while Idn is t h e driving p u l s e f o r Sn~ a n d Sn2. Id, t o Id6 drive the s w i t c h e s S~ to $6 while IdT to Ida2 c o n t r o l t h e s w i t c h e s SNs+ 1 to Sans in t h e i n v e r t e r circuit in Fig. 1.
4. Conclusion
In this w o r k an i n v e r t e r f o r a p h o t o v o l t a i c p o w e r s y s t e m w a s a n a l y s e d in detail. It w a s f o u n d t h a t the full b r i d g e i n v e r t e r circuit is p r e f e r a b l e b e c a u s e o f the r e d u c t i o n in t h e n u m b e r o f s t o r a g e b a t t e r i e s , w h i c h are a v e r y e x p e n s i v e e l e m e n t . In o r d e r to e l i m i n a t e o r to r e d u c e t h e filtering c o m p o n e n t s a n d the t r a n s f o r m e r s , w e u s e d a s t e p p e d s i n e w a v e inverter. A c o m p u t e r p r o g r a m w a s d e v e l o p e d f o r a n a l y s i s a n d d e s i g n o f this inverter. T h e h a r m o n i c efficiency i n c r e a s e s with the n u m b e r o f stairs u p t o six stairs a n d a f t e r t h a t it s a t u r a t e s . T h e f u n d a m e n t a l c o m p o n e n t s will b e slightly r e d u c e d w h i c h m e a n s a n i n c r e a s e in t h e n u m b e r o f b a t t e r i e s f o r the s a m e o u t p u t v o l t a g e f o r h i g h e r Ns values. T h e h a r m o n i c d i s t o r t i o n d e c r e a s e s with Ns to a l l o w a b l e v a l u e s f o r m o s t l o a d s if Ns >_ 6. T h e d e p e n d e n c e o f ~/ o n t h e relative d e v i a t i o n o f t h e stair w i d t h s f r o m t h e i r o p t i m u m v a l u e s s h o w s a v e r y slight d e c r e a s e in ~? f o r 10% deviation. This r e l a x e s t h e p r e c i s i o n o f t h e t i m i n g c o n t r o l circuit.
578
I°°l Idn
I
I T/2 tI tI
Idl'id71h
Fl,fl T/2 t2 t2 N,n
Id2JdI fl
fl
A dlI
]dS"IdllI
t2
N
, _
T/2
Id3'Id91 t3 I d4' I
l_ T
tn
tn
A
A
T/2
t5
i
t5
R
I
A tn
i
R t5
t5
~
R
T/2
t6
Id6' Zdl21
n t n
T/2
V]
t3
t3
J
I
T>~
t6
l
I
i
Fig. 13. The timing diagram of the driving pulses of the switches for N , = 6.
The main characteristic o f p o w e r s w i t c h e s suitable for this application w e r e w o r k e d out. It is preferable to u s e fully c o n t r o l l e d d e v i c e s with g o o d s w i t c h i n g p e r f o r m a n c e to d e c r e a s e the lost p o w e r in the circuit. The l o s s efficiency w a s d e t e r m i n e d and it w a s f o u n d that it is nearly i n d e p e n d e n t o f t h e n u m b e r o f s t a g e s but varies with t h e c o n t r o l m o d e o f t h e p o w e r switches. The overall efficiency o f t h e inverter i n c r e a s e s with Ns, r e a c h i n g a c o n s t a n t v a l u e for Ns > 6. G o o d a g r e e m e n t w a s f o u n d b e t w e e n our c a l c u l a t e d v a l u e s for the efficiency and p u b l i s h e d e x p e r i m e n t a l results. Further studies are required c o n c e r n i n g t h e i n t r o d u c t i o n o f interruption p e r i o d s b e t w e e n t h e p u l s e s and their effect o n the p e r f o r m a n c e o f the inverter. Capacitive and inductive l o a d s w e r e n o t treated here. In addition, the inverter circuit m u s t be p r o v i d e d with f r e q u e n c y and v o l t a g e c o n t r o l as well as p r o t e c t i o n e l e m e n t s . A practical inverter circuit is u n d e r d e v e l o p m e n t to e v a l u a t e it u n d e r real w o r k i n g c o n d i t i o n s .
Acknowledgment The a u t h o r t h a n k s Eng. M. El Znaedi for the p r o g r a m m i n g .
579
References 1 W. Anis, Analysis of a three-level bridge inverter for photovoltaics, Sol. Cells, 25 (1988) 255-263. 2 J. Schmitt and R. Schaetzle, Simple transformerless inverter with automatic grid, tracking and negligible harmonic content for utility interactive photovoltaic systems, Proc. 4th European Community Photovoltaic Solar Energy Conf., Tresta, I982, Reidel, Dordrecht, 1982, pp. 3 1 6 - 3 1 9 . 3 M. El-Znaedi, Computer-aided analysis of inverters for photovoltaic power system, Project Report, 1989 (Electrical Engineering Department, College of Engineering, King Saud University, Riyadh). 4 FORTRAN Subroutines f o r Mathematics and Studies, Vol. 4, 1985, edn. 9.2, International Mathematical and Statistical Libraries, Houston, TX.
559
Computer-aided analysis of stepped sinewave inverters Abdelhalim Zekry Electrical Engineering Department, College of Engineering, King Saud University, Riyadh (Saudi Arabia) (Received July 25, 1990)
Abstract A stepped sinewave full-bridge inverter was analysed theoretically to determine its optimum working conditions, its minimum number of components, the characteristic of its constituent components and control circuit of its power switches. A computer program, written in FORTRAN-77, was developed to analyse this inverter circuit. The program prompts for input data such as the characteristics of the power and control circuit elements, the output fundamental frequency, and the number of available batteries and their voltage. The output of the program is designed to give the optimum stair widths, the loss, the harmonic and the total efficiency, the harmonic distortion, the sensitivity of the efficiency and fundamental voltage to the variation in stair width, and the energy withdrawal from each battery for the power switches controlling the circuit. Interesting design data for this circuit were also obtained.
1. I n t r o d u c t i o n One o f the main units in a p h o t o v o l t a i c p o w e r s y s t e m is the d . c . - a . c . c o n v e r t e r . Different inverter circuits suitable for p h o t o v o l t a i c p o w e r s y s t e m s have b e e n d e v e l o p e d [ 1, 2]. The m o s t efficient a n d at the s a m e time the c h e a p e s t inverter is t h a t p r o p o s e d b y S c h m i t t and Schaetzle [2 ]. A s y s t e m a t i c analysis o f t h e s e inverters t o d e t e r m i n e their o p t i m u m p e r f o r m a n c e a n d their o p t i m u m d e s i g n h a s n o t yet been carried out. In this work, a general inverter circuit is i n t r o d u c e d a n d a n a l y s e d systematically. A design e x a m p l e is also given. The p h o t o v o l t a i c g e n e r a t o r s u p p l y i n g the inverter has the following features: (i) it is a divided s o u r c e , n o r m a l l y in the f o r m o f a large n u m b e r o f batteries; (ii) it is e x p e n s i v e (this applies to b o t h the p h o t o v o l t a i c a r r a y a n d the c h e m i c a l b a t t e r y storage). T h e s e t w o i m p o r t a n t f a c t o r s will b e t a k e n into c o n s i d e r a t i o n in designing the inverter.
2. A n a l y s i s
of the inverter
circuit
A g e n e r a l s t e p p e d sinewave full-bridge inverter circuit is s h o w n schematically in Fig. 1. It c o n s i s t s o f (2Ns + 4) p o w e r switches with Ns the n u m b e r o f b a t t e r y stages. E a c h b a t t e r y s t a g e m a y c o n t a i n m o r e than one b a t t e r y
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560 ! NsVs ~ SNs_i" SNs+~ T SNs-I /
\ Spl S2Ns-~
~Sn I
Si j
RL I
S2Ns-~
, |
$
2
!
~
SI_/
Sp2
Sn2
~V S
SNs+~
Fig. 1. Stepped sinewave full-bridge inverter circuit.
, V0
NsVs
VS
Ai
-i;;i ~11
-Vs
~
i
i sta i r
i b', i , ~pulse
I
l
angle O _
!:ii
-i Vs -Ns Vs Fig. 2. Output with stacked k-pulses having different pulse widths (k = 2 N ~ - 1 ) .
and has an e.m.f. Vs. With p r o p e r control signals for the power switches, this circuit can generate the periodic output voltage Vo(O) shown in Fig. 2. It is in the form of a stepped sinewave. For the positive half-wave Spl and Sp2 are on while Sn, and S,2 conduct in the negative half-cycle only. Each half is c o m p o s e d of Ns stairs stacked on top of each other. The width of the ith stair is Li where i is a running index varying from 1 to AT,. The ith stair is delayed by a phase angle Ai as shown in Fig. 2. The positive stairs are symmetrical around 0= 7r/2 and the negative stairs around O= 3~r/2. The positive half-wave can also be obtained by stacking k-pulses having different pulse widths side by side where k = ( 2 N s - 1 ) . The width of the ith pulse is X~ in radiance. The negative half-wave is identical with the positive half-wave. In generating the ith and ( 2 N s - i ) t h pulses, only the switch St must be on, t oget her with Sp, and Sp2. The group o f switches from SN.+, . . . t o S2Ns pr oduces the pulses of the negative halfcycle. The switching circuit operates as follows. In the positive half-cycle Spl and Sp2 are on and all other switches are off. The pulse X, is p r o d u c e d by putting S~ on during the duration of this pulse as shown in Fig. 2. Directly before putting $2 on to produce the pulse X2, one has to put off S,. To
561 produce the ith pulse the switch S~ is put on directly after making S~_ ~ off. The ith pulse in the falling part of the positive half-cycle is produced by again putting Si on after putting Si+~ off for a duration of X~. It should be noticed that battery 1 in the string delivers the largest current in the positive half-cycle. In the negative half-cycle S,I and S,2 are on. The first pulse i = - 1 in the falling part of the negative half-cycle is produced by making SN.+, on, the second pulse i = - 2 by switching SN~+~ on after switching SN~., off. The ith sample is produced by the switch SN~+i. In the rising part of the negative cycle the - i t h pulse is also produced by putting SN~., on while all other switches are off except Snl and Sn2. It must be mentioned that in the negative half-cycle the largest charge will be delivered by the uppermost battery stage in the string Ns, in contrast with the positive half-cycle. In this way charge equilization among all batteries results. If switching spikes or noise are present, they will be of high frequency and can easily be filtered by connecting a relatively small capacitor across the load. If any short occurs across any battery stage it will be eliminated by melting because of the capabilities of chemical batteries to supply very large surge currents. This means that the batteries will probably not be damaged. In the output waveform only two pulses will not be produced. If the number of stages is large the effect of shorting one battery stage is not serious. A switch failure, which means a switch is always off, would lead to the disappearance of two pulses in a half-cycle only, i.e. the same effect as in the previous case. In the practical circuit one has to add protection as well as monitoring elements. The inverter described here can be used directly in stand-alone photovoltaic systems. This inverter is not regulated and its output voltage varies according to the state of the battery charges. Consequently, loads accepting these voltage variations can be utilized with this inverter type. However, by synchronizing the timing control circuit with the utility voltage waveform, and providing the inverter with a means of voltage control and safety circuits, it can be coupled with the utility. The voltage control can be carried out by (i) controlling the input voltage to the inverter, (ii) controlling the voltage within the inverter or (iii) controlling the voltage delivered by the inverter. Methods (i) and (ii) could be uneconomical, especially when the output voltage varies over a wide range because one has to make full use of the batteries. For limited load voltage variation, however, they could be utilized. Method (iii) would be preferable for large variations in the input or load voltage. The basic concept used to control the output voltage within the inverter is modulation of the pulse width where the durations of both the positive and negative pulses are varied from zero to half the period of one complete cycle. Here, the height and the width of each stair can be controlled separately. To avoid the need of a high voltage photovoltaic array to charge the batteries as the input to the inverter, one can feed each battery separately through a photovoltaic battery charge controller. -
562 The circuit must satisfy the following r e q u i r e m e n t s : (1) high efficiency, (2) m i n i m u m h a r m o n i c c o n t e n t in the o u t p u t voltage, (3) equal e n e r g y withdrawal from e a c h battery, (4) simplicity of the inverter circuit, (5) low cost, (6) ease of control o f the p o w e r switching circuit, (7) reliability o f operation. R e q u i r e m e n t s ( 4 ) - ( 7 ) can be fulfilled by reducing the n u m b e r of switches and by avoiding e x p e n s i v e c o m p o n e n t s such as c h o k e s and t r a n s f o r m e r s . R e q u i r e m e n t s (1) and (2), however, can be only realized by increasing the n u m b e r o f switches. T h e r e f o r e , all r e q u i r e m e n t s m a y not be satisfied simultaneously. B e c a u s e o f the relative i m p o r t a n c e o f high efficiency in c o m p a r i s o n with all the o t h e r r e q u i r e m e n t s , this must be given priority. This will be clear if we think a b o u t the capital cost p e r watt for the solar array and the battery storage. In fact, o n e has to avoid the loss of any joule g e n e r a t e d in the solar p o w e r system, while satisfying all o t h e r r e q u i r e m e n t s by using high quality c o m p o n e n t s and a d v a n c e d t e c h n o l o g y to a s s e m b l e t h e s e devices. In this way, a reliable inverter can be obtained. In summary, the inverter cost per watt m u s t be k e p t less than that of the array and storage.
2.1. Inverter e ~ c i e n c y The efficiency ~? is defined by the ratio o f the f u n d a m e n t a l o u t p u t p o w e r P0: to the p o w e r input to the inverter Pi P0~ . = -Pi
(1)
Multiplying the right-hand side by (Po/Po) where Po is the total o u t p u t power, ~? can be rewritten in the form
(e) with 77L:
T~h
Po
(2a)
Pi
Po~
(2b)
Po
w h e r e ~h is the h a r m o n i c efficiency and ~?L is the efficiency due to p o w e r loss in the inverter circuit. It will be t e r m e d simply the loss efficiency. The allowable h a r m o n i c c o n t e n t in the o u t p u t voltage is d e t e r m i n e d b y the n a t u r e o f the load. T h e r e f o r e , we split the efficiency into two efficiencies ~?L and ~Th t o reflect this.
2.1.1. The harmonic e ~ c i e n v y In this section we shall s t u d y the h a r m o n i c efficiency, the h a r m o n i c c o n t e n t in the o u t p u t voltage and the f u n d a m e n t a l voltage, assuming resistive load.
563
Denoting the effective value of the output voltage by Vrms and the magnitude of its fundamental c o m p o n e n t by V,m, the harmonic efficiency can be expressed by V2m
~?h-- 2V2r~
(3)
For the stepped sinewave shown in Fig. 2 one obtains, according to the Fourier expansion, the nt h component, Vn of the output voltage [3].
N. . [nL,~
4Vs •
Vm~ =
n=
sml--/-I
n~T i=l
\z/
1, 3, 5, ...
(4)
The mean square output voltage is given by ~r
V~m~= -7T
v 2(0) dO
(5)
0
2V~ Ns
~: i ~ x~
V~m~ =
(6)
77" i = 1
Equation (6) can be rewritten in the form V~m~=
77
(2i
)
- 1)L~ - (N~(LNJ2)}
]
(7)
Substituting eqn. (4) for n = 1 (the fundamental) and eqn. (7) in eqn. (3) we obtain
sin(LJ2) 8
\i=1
nh =
(8)
~r [ (,~ (2i--1)Li)--{N2,(LNJ2)) ] Equation (8) relates the harmonic efficiency to the n u m b e r of stairs Ns and their widths L1, L2, . . . , Li . . . . . L~¢,. It has be en implicitly assumed that the heights of the stairs are equal. Consequently, for a given n u m b e r of stairs Ns one can vary the stair widths L~ to make the stepped wave a p p r o a c h a pure sinusoidal wave. Under this condition the harmonic will be minimum and the efficiency b e c o m e s maximum. The harmonic content (HC) of the output is then given by HC = [1 - ~h] °'5
(8a)
Before obtaining the m a x i m u m efficiency one can decide different constraints concerning the relationships bet w e e n the different stair widths. The maximum efficiency will be calculated f or three different regimes.
Control mode 1, CM1. There is no constraint on the variation of the stair width e x c e p t that it m us t be smaller than Pi. This case will be t e r m e d free control mode.
564
Control m o d e 2, CM2. This is the regular t r a p e z i u m with two variables s h o w n in Fig. 3. F o r any given n u m b e r o f stairs t h e r e will be only two optimization p a r a m e t e r s . Control m o d e 3, CM3. This is the regular t r a p e z i u m with one variable base s h o w n in Fig. 4. F o r a n y given n u m b e r of stairs t h e r e will be only one optimizing p a r a m e t e r . Optimization m o d e s 2 and 3 have the a d v a n t a g e that the control circuit o f the p o w e r switches is greatly simplified and can be e x e c u t e d using far fewer c o m p o n e n t s b e c a u s e of the equal on-times of m o s t switches. Moreover, these m o d e s c o u l d be beneficial for e n e r g y equalization a m o n g the batteries supplying the inverter. Their d r a w b a c k is the r e d u c t i o n in h a r m o n i c efficiency for the same n u m b e r of stairs. H o w g r e a t this effect is will be seen in the n e x t section w h e r e the results are presented. 2.1.2. C i r c u i t loss e ~ c i e n c y The circuit loss efficiency is limited by the c o n v e r s i o n of electrical e n e r g y into heat in all circuit c o m p o n e n t s of the inverter including the p o w e r switches,
Vo(O)
,1,'1 Xl
X1
;7/,, x:lXl[
/x 1 x, ix, X0
\\
I
Xo
:~}2
Fig. 3. Control mode 2: each half-wave is in the form of a regular trapezium with two variables X0 and X~ for Ns = 6.
IVo(O) //1 //1 ×I //1 xl //1 Xl Xl (./,.,1/'1 Xl
xl
L~
Xl I\\\ Xl \ ~ X \ l
X1 oXl ~/2I Fig. 4. Control mode 3: each half-wave is in the form of a regular trapezium with one variable XI, for Ns = 6.
565 the control circuit and the internal resistances of the storage batteries supplying the inverter. Our p ur pos e is to reduce the power dissipated in each c o m p o n e n t to obtain as high an efficiency as possible. The power dissipated in the power switches is responsible for the greatest proportion of power loss and hence nearly limits the circuit loss efficiency. Assuming that the current in the switch Si is isw(t) and the voltage across it is vsw(t), the power dissipated in the switch is given by T
Psw =
1 r
Ji w(t)
dt
(9)
0
According to the typical switching waveform in Fig. 5, the power dissipated in a switch Psw can be divided into three parts according to the state of the switch, the off, the on and the dynamic pow e r Pon, Po, and Pdy respectively,
i.e.
Psw=Po,f-i-Pdy-i-Pon
(10)
Since the off-voltage Vo~ and the off-current Ion are independent of time, the off-power can be expr e s s e d by Po~ =
oN orf~ T ]
(11)
where ton is the off-time in one cycle with period T. The on-power during the on-time ton is given by
where Ion is the on-current and Von is the on-voltage. Both are independent o f time. In the transition from the on-state to the off-state and v i c e v e r s a the voltage and current can be considered to vary approximately linearly with time as indicated in Fig. 5. Accordingly, one can write the dynamic p o w e r
-V
Pay-
I
(t~+tf~
off on~ 6 T }
(13)
where tr and tf are the rise and fall times respectively. A c o m m o n p r o p e r t y of all electronic p o w e r switches is that they behave nearly ideally in the offstate. Therefore, their pow er consumption in the off state can be neglected.
Vsw(t)
isw(t) ........
' k
l ~ ..........
!Jr !~ ton _! tf_!_ toff
] 0
.!
Fig. 5. T y p i c a l v o l t a g e a n d c u r r e n t w a v e f o r m s o f a p o w e r s w i t c h .
566
Consequently,
P~w = P,,, +Pay
(14)
After i n t r o d u c i n g the basis for calculating the p o w e r c o n s u m e d in a p o w e r switch we are n o w able to d e t e r m i n e the p o w e r c o n s u m p t i o n in the inverter circuit of c o n c e r n ill Fig. 1. A c c o r d i n g to eqns. (12), (13) a n d (14), the voltage a n d c u r r e n t w a v e f o r m s for e a c h switch of the circuit m u s t be d e t e r m i n e d o v e r o n e c o m p l e t e cycle. F o r distinct division of the v o l t a g e o v e r p o w e r s w i t c h e s c o n n e c t e d in series, we only a s s i g n e d p o l a r i t y r e v e r s a l f u n c t i o n s for the s w i t c h e s Sp~, Sp2, Sn~ and S,2 in the bridge. In the p o s i t i v e half-cycle Sp~ and Sp~ will always b e on while Sn, and S,e will be on in the n e g a t i v e cycle. The r e m a i n i n g s w i t c h e s in the circuit c o n t r o l the c u r r e n t flow to the load f r o m the batteries. B e c a u s e of s y m m e t r y , the p o w e r c o n s u m e d in the positive half-cycle will be equal to t h a t c o n s u m e d in the n e g a t i v e half-cycle, p r o v i d e d the s w i t c h e s a r e identical. H e n c e , it is only sufficient to d e t e r m i n e the p o w e r c o n s u m p t i o n in the p o s i t i v e half-cycle. A c c o r d i n g to the s e c o n d Kirchoff law, the v o l t a g e a c r o s s the ith switch Si is given by:
(15)
v~i(t) = i ys - Vo(t)
w h e r e vo(t) is the load v o l t a g e o f the w a v e f o r m s h o w n in Fig. 2. T h e v o l t a g e w a v e f o r m o f the i t h switch is s h o w n in Fig. 6. It c a n b e s e e n f r o m this figure t h a t in the a s c e n d i n g p a r t o f t h e load voltage, the off-voltage of e a c h switch is equal to Vs. In the d e s c e n d i n g p a r t of the positive half-cycle the off-voltage is - V s . This m a y not b e allowed for real e l e c t r o n i c switches. In fact the s w i t c h e s r e m a i n off until t h e v o l t a g e a c r o s s t h e m h a s p r o p e r polarity. This m e a n s S~ ÷ ~ m u s t be t u r n e d off b e f o r e Si is t u r n e d on. One m u s t wait at least until the off-voltage c r o s s e s zero value to p o s i t i v e values. C o n s e q u e n t l y , the off-voltage in the d e s c e n d i n g p a r t c a n t a k e v a l u e s f r o m zero to i Vs. P o w e r saving could b e a c h i e v e d if S~ w e r e t u r n e d on at zero off-voltage, b u t this condition could be v e r y difficult to fulfil in p r a c t i c e . F o r the calculations, we shall a s s u m e t h a t the off-voltage h a s the s a m e v a l u e as in the a s c e n d i n g case, i.e. Volt= Vs. ~J'sw iV s
x1 V_V-'j
Li
K
-I
i-Ns)V,
Fig. 6. V~, waveform for switch i in the positive half-cycle.
567
The voltage ratings of the p o w e r switches could be d e t e r m i n e d f r o m eqn. (15), i.e. the m a x i m u m positive voltage Vsim=i ITs since Vo=O and the m a x i m u m value o f the negative voltage is given by
V~m = [ ( i V s - N s V s ) l Now it is required to d e t e r m i n e the o n - c u r r e n t in the switches for the positive half-cycle as was d o n e for the voltage. In the on-state, the switch will be m o d e l l e d by the circuit s h o w n in Fig. 7, where, Rsw is the on-resistance o f the switch and Vswo is the cut-in voltage o f the switch. Both are c o n s i d e r e d i n d e p e n d e n t o f the c u r r e n t in the switch. Let us a s s u m e that the load r e s i s t a n c e is RL and the internal r e s i s t a n c e o f the chemical b a t t e r y is Rs. The o n - c u r r e n t Ioni in the ith switch can b e e x p r e s s e d by
iVs-3Y~o
(16)
Ioni = iRs + 3R~,, +RL with
Rs =RBNB/Ns
(17)
w h e r e NB is the total n u m b e r o f the b a t t e r i e s and Rs is the resistance of a b a t t e r y stage. This e q u a t i o n is valid only for the s t e p p e d sinewave full wave inverter. F o r the tri-state inverter, i.e. Ns = 1 the o n - c u r r e n t is given by: Zon =
V s - 2V~wo
(18)
Rs + 2 R ~ + R L
The voltage a c r o s s switch S~ in the on-state follows directly f r o m the equivalent circuit o f Fig. 7, i.e.
Voni = V~o +IoniRo,
(19)
Now the v o l t a g e s and c u r r e n t s can be substituted in eqn. (14) to find the p o w e r c o n s u m e d in switch i, and b y s u m m i n g o v e r all the switches we obtain the total p o w e r loss Pst. F o r the tri-state inverter it follows t h a t Psi = 4 Von/on ~ - -
+ Von/on - ~
(20)
with /on =
V - 2V~o Rs + 2 R ~ + R L !
•
AAA~
Ion
Ron~Rsw
II !
:
Von~Vswo
Fig. 7. Equivalent circuit o f t h e p o w e r s w i t c h in the o n state.
568
Von= Vswo+IonRo, V Yof f --
2
ton~2 =X1 is the half-pulse width The power lost in the stepped sinewave inverter is consequently given by [3]:
Pst=
-~- V o n i l o n i - ~
2
--"
--
with Vo,= V~ and toni/T=Xi/vr. Using eqns. (2a), (6) and (20) or (21) we can calculate the loss efficiency.
2.2. Amp~re hours from each battery Since the voltage of the batteries is constant, the ampere hour supplied by each battery in one complete cycle is the integration of its discharge current over its operating time. The battery stage designated i operates in the positive half-cycle for a period corresponding to N~
iLl+ ~ L i i+l
and in the negative half-cycle for the period corresponding to Ns
(Ns-i+l)Lgs-i+l+
~
Li
Ns-i+2
It follows that the total operating time of the ith battery stage corresponds to Ns
LT(i)=iL,+ ~ L i + ( g s - i + 1) Ln~-i+l i+I Ns
+
~
L~
(22)
Ns--i+2
We are interested in the ratio DSR1 (i) of the discharge period of the battery stage i to the stage i = 1, i.e.
Lr(i)
DSRI(i) = - LT(1)
(23)
3. C o m p u t e r - a i d e d a n a l y s i s o f t h e i n v e r t e r In this section we present the results of the analysis introduced in the previous section for the stepped sinewave full-bridge inverter. The harmonic
569 efficiency ~h will be maximized for a given num ber of stairs and total battery voltage V by finding the optimum widths of the stairs. Under optimum conditions, the fundamental c o m p o n e n t of the output, the energy delivered by each battery and the loss efficiency will be determined. In addition, the sensitivity of the efficiency to variation in stair widths will be defined. This is especially important for the design of the control circuit of the power switches since the accur a c y of the timing circuit could largely affect the stair widths and consequently the efficiency. We also need to calculate the harmonic c o m p o n e n t s and their overall effective value.
3.1. Solution m e t h o d It is clear from eqn. (8) that the harmonic efficiency ~/l, is a non-linear function o f the widths LI, Lz, ... Li . . . . L N s . Therefore, to determine the maximum efficiency, an optimization technique is needed for such a multidimensional non-linear optimization problem. A numerical method based on a QUASI-NEWTON algorithm [4] was utilized. It is based on nulling the Jacobian matrix, followed by inspection of the second derivative matrix for an extremum. A FORTRAN subroutine ZXMIN based on this algorithm as a part of the IMSL Library is available. We developed a FORTtLAN program to calculate the following quantities: the optimum widths of the stairs Li, the widths of the pulses Xi, the harmonic efficiency ~?h, the d e p e n d e n c e of the harmonic efficiency on the stair widths, i.e. the sensitivity of the efficiency to variations in the stair widths, the relative amplitudes of the fundamental RI=Vo~/VsN~, the total harmonic content, the conduction period of each pow e r switch, the maximum forward and reverse voltages on each power switch, the pulse widths of the control pulses for the p o w er switches, the loss efficiency, the overall efficiency and the energy supplied by each battery. The following data must be available: (i) the n u m b e r of batteries NB, (ii) the voltage of each battery VB, (iii) the internal resistance of the battery Rm (iv) the num ber of stages Ns, (v) the frequency of the output voltage, (vi) the load resistance RL, (vii) the onresistance o f the switch Rsw and its cut-in voltage V~o, (viii) the rise and fall-times of the pow e r switch, tr, tr, (ix) the control m ode CM, (x) the nominal discharge current of the battery Im (xi) the maximum on-current of the p o wer switch which must be equal to or greater than IB. The flow chart of the program developed for computer-aided analysis o f the inverter with the above-mentioned inputs and outputs is shown in Fig. 8. 3.2. Results In this section we present the most interesting results of the computeraided analysis in or de r to work out the optimum operating conditions of an inverter havingNB = 14, VB= 12 V, RB= 0.01 12, the nominal discharge current o f the battery I B - - 10 A and the f r equency of the output f = 5 0 Hz. At first the minimum load resistance is calculated from RL=NBVB/IB. The m a x i m u m on-current of the power switch is IB and its maximum off-voltage in either
570
I ~.... I l output fund. f r e q . , the load r e s i s t a n c e , number of a v a i l a b l e b a t t e r i e s and t h e i r characteristics
(a)
l
>
L
Loop: Control mode # i to 3 [ i
t
(b)
Loop: (Ns " I to NB)
I I
Calculate: B a t t e r y s t a g e v o l t a g e I and r e s i s t a n c e , V$ and Rs
L
f t Calculate the initial values of the parameters ( X t . t ' I , 2 , . . , , N s}
.I Form the function of the' harmonic efficiency
E
i Call "ZXMIN" To: find the "mini~" of the previous funetlon
Calculate: i. The optimum 2. The optimum 3. The optimum i
stair width L i delay A i pulse width X I - 1,2,3,...,N s
(in rad.) (in tad.) (in tad.)
Calculate The pulse width t i (~n msec.) I - 1,2,3,...,N B
(c)
I I C a l c u l a t e the optimum p a r a ~ t e r s
[
(~L' ~W ~' Rl' aC, krl)
! I
i . Calculate the s e n s i t i v i t y
of q to the l v a r i a t i o n of Lt, 1 - 1 , 2 , 3 , . . . . , Ns 2. C a l c u l a t e the s e n s i t i v i t y of r e l a t i v e fundamental v o l t a g e R1 t o the v a r i a t i o n of L i , i " 1,2,3,..., Ns
(b) ~
(a) •
b
[
L~***.~..... ~*'*~*'~"1 1,
l ~,o, 1
Fig. 8. n o w chart of computer-aided analysis of the inverter.
571
polarity is V o , = N B V B. T h e r e are m a n y p o w e r e l e c t r o n i c devices which can be u s e d as switches in the p o w e r switching circuit, e.g. bipolar p o w e r transistors, gate-turn-off thyristors and VMOS transistors. T h e i r switching times are m u c h less t h a n their on-time. A typical 10 A bipolar t r a n s i s t o r has Vs~o=0.1 V, an o n - r e s i s t a n c e Rsw=O.07 12 and t r = t r = 5 0 ~s. 3.2.1. T h e i n v e r t e r ejTiciency Figure 9 shows the m a x i m u m h a r m o n i c efficiency as a f u n c t i o n o f the n u m b e r of stairs Ns k e e p i n g the overall b a t t e r y voltage c o n s t a n t for the t h r e e control m o d e s t e r m e d l , 2 and 3. It is clear f r o m this figure that as N~ increases, the h a r m o n i c efficiency b e c o m e s larger, a p p r o a c h i n g unity for Ns > 6. C o n s e q u e n t l y it is n e c e s s a r y to m a k e N~ h i g h e r for l o w e r costs. This is valid for the t h r e e control m o d e s . This can b e easily u n d e r s t o o d b e c a u s e as the step size d e c r e a s e s , the synthetic sinewave a p p r o a c h e s a p u r e sinewave with f u n d a m e n t a l f r e q u e n c y . As h a r m o n i c s are generally related to step size, t h e y are related to the individual b a t t e r y voltage and the p e a k voltage. The h a r m o n i c efficiency ~?h also d e p e n d s on the control m o d e . It has the highest values for the free c o n t r o l m o d e as the n u m b e r of w a v e f o r m adjusting p a r a m e t e r s is the largest. It is interesting to n o t e that as N~ i n c r e a s e s the difference in the efficiency decreases, a p p r o a c h i n g a l m o s t the s a m e value for N~>_6. The loss efficiency is also s h o w n in Fig. 9 as a function of the n u m b e r o f stairs N~ for the different control m o d e s . This result is v e r y interesting. W e see from the figure that the loss efficiency d e c r e a s e s at first slightly with N~, then b e c o m e s constant. It is m o r e o r less i n d e p e n d e n t o f the n u m b e r
I
96
.96
"G
~-.94
.92
.9
0
8
lo
1'2
Number of Battery Stages (NS) Fig. 9. ~?h, ~?L a n d 7; v s . N s for different
control modes
CM1, CM2 a n d CM3.
14
572 o f stages. This is attributed to the negligibly small d y n a m i c loss which c a u s e s the d e p e n d e n c e o f ~?L on/Vs. W e can also see f r o m the figure, that in c o n t r a s t to the h a r m o n i c efficiency, which i n c r e a s e s f r o m m o d e 3 to m o d e 1, the loss efficiency d e c r e a s e s f r o m m o d e 3 to m o d e 1. As a result the overall efficiency s h o w n in the same figure is nearly i n d e p e n d e n t of c o n t r o l mode. It i n c r e a s e s f r o m 0.9 for the bridge inverter w h e r e N~ = 1 to a plateau value of 0 . 9 6 5 at N~>_6. T h e s e values are in g o o d a g r e e m e n t with the m e a s u r e d efficiencies o f 0.88 for N~= 1 b y Anis [ l l and 0.94 for N~-= 16 by Schmitt and Schaetzle [2]. The slightly lower m e a s u r e d efficiencies are a result of losses in their p o w e r switches being larger t h a n t h o s e c o n s i d e r e d here. A c c o r d i n g to these results, one has to k e e p N~ a r o u n d 6 and o p e r a t e the i n v e r t e r in control m o d e 3.
3.2.2. D e p e n d e n c e o f the total eJficiency on the d e v i a t i o n s o f s t a i r w i d t h s f r o m their o p t i m u m values This is an interesting p a r a m e t e r for the design o f a c o n t r o l circuit for the p o w e r switches. It d e t e r m i n e s h o w a c c u r a t e the pulse widths o f the c o n t r o l signals m u s t be. T h e r e f o r e , it could affect the cost o f p r o d u c i n g t h e s e pulses if circuit c o m p o n e n t s with larger t o l e r a n c e s can be utilized. Figure 10 shows the total efficiency ~? as a f u n c t i o n of the relative deviation in Li f r o m their o p t i m u m values for different n u m b e r s of stairs for c o n t r o l m o d e s 1, 2 and 3. Fortunately, it s e e m s that ~ has a b r o a d m a x i m u m . This is m o r e p r o n o u n c e d for the efficiency c u r v e s o f c o n t r o l m o d e 3 (Fig. 10(c)). If one allows a 0.2% d e c r e a s e f r o m the m a x i m u m efficiency one can d e t e r m i n e f r o m t h e s e c u r v e s the permissible p e r c e n t a g e deviation in stair widths. H o w e v e r , it is clear f r o m t h e s e figures that a deviation o f a b o u t 10% in L, c o u l d b e allowed without any significant loss in efficiency. T h e r e f o r e , high p r e c i s i o n timing c o m p o n e n t s in the control signal g e n e r a t o r are n o t needed. 3.2.3. The F o u r i e r c o m p o n e n t s of the output, the f u n d a m e n t a l a n d the h a r m o n i c content T h e relative m a g n i t u d e of the f u n d a m e n t a l and the total h a r m o n i c c o n t e n t are s h o w n in Fig. 11 for different values of N~ u n d e r different c o n t r o l m o d e s . F r o m Fig. l l ( a ) we can o b s e r v e t h a t as Ns increases, the f u n d a m e n t a l c o m p o n e n t d e c r e a s e s slowly. T h e r e f o r e , one n e e d s a slightly larger overall b a t t e r y voltage for the s a m e o u t p u t voltage w h e n increasing Ns. T h e situation is e v e n w o r s e for c o n t r o l m o d e s 1 and 2 since their f u n d a m e n t a l c o m p o n e n t s are smaller t h a n t h a t of m o d e 3. It is clear f r o m Fig. 11 (b) that the h a r m o n i c c o n t e n t d e c r e a s e s as e x p e c t e d b y increasing the n u m b e r o f stairs. Control m o d e 1 results in the lowest h a r m o n i c content. F o r N s = 6 the h a r m o n i c c o n t e n t s are 6.1%, 6.8% and 9.1% for control m o d e s 1, 2 and 3 respectively, w h i c h c o u l d be a c c e p t a b l e for m a n y loads. B e c a u s e o f the slow d e c r e a s e in h a r m o n i c c o n t e n t for Ns > 6, one would have to i n c r e a s e N~ excessively to r e d u c e the h a r m o n i c s further. In fact, the h a r m o n i c c o n t e n t is inversely p r o p o r t i o n a l to /Vs. If h a r m o n i c c o n t e n t less t h a n t h o s e c o r r e s p o n d i n g to
573 n% I00,
%
100.
Total 98. efficiency
9aI 961
Ns=6 Ns=4
941
Ns=2
~ N s = 2
92"
92 NS:I
-0.2
-0.I
0
o.1
,
o.z
-olz
Fr= ALi/L i
-o:1
90 011
O.J2
Relative deviation of the step width Fr: ALi/L i
(a)
(b) i00 ,n % (Total efficiency) 98 _
~Ns=6 Ns:4 Ns:2
92
9O
-o:1 (c)
o
o11
o:2
Fr = ALi/L i
Fig. 10. Total efficiency as a function of control mode 2, and (c) control mode 3.
(ALJL,) at different N~: (a) control mode 1, Co)
Ns= 6 are required, it w o u l d b e m o r e e c o n o m i c a l to u s e filters. F o r filter design, the relative a m p l i t u d e s of the h a r m o n i c s are given in T a b l e 1 for t h e different control m o d e s . W e c a n see t h a t the a m p l i t u d e s o f the h a r m o n i c s a r e relatively l a r g e a n d d e c a y slowly with the o r d e r o f the h a r m o n i c n. In addition, the h a r m o n i c s a l t e r n a t e in sign.
3.2.4. Energy delivered by each battery B e c a u s e w e are i n t e r e s t e d in t h e relative d i s c h a r g e e n e r g y a n d n o t in a b s o l u t e values, t h e d i s c h a r g e t i m e s o f all b a t t e r i e s are r e l a t e d to the o p e r a t i n g t i m e of t h e b a t t e r i e s in s t a g e one. This ratio is t e r m e d DSRi. It is c a l c u l a t e d f r o m eqns. (22) a n d (23) a n d p l o t t e d in Fig. 12 f o r the different c o n t r o l m o d e s t a k i n g the n u m b e r o f s t a g e s Ns as 6. F o r all c o n t r o l m o d e s , t h e b a t t e r i e s in t h e m i d d l e o f the b a t t e r y string deliver m o r e e n e r g y t h a n t h o s e a t t h e e n d s o f t h e string. F o r c o n t r o l m o d e 2 t h e distribution o f t h e d i s c h a r g e e n e r g y is b e t t e r t h a n f o r m o d e 1, a n d is the b e s t for c o n t r o l m o d e 3. H e n c e c o n t r o l m o d e 3 is a d v a n t a g e o u s f r o m this r e s p e c t o v e r the o t h e r p r o p o s e d control modes.
3.2.5. Pulses f o r the control circuit T h e f u n c t i o n o f the c o n t r o l circuit is to switch the p o w e r s w i t c h e s in t h e i n v e r t e r on a n d off in the p r o p e r t i m i n g s e q u e n c e to g e n e r a t e the s t e p p e d
574 1.1702
1.1315-
Y:
1.0930 \
\
\
~ CM3
1.0544
5
CMI
\
1.015B
CM2
.97721 o
g~
.93860
.9
5
0 (a)
10
15
Number of Battery Stages (NS)
.3
.25 -
.2 ,3 .15 ,.?o c
cM3___Z___ c.~,K
.05
0 0 (b)
J
I
I
I
i
i
2
4
6
8
10
12
4
Number of Battery Stages (NS)
Fig. 11. (a) R a t i o o f f u n d a m e n t a l c o m p o n e n t to t h e total b a t t e r y v o l t a g e R1 a s a f u n c t i o n o f t h e n u m b e r o f t h e b a t t e r y s t a g e s Ns. (b) T h e overall h a r m o n i c c o n t e n t vs. N . for different control modes.
N,=I
1.17007 -0.14744 -0.11211 0.17354 -0.12371
1.17023 -0.14783 -0.11173 0.17341 -0.12392
1.17023 -0.14783 -0.11173 0.17341 -0.12392
n
CM1 1 3 5 7 9
CM2 1 3 5 7 9
CM3 1 3 5 7 9
1.11256 -0.04488 -0.13217 0.07553 -0.00594
0.95898 0.04557 0.09665 -0.06323 -0.09836
1.09490 -0.04255 -0.05591 - 0.03537 0.03706
N,=2
1.08732 -0.01419 -0.11153 0.03175 0.00188
0.93840 0.09114 0.04224 -0.06933 0.01411
1.06497 - 0.01958 -0.02745 - 0.02955 -0.01342
Ns=3
1.07408 -0.00192 -0.09570 0.01301 0.00044
0.94322 0.09762 0.01030 -0.04594 0.01439
1.04931 -0.01139 -0.01579 - 0.01949 -0.01668
Ns---4
1.06595 0.00431 -0.08474 0.00324 -0.00119
0.94816 0.09605 - 0.00311 -0.03117 0.00466
1.03931 -0.00729 -0.01002 - 0.01290 -0.01381
Ns=5
1.06113 0.00716 -0.07709 -0.00234 - 0.00214
0.95519 0.09138 -0.01089 -0.02348 -0.00330
1.03287 -0.00508 -0.00702 -0.00927 - 0.01087
Ns=6
1.05688 0.00988 -0.07119 - 0.00629 -0.00308
0.95868 0.08794 - 0.01372 - 0.01908 -0.00794
1.03053 - 0.00490 -0.00641 - 0.00761 - 0.00870
N~=7
1.05421 0.01115 -0.06683 - 0.00889 -0.00357
0.96950 0.07904 -0.01889 -0.01604 -0.01343
1.02485 -0.00275 -0.00430 - 0.00553 - 0.00645
Ns=8
1.05212 0.01208 -0.06337 -0.01082 - 0.00392
0.97166 0.07654 -0.01913 - 0.01537 - 0.01444
1.02271 -0.00261 -0.00326 - 0.00429 - 0.00564
N~=9
The r a t i o Rn of th e h a r m o n i c a m p l i t u d e s r e l a t i v e to t h e t o t a l b a t t e r y v o l t a g e N , Vs for di ffe re nt c o n t r o l m o d e s CM1, CM2 an d CM3
TABLE 1
to
1.2902 0.7145 0.4937 0.3754 0.3051 0.2572 0.2214 0.1956 0.1675
1.2892 0.7189 0.4576 0.3191 0.2284 0.1692 0.1159 0.0826
1.2892 1.0264 0.8115 0.6642 0.5604 0.4833 0.4254 0.3793 0.3421
Ns
CM1 1 2 3 4 5 6 7 8 9
CM2 1 2 3 4 5 6 7 9
CM3 1 2 3 4 5 6 7 8 9
14.843 1.0264 0.8115 0.6642 0.5604 0.4833 0.4254 0.3793 0.3421
14.8433 1.6077 1.1473 0.8830 0.7186 0.6031 0.5213 0.4060
7.4195 1.6102 1.0396 0.7782 0.6221 0.5203 0.4445 0.3892 0.3497
tl
11.789 0.8115 0.6642 0.5604 0.4833 0.4254 0.3793 0.3421
7.1387 1.1473 0.8830 0.7186 0.6031 0.5213 0.4060
5.3506 1.2765 0.8577 0.6602 0.5355 0.4555 0.3994 0.3574
t2
10.2615 0.6642 0.5604 0.4833 0.4254 0.3793 0.3421
6.2320 0.8830 0.7186 0.6031 0.5213 0.4060
4.3805 1.0907 0.7485 0.5903 0.4832 0.4173 0.3654
t3
9.3732 0.5604 0.4833 0.4254 0.3793 0.3421
0.6031 0.5213 0.4060
0.7186
5.8719
3.7957 0.9705 0.6711 0.5304 0.4445 0.3880
t4
8.7921 0.4833 0.4254 0.3793 0.3421
5.6274 0.6031 0.5213 0.4060
3.3870 0.8785 0.6085 0.4900 0.3969
t5
8.4016 0.4254 0.3793 0.3421
5.5267 0.5213 0.4060
3.0944 0.8015 0.5653 0.4611
t6
8.0890 0.3793 0.3421
5.4039 0.4060
2.9099 0.7566 0.5475
t7
7.8634 0.3421
0.4060
2.6840 0.6903
t8
7.6836
5.3846
2.5525
t9
The o p t i m u m pulse widths t~ in milliseconds as a function of the n u m b e r of battery s t a g e s Ns for different control m o d c s CM1, CM2 and CM3
TABLE 2
577 1.16
CMll
1.14 .o
I . 12
1.10 1.08
~-
1.06
g
1.04
=~
i.o2
1. 001 2
3
4
6
Number of battery stages
Fig. 1 2. Discharge amp~re-hour ratio DSRi of any battery stage I relative to battery stage 1
for a number of battery stages Ns= 6 at different control modes.
sinewave. O n c e t h e f r e q u e n c y o f t h e a.c. o u t p u t v o l t a g e a n d t h e p o w e r switch a r e d e t e r m i n e d o n e c a n easily d e d u c e the driving p u l s e s for e a c h switch in t h e inverter. T h e d e v e l o p e d c o m p u t e r p r o g r a m s u p p l i e s t h e d a t a r e q u i r e d to design the c o n t r o l circuit. T a b l e 2 gives c o m p l e t e t i m i n g i n f o r m a t i o n at different v a l u e s o f Ns f o r c o n t r o l m o d e s 1, 2 a n d 3 a s s u m i n g f = 5 0 Hz. As a d e m o n s t r a t i o n , t h e c o n t r o l p u l s e s b a s e d on t h e s e t a b l e s are s h o w n in Fig. 13 f o r Ns = 6. Idp is the driving p u l s e f o r Sp~ a n d Sp2 while Idn is t h e driving p u l s e f o r Sn~ a n d Sn2. Id, t o Id6 drive the s w i t c h e s S~ to $6 while IdT to Ida2 c o n t r o l t h e s w i t c h e s SNs+ 1 to Sans in t h e i n v e r t e r circuit in Fig. 1.
4. Conclusion
In this w o r k an i n v e r t e r f o r a p h o t o v o l t a i c p o w e r s y s t e m w a s a n a l y s e d in detail. It w a s f o u n d t h a t the full b r i d g e i n v e r t e r circuit is p r e f e r a b l e b e c a u s e o f the r e d u c t i o n in t h e n u m b e r o f s t o r a g e b a t t e r i e s , w h i c h are a v e r y e x p e n s i v e e l e m e n t . In o r d e r to e l i m i n a t e o r to r e d u c e t h e filtering c o m p o n e n t s a n d the t r a n s f o r m e r s , w e u s e d a s t e p p e d s i n e w a v e inverter. A c o m p u t e r p r o g r a m w a s d e v e l o p e d f o r a n a l y s i s a n d d e s i g n o f this inverter. T h e h a r m o n i c efficiency i n c r e a s e s with the n u m b e r o f stairs u p t o six stairs a n d a f t e r t h a t it s a t u r a t e s . T h e f u n d a m e n t a l c o m p o n e n t s will b e slightly r e d u c e d w h i c h m e a n s a n i n c r e a s e in t h e n u m b e r o f b a t t e r i e s f o r the s a m e o u t p u t v o l t a g e f o r h i g h e r Ns values. T h e h a r m o n i c d i s t o r t i o n d e c r e a s e s with Ns to a l l o w a b l e v a l u e s f o r m o s t l o a d s if Ns >_ 6. T h e d e p e n d e n c e o f ~/ o n t h e relative d e v i a t i o n o f t h e stair w i d t h s f r o m t h e i r o p t i m u m v a l u e s s h o w s a v e r y slight d e c r e a s e in ~? f o r 10% deviation. This r e l a x e s t h e p r e c i s i o n o f t h e t i m i n g c o n t r o l circuit.
578
I°°l Idn
I
I T/2 tI tI
Idl'id71h
Fl,fl T/2 t2 t2 N,n
Id2JdI fl
fl
A dlI
]dS"IdllI
t2
N
, _
T/2
Id3'Id91 t3 I d4' I
l_ T
tn
tn
A
A
T/2
t5
i
t5
R
I
A tn
i
R t5
t5
~
R
T/2
t6
Id6' Zdl21
n t n
T/2
V]
t3
t3
J
I
T>~
t6
l
I
i
Fig. 13. The timing diagram of the driving pulses of the switches for N , = 6.
The main characteristic o f p o w e r s w i t c h e s suitable for this application w e r e w o r k e d out. It is preferable to u s e fully c o n t r o l l e d d e v i c e s with g o o d s w i t c h i n g p e r f o r m a n c e to d e c r e a s e the lost p o w e r in the circuit. The l o s s efficiency w a s d e t e r m i n e d and it w a s f o u n d that it is nearly i n d e p e n d e n t o f t h e n u m b e r o f s t a g e s but varies with t h e c o n t r o l m o d e o f t h e p o w e r switches. The overall efficiency o f t h e inverter i n c r e a s e s with Ns, r e a c h i n g a c o n s t a n t v a l u e for Ns > 6. G o o d a g r e e m e n t w a s f o u n d b e t w e e n our c a l c u l a t e d v a l u e s for the efficiency and p u b l i s h e d e x p e r i m e n t a l results. Further studies are required c o n c e r n i n g t h e i n t r o d u c t i o n o f interruption p e r i o d s b e t w e e n t h e p u l s e s and their effect o n the p e r f o r m a n c e o f the inverter. Capacitive and inductive l o a d s w e r e n o t treated here. In addition, the inverter circuit m u s t be p r o v i d e d with f r e q u e n c y and v o l t a g e c o n t r o l as well as p r o t e c t i o n e l e m e n t s . A practical inverter circuit is u n d e r d e v e l o p m e n t to e v a l u a t e it u n d e r real w o r k i n g c o n d i t i o n s .
Acknowledgment The a u t h o r t h a n k s Eng. M. El Znaedi for the p r o g r a m m i n g .
579
References 1 W. Anis, Analysis of a three-level bridge inverter for photovoltaics, Sol. Cells, 25 (1988) 255-263. 2 J. Schmitt and R. Schaetzle, Simple transformerless inverter with automatic grid, tracking and negligible harmonic content for utility interactive photovoltaic systems, Proc. 4th European Community Photovoltaic Solar Energy Conf., Tresta, I982, Reidel, Dordrecht, 1982, pp. 3 1 6 - 3 1 9 . 3 M. El-Znaedi, Computer-aided analysis of inverters for photovoltaic power system, Project Report, 1989 (Electrical Engineering Department, College of Engineering, King Saud University, Riyadh). 4 FORTRAN Subroutines f o r Mathematics and Studies, Vol. 4, 1985, edn. 9.2, International Mathematical and Statistical Libraries, Houston, TX.