METHOD FOR CALCULATING THE ENERGY CHARACTERISTICS AND SOLAR BATTERY PARAMETERS OF HIGH-VOLTAGE POWER SUPPLY SYSTEMS

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Introduction. Ensuring the long term of the space- craft (SC) active lifetime (TAL) is one of priority scien- tific, technical and economic tasks which solution de- mands a comprehensive fundamental approach during the development and creation of SC power supply systems (PSS). The main primary energy source in SC PSS are solar batteries (SB) designed on the basis of Si or GaAs solar cells having non-linear I-V and V-W characteristics with a pronounced maximum of the generated power, deter- mined by an operating voltage level that proves the expe- diency of maximum power point tracking (MPPT) mode of SB in PSS application [1-4]. Due to the increase of output voltage of the SC load power bus up to 100 V, new technical requirements to the ways of coordination and service conditions of energy sources in PSS have been created. This is due to the possibility of electrostatic dissharp-variable load curve and the significant number of shadow areas on the orbit. Therefore for the correct plan- ning of PSS target equipment operation it is expedient to carry out the calculation of SB generated power in all operating modes of SC PSS during TAL. The SB current parameters, taking into account the graphs of their illumination F and temperature t, are cal- culated on the experimental solar cells I-V characteristics of any area provided by their manufacturers. The SB mathematical model set by three points is the basis for the developed mathematical model: of an open circuit (OC) voltage UOC, short circuit (ShC) current IShC, optimum values of SB current IMPP and voltage UMPP [8]. The SB mathematical model depending on temperature t and il- lumination F is described as ISB (USB , t, F ) = IShC (t, F ) ´ charges between photodiodes chains and current collec- é UOC (t ,F )-USB ù tion elements of the SB and, as a result, to the need for the ´ ê1- æ1- IMPP (t, F ) öUOC (t ,F )-UMPP (t ,F ) ú , (1) maximum level of SB open circuit (OC) voltage operation limitation reached when the SC leaves the orbit shadow ç ø ê ëê è IShC (t, F ) ÷ ú úû areas [5-7]. Application of other solar cells types in PSS is possible in case of feasibility of their use realized by means of calculation of the generated by SB capacities during TAL providing an energy balance in PSS, rational distribution of energy flows and prevention of PSS work emergency operation. The developed technique provides the reduction in calculation time of power characteristics and the SB pawhere USB - current value of SB voltage; IShC(t,F) - SB short circuit current; IMPP(t,F) - optimum value of SB current on I-V characteristics at SB MPPT mode in PSS; UMPP(t,F) - optimum value of SB voltage on I-V charac- teristics at SB MPPT mode in PSS; UOC(t,F) - SB open circuit voltage. The SB open circuit voltage depending on t and F is calculated by the formula rameters of high-voltage SC PSS which is obtained by the decrease in the number of iterations directly proportional UOC (t, F ) = kUn ×UOC (t , F ) × (1+ 0, 01×b)(t-t1 ) , (2) 1 1 to the quantity of correction factors to be calculated. The technique can be applied when calculating options of ground and onboard PSS with various load curves and service conditions for the purpose of realization of energy flows in PSS rational redistribution and to impose its ECE and SB design requirements. Mathematical model of solar battery. The SB elec- tric parameters during SC service change significantly. where kUn - the correction coefficient of voltage deter- mined by SB I-V characteristics abscissa axis, considering its illumination influence; F1 - SB nominal illumination value; UOC(t1,F1) - SB open circuit voltage at nominal values t1 and F1; β - temperature coefficient of SB open circuit voltage. The SB short circuit current depending on t and F is determined by the formula The considerable influence on SB I-V and V-W character- istics is rendered by temperature and illumination. So, in IShC (t, F ) = kIn × IShC (t , F ) × (1+ 0, 01×a)(t-t1 ) , (3) 1 1 low-orbit SC the SB panels temperature changes in the range from -90 to +80 0С, herewith the SC can have the where kIn - the correction coefficient on current deter- mined by SB I-V characteristics ordinate axis, considering its illumination influence; IShC(t1,F1) - SB short circuit k = UOCn+1 (t1, Fn+1 ) + current at nominal values t1 and F1; α - temperature coefficient of SB short circuit current. Un U OC1 (t1, F1 ) The optimum value of SB current is calculated taking into account t and F as follows éæ UOCn (t1, Fn ) - UOCn+1(t1, Fn+1) ö ù êç ÷ ú U (t , F ) U (t , F ) + êç OC1 1 1 OC1 1 1 ÷ × (F - Fn+1 )ú . (6) k × I (t , F ) × (1+ 0, 01×l)(t-t1 ) êç Fn - Fn+1 ÷ ú MPP ê ú I (t,F ) = In MPP 1 1 , (4) (1+ 0, 01×n)(t-t1 ) ç ÷ ëè ø û where IMPP(t1,F1) - optimum value of SB current at t1 and F1; λ - temperature coefficient of SB maximum generated power, i. e. power in an optimum point (MPPT mode); ν - temperature coefficient of the optimum value of SB voltage. Correction coefficient of current kIn is determined in the same way by an abscissa axis of the experimental so- lar cells I-V characteristics and taking into account the use of solar cells short circuit current values: k = IShCn+1 (t1, Fn+1 ) + The optimum value of SB voltage taking into account t and F is calculated according to the formula In I ShC1 (t1, F1 ) U (t, F ) = k ×U (t , F ) × (1+ 0, 01×n)(t-t1 ) , (5) éæ IShCn (t1, Fn ) - IShCn+1 (t1, Fn+1 ) ö ù MPP Un MPP 1 1 êç I (t , F ) I (t , F ) ÷ ú where UMPP(t1,F1) - value of SB optimum voltage on I-V + êç ShC1 1 1 ShC1 1 1 ÷ × (F - Fn+1 )ú . (7) ê ú êç Fn - Fn+1 ÷ ú characteristics at t1 and F1. Correction coefficient of voltage kUn is determined as follows: 1. On the experimental solar cells I-V characteristics of any single area (fig. 1), obtained at different illumina- tion levels F1…Fn of solar cells and at some nominal temperature t1, values of solar cells open circuit voltages UOC1(t1,F1) … UOCn(t1,Fn) are determined by the abscissa axis. 2. The correction coefficients kU1...kUn, reflecting the relative change of solar cells open circuit voltage which is in the range between UOCn(t1,Fn) and UOCn+1(t1,Fn+1) de- pending on levels of their illumination are calculated ac- cording to the formula: ç ÷ ëè ø û Calculation method of spacecrafts high-voltage power supply systems. Calculation of SB parameters begins with the choice of SC PSS structure and formation of the mathematical description of PSS operation modes taking into account the ECE coefficients of energy effi- ciency (CE) by researching the processes of energy flows in PSS distribution depending on a ratio of generated by energy sources and power consumption load [9-11]. For example, for parallel-serial (PS) PSS the current values of SB power (РSB), AB charge power (РAB_C), AB discharge power (РAB_DC) and load power (РVR) are calculated ac- cording to tab. 1. Fig. 1. I-V characteristics of the solar cell Рис. 1. Вольт-амперные характеристики ФЭ Table 1 Current values of SB, AB and load power in PSS Load power supply mode from SB (VR) PLOAD = PSB × hVR Load power supply mode from SB and AB charge (VR + C) P = PLOAD + PAB_C SB h h VR C Load power supply mode from SB and AB discharge (VR + DC) PLOAD = PSB × hVR + PAB_DC × hDC × hAB Load power supply mode from AB (DC) PLOAD = PAB_DC × hDC × hAB The following designations are introduced in tab. 1: ηVR - voltage regulator CE, ηC - charger CE, ηDC - dis- charger CE, ηAB - AB efficiency. The load curve of SC for some given period of time of T which, as a rule, has a cyclic character is formed. Calculation of total energy WLOAD_sum consumed by load during T is carried out ac- cording to the formula: Подпись: n WLOAD_sum = åd ti × PLOAD (ti ) , (8) i=1 where n - the number of areas on Т, during which the current value of load power PLOAD(τi) is invariable; i - ordinal value of T n-area; dτi - is a time period on T dur- ing which PLOAD(τi) remains invariable. Load curves of SB illumination F(τj) and temperature t(τj) are formed. Initial values of solar cells parameters of any single area at nominal parameters of temperature and illumination are introduced: IMPP(t(τ1), F(τ1)), UMPP (t(τ1), F(τ1)), UOC(t(τ1), F(τ1)) and IShC(t(τ1), F(τ1)), α, ν, λ и β. Further calculation of solar cells parameters current values taking into consideration SB F(τj) and t(τj) curves according to formulas (2)-(7) is carried out and calculawhere r - the number of areas on Т, in which PSS oper- ates in modes VR + DC or DC; b = 1…r - r-area ordinal number on Т; UAB_DC(τb) - AB discharge voltage in modes VR + DC or DC; s - the number of areas on Т, in which PSS operates in VR + C or VR modes; c = 1... s - s-area ordinal number on Т; UAB_C(τc) - AB voltage in the charge mode. AB power characteristics, including the current values of AB charge and discharge currents and AB nominal power, counted taking into consideration the accepted maximum AB charge depth, are calculated [1; 4; 11]. If discharge/charge currents of AB do not meet the technical requirements to AB, ECE and PSS in general, then their correction is implemented at the given level by calculation of correction coefficient kW2 which provides proportional change of PSB(t(τk),F(τk)) on T areas with SB maximum generated power for the purpose of providing PSS energy balance. In this case, calculation values of SB current power is carried out on condition: Подпись: p f åd tk × PSB (t(tk ), F (tk )) = åd te × PSB (t(te ), F (te )) + tion of total value of the solar cells generated energy Wsolar cell_sum during T is similar to WLOAD_sum computak =1 q e=1 tional method. The current values of the SB generated power on con- dition of equality of the consumed load and the SB gener- ated power according to the formula (9) by determination of correction coefficient kW as the ratio of WLOAD_sum to Wsolar cell_sum are calculated. The k-areas on the T period corresponding to their invariable current values are de- fined PSB (t(tk ), F (tk )) = Psolar cell (t(tk ), F (tk )) × kW . (9) Calculation of an energy balance in SC PSS is carried out by calculation of the current and total values of energy and discharge QAB_DC_sum and charge QAB_C_sum power of AB calculated taking into account PSS ECE CE, and de- termination of correction coefficient kAB providing pro- portional increase in the current values of SB power for + å d th × PMPP (t(th ), F (th )) × kW 2, (11) h=1 where p - total number of k-reas on Т; f - the number of areas on Т without SB MPPT realization in PSS; e = 1…f - f-area ordinal number on Т; dτe - time period on Т, during which the current value of SB generated power PSB(t(τe),F(τe)) remains invariable in PSS without SB MPPT realization; q - the number of areas with SB MPPT realization in PSS on Т; h = 1…q - q-area ordinal number on Т; dτh - time period on Т, during which cur- rent optimum value of SB generated power PMPP(t(τh),F(τh)) remains invariable in PSS with SB MPPT. The SB parameters in PSS are calculated. The coeffi- cient of proportional increase in the solar cells initial pa- rameters is calculated [11] each k-rea on T, on which the current values of SB gener- ated power are more than zero. The total value of SB genkIU = . (12) erated energy (WSB_sum) is calculated similarly to computa- tional method WLOAD_sum. For example, for PSS PS equation for energy balance calculation is as follows QAB_DC_sum = QAB_C_sum = For restriction of SB actual value of current or voltage level on T the value of the restricted level of the actual parameter value is set, and also the level of illumination and temperature at which this level of restriction should not be broken and the correction coefficient of restriction klim is calculated. For example, in high-voltage SC PSS on the condition of the maximum SB OC voltage level UOC_max restriction the coefficient of restriction klim is calculated as: U (t(t ), F (t )) × k r = × t = klim = OC k k IU , (13) å b=1 d b PSB (t(tc ), F (tc )) × kAB (tc ) ×hC (tc ) - UOC _max where UOC(t(τk), F(τk)) - current value of solar cells OC voltage at minimum values of t and F. Solar cells initial parameters are corrected taking into - s PLOAD (tc ) ×hC (tc ) = å hVR (tc ) × d t , (10) account kIU and klim, what allows to calculate SB parame- ters and their I-V and V-W characteristics accounting с=1 U AB _ C (tc ) с F(τj) and t(τj) curves by application of the developed SB mathematical model. At the same time solar cells non-restricted parameters are multiplied by kIU and klim. The klim placement in the denominator means the SB pa- rameter restriction. It is also necessary to consider coher- ence of solar cells parameters change. For example, at restriction of the allowed maximum level of SB OC volt- age reached at a minimum temperature of its panels the level of SB optimum voltage is also corrected. Results of mathematical modeling of the spacecraft high-voltage power supply system. Calculation of SB power characteristics and parameters was executed for high-voltage parallel-serial SC PSS (100 V) with SB MPPT [10] operating either in the mode of a simultaneous power supply load from the SB and the AB charge, or in the mode of a simultaneous power supply load from SB and the AB discharge. Arbitrarily composed SC load curve PLOAD(τ) and the solar cells generated power curve obtained by using the developed mathematical model of SB taking into account solar cells initial parameters at t1 = 25 0С: IShC(t1,F1) = = 5.83 А, UOC(t1,F1) = 46.2 V, IMPP(t1,F1) = 5.43 А, UMPP(t1,F1) = 37.7 V, β = -0.3, λ = -0.39, α = 0.04, ν = -0.4 and SB temperature t(τ) and illumination F(τ) curves are shown in fig. 2. During the calculation it was obtained that Wsolar cell_sum = 314.33 W·h. For providing an energy bal- ance in PSS at WLOAD_sum = 9883.33 W·h and taking into account the restriction of allowed maximum level of SB OC voltage reached at a SC exit from the Earth’s shadow is 180 V, the correction coefficient kW = 31.443, the cor- rection coefficient kAB = 1.158, the coefficient of propor- tional increase in solar cell initial parameters kIU = 36.403 and the correction coefficient klim = 2.097. The current values of the SB generated capacities PBS_cur are shown in tab. 2. Fig. 2. The load curve PLOAD(τ) and the solar cell generated power curve Psolar cell(τ) in the SC PSS with the MPPT Рис. 2. Графики нагрузки PН(τ) и генерируемой ФЭ мощности PФЭ(τ) в СЭП КА с ЭРМ БС Table 2 Parameters of the solar battery of the spacecraft high-voltage power supply system τk, minute PLOAD_cur, W Psolar cell_cur, W PBS_cur, W UOC(t,F), V IMPP(t,F), А UMPP(t,F), V 0 4000 247.37 9005.03 121.90 93.76 96.04 19 10000 247.37 9005.03 121.90 93.76 96.04 26 4000 247.37 9005.03 121.90 93.76 96.04 35 5500 247.37 9005.03 121.90 93.76 96.04 38 5500 0.00 0.00 - - - 43 6500 0.00 0.00 - - - 62 6500 254.79 14944.29 180 92.55 161.47 63.5 6500 256.32 14096.72 172.09 92.69 152.08 64 6500 257.85 13297.24 164.53 92.83 143.24 65 6500 259.33 12543.09 157.30 92.97 134.92 66.5 6500 260.98 11603.68 148.15 93.15 124.57 68 6500 261.54 10945.59 141.64 93.29 117.33 69.5 6500 260.83 10324.81 135.42 93.43 110.51 71 6500 257.78 9737.02 129.45 93.57 104.06 74 6500 247.37 9005.03 121.90 93.76 96.04 76 16500 247.37 9005.03 121.90 93.76 96.04 81 6500 247.37 9005.03 121.90 93.76 96.04 85 4000 247.37 9005.03 121.90 93.76 96.04 The maximum rated power of a voltage regulator in the channel of energy transformation from the SB and an AB charger are defined by the SB maximum generated power value on T and are equal to 14.94 kW. The maxi- mum rated power of the discharger is defined by the SC maximum power of load, depends on the discharger CE and is equal to 17.37 kW. PSS mass decrease is reached by the development of new circuit realization of SC PSS ECE with the increased values of CE [12-15] and by the research of ways of en- ergy flows rational distribution in PSS. For example, the change of SB MPPT mode use algorithm at the SC exit from shadow areas on the orbit, will allow to reduce the ECE maximum rated power, and, as a result, the mass of PSS in general. Conclusion. The developed solar battery mathemati- cal model is based on the values of initial and experimen- tal parameters of photoelectric cells use and provides calculation of SB I-V and V-W characteristics taking into account arbitrary assigned values of illumination and temperature regardless of their manufacture tech- nology. The offered method of calculation of SB power char- acteristics and parameters in PSS based on the use of cor- rection coefficients provides calculation and the possibil- ity of energy flows redistribution in the system for the purpose of PSS weight-dimension characteristics de- crease, and allows to obtain SB initial parameters, on condition of ensuring an energy balance in PSS, account- ing the given service conditions and load curves of the SC. Based on calculation results, carried out according to the developed technique, the requirements for solar batter- ies and PSS energy-converting equipment designing can be formulated and imposed. The developed method of SB initial parameters calcu- lation allows defining the requirements to their design in PSS taking into consideration the limitation of the actual values of SB currents and/or voltage maximum or mini- mum level.

About the authors

M. M. Chernaya

Tomsk State University of Control System and Radio Electronics

Email: cmm91@inbox.ru
40, Lenina Av., Tomsk, 634050, Russian Federation

References

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