CC BY
157
UDK 621.3.088.7
Siberian Journal of Science and Technology. 2018, Vol. 19, No. 2, P. 281-292
CONTROL PROCESS ABSOLUTE STABILITY ANALYSIS OF CHARGE-DISCHARGE DEVICE WITH LOAD CONVERTER IN CONSTANT POWER MODE
E. A. Kopylov*, D. K. Lobanov, E. A. Mizrakh
Reshetnev Siberian State University of Science and Technology 31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660037, Russian Federation *E-mail: [email protected]
To reduce life time testing period of lithium-ion accumulator (LIA) special dynamic stress test (DST) is widely used. Lithium-ion accumulator dynamic stress test requires automatic charge-discharge devices (CDD) which provides necessary DST technological parameters with required precision. Authors developed charge-discharge devices with load converters (CDD-LC), which allow to reproduce required charge-discharge modes of high-power LIA automatically.
LIA cyclic charge-discharge with constant power pulses is the most difficult mode of DST. In this case, control system became nonlinear and time variant due to computation of signal power as multiply of LIA voltage and current.
Authors studied mathematical model of electromagnetic processes of CDD-LC in LIA power stabilization mode, formulated requirements to power stabilization control loop quality parameters, synthesized correction devices providing necessary control quality, studied CDD-LC control process absolute stability with Naumov-Tsypkin in LIA power stabilization and regulation modes.
Keywords: lithium-ion accumulator, capacity, power, charge-discharge, control system, load converter, correction device, absolute stability.
Сибирский журнал науки и технологий. 2018. Т. 19, № 2. С. 281-292
АНАЛИЗ АБСОЛЮТНОЙ УСТОЙЧИВОСТИ ПРОЦЕССОВ УПРАВЛЕНИЯ ЗАРЯДНО-РАЗРЯДНЫМ УСТРОЙСТВОМ С НАГРУЗОЧНЫМ ПРЕОБРАЗОВАТЕЛЕМ В РЕЖИМЕ СТАБИЛИЗАЦИИ МОЩНОСТИ
Е. А. Копылов*, Д. К. Лобанов, Е. А. Мизрах
Сибирский государственный университет науки и технологий имени академика М. Ф. Решетнева Российская Федерация, 660037, г. Красноярск, просп. им. газ. «Красноярский рабочий», 31
*Е-таП: [email protected]
Для сокращения сроков ресурсных испытаний литий ионных аккумуляторов (ЛИА) применяют методики динамического стрессового тестирования (ДСТ). Для проведения ДСТ ЛИА необходимы автоматические зарядно-разрядные устройства (ЗРУ), обеспечивающие с заданной точностью требуемые параметры технологических режимов ДСТ ЛИА. Разработаны зарядно-разрядные устройства с нагрузочным преобразователями (НП), позволяющими автоматически воспроизводить требуемые режимы заряда-разряда ЛИА большой емкости.
Наиболее сложным режимом является циклический заряд-разряд ЛИА импульсами постоянной мощности разной величины и длительности. В этом случае система управления ЗРУ становится нестационарной нелинейной вследствие того, что мощность сигнала вычисляется как произведение тока на напряжение ЛИА.
Рассмотрена математическая модель электромагнитных процессов ЗРУ-НП в режиме стабилизации мощности заряда-разряда ЛИА, сформулированы требования к показателям качества управления контура стабилизации мощности, проведен синтез корректирующих устройств, обеспечивающих требуемое качество управления, исследована по методу Наумова-Цыпкина абсолютная устойчивость процессов управления ЗРУ-НП в режимах регулирования и стабилизации мощности заряда-разряда ЛИА.
Ключевые слова: литий-ионный аккумулятор, ёмкость, мощность, заряд-разряд, система управления, нагрузочный преобразователь, корректирующее устройство, абсолютная устойчивость.
Doi: 10.31772/2587-6066-2018-19-2-281-292
Nomenclature KVS Second voltage sensor transfer ratio
Boost converter input current in point of lin- KVSa Accumulator voltage sensor transfer ratio
Ia* n Transformer ratio
ear decomposition Ua* Accumulator voltage in point of linear de-
IL2* Reactor L2 current in point of linear decom- composition
position Uin*FB Full-bridge converter input voltage in point of
KCS Current sensor transfer ratio linear decomposition
WfbPS (s) Feedback loop transfer function of power
WOLV(s)
WOLP(s)
WPS (s)
WP(s)
WPWM1(s)
WPWM2(s)
WU(s)
Za(s)
Zload (s)
ZPS (s)
Ala
AIin_FB
AIL2
AIload(s)
AUa(s)
AUaIdl
AUin
AUinFB
AUref FB(s)
AUload(s) AUPS(s) AUref PS(s) AP(s) APref(s)
Ayl(s) Ay2(s) yl*
y2*
Open voltage loop transfer function Open power loop transfer function Open power source loop transfer function Power regulator transfer function Power controller PWM transfer function Voltage controller PWM transfer function Voltage regulator transfer function Accumulator impedance Load impedance Power source impedance Increment of accumulator current Increment of full-bridge converter input current
Increment of reactor L2 current Increment of load current Increment of accumulator voltage Increment of accumulator idling voltage Increment of boost converter input voltage Increment of full-bridge converter input voltage Increment of full-bridge converter reference voltage
Increment of load voltage Increment of power source voltage Increment of power source reference voltage Increment of accumulator power Increment of power controller reference power
Increment of boost converter duty cycle Increment of full-bridge converter duty cycle Boost converter duty cycle in point of linear decomposition
Full-bridge converter duty cycle in point of linear decomposition
Introduction. Reducing life time testing period of LIA can significantly accelerate and reduce the cost of design and development of lithium-ion accumulator battery (LIAB) and electrical power system (EPS) of spacecraft. To reduce life time testing period of LIA, standards are developed: GOST R IEC 62660-1-2014, GOST R IEC 61427-1-2014 [1; 2], in which the LIA life time tests are based on the dynamic stress testing (DST) method. Reduction of the terms for life time tests with DST is achieved by increasing the values of the attributes (constant current, voltage and capacity) of the charge / discharge up to the maximum values set by the manufacturer.
To automate the electrical tests of LIA, including life time tests with DST, the authors developed a chargedischarge device with a load converter (CDD-LC) [3-7] with a pulse-width method of regulation, which due to the original topology of the LC [3-7], has the following advantages in comparison with the known ones [8-12]:
- the possibility of providing the required values of the attributes of the DST LIA of a large capacity;
- extended range of testing currents of LIA (0.1 A-160 A);
- the possibility of LC power surplus recuperation in a direct current network of an uninterruptible power supply.
CDD-LC [3-7] in the regime of charge / discharge LIA power stabilization can be represented as two interconnected control loops: the power stabilization loop and
the input voltage stabilization loop of the bridge transformer converter (BTC).
The questions of static and dynamic analysis and synthesis of CDD-LC with stabilization of charge/discharge LIA current are considered in [13; 14]. In this case, pulsed electromagnetic processes in CDD-LC are described by continuous differential equations, which is possible on the basis of Kotelnikov-Shannon sampling theorem [15; 16].
The most complicated mode of DST is the cyclic charge-discharge of LIA by pulses of constant power of different magnitude and duration. In this case, the power management system of the CDD becomes time variant nonlinear, because the charge / discharge power is calculated as the product of the current by the voltage of LIA. The charge / discharge power of LIA at DST varies over a wide range and, accordingly, the nonlinear characteristic of the CDD-LC is regulated, which requires an investigation of the control system absolute stability.
Let us consider the stability of each stabilization loop.
Power stabilization loop. According to the structural scheme [7; 13; 14], the block diagram (fig. 1) and the equivalent scheme [13], the electromagnetic processes in the mode of CDD-LC charge power stabilization can be described by the following systems of differential equations:
A Ups (s) = (AUref _ps (s) - Д Un (s) • Wfb PS (s)) x x Wps(s)-Zps(s)•Ala(s), AUm(s) = AUps (s) - AUa (s) - Za (s) • AIa (s), Ay I (s) = APS (s) • KCs • Kvsa • Wp (s) • Wpwmi (s), Ay2(s) = AUS (s) • Kvs • Wu (s) • WpwM2 (s), AUin fb (s) = A Un (s) - AIa (s) • (Rli + LLX • s) +
+ AUn _ FB (s) •Yl + AUn _ FB -^sX
(l)
AIa (s) = Ia • Ay, (s) + AIa (s) • Y, + A/,n_fb (s) +
+ AUn_FB • s • Q, AUload (s) = (AUin_FB (s) • Y2 + Uin_FB • AY2 (s)) •n -
-AIl 2 (s) • (R2 + L2 • s),
AIin FB (s) = (AIL2(s) • y2 + IL*2 • AY 2 (s)) •
Al2(s) = AIw (s) + AUload (s) • s • C2, AU a (s) = AUa d (s) + Za (s) •AIa (s), APa (s) = AUa (s) • I'a +AIa (s) •U*,
Aload (s) = AUW (s)/Zload (s)-
Considering the power stabilization loop closing equa-
tions
Aps=Ap.ef ( s) -Apa (s),
(2)
and the stabilization loop of the input voltage of the BTC
AUs=AU,n_FB (s) -AUf FB (s):
(3)
we will compose the functional diagram of the CDD-LC with closed stabilization loop in the charging mode of the battery (fig. 1).
In the discharge mode of the battery with constant power, it is necessary to change the plus sign to the minus sign in the functional diagram (fig. 1) before the AUaidl increment of the open circuit voltage.
Fig. 1. Functional scheme of CDD-LC linearized model in dynamical mode for LIA constant power stabilization
Рис. 1. Функциональная схема линеаризованной модели ЗРУ-НП-РН в динамическом режиме при стабилизации зарядной мощности аккумулятора
Table 1
Resistance of resistors of CDD-LC mathematical model
RPS, Ohm Ra, Ohm Rw a, Ohm RL2, Ohm Rioad, Ohm RL1, Ohm
9.340-3 240-3 340-3 0.33 3 5.340-3
Table 2
Values of reactive elements of CDD-LC mathematical model
Lps, |HY Cps, |F La, |HY Lw a, |HY L2, |HY C2, IF C1, |F Lload, |HY Lb |HY
11 25 1.5 2 60 220 1050 23 31.3
Table 3
TF expressions of CDD-LC mathematical model
Wps(s) Wfhps (s) Wpwmi(S) WpwMi(s)
99 1 + s • 1.59 •Ю-4 1 11 1П-3 -3.3 • 10-6. s 3.740 e 2.840-4 e-12-510-6s
Table 4
Expressions of the impedances of CDD-LC mathematical model
Zps(s) za(s) Zw a (s) Zload(s)
rps + s • lps 1 + s •r •C + ?2 l •C 1 Ts rps cps s lps cps Ra+ sLa Rw_A+ s Lw_a Rlod+ s Lload
To analyze the stability of the power stabilization loop, we find the transfer function (TF) of the open loop (OL)
Wol_p(s) = APa(s)/APre/(s).
For this reason, in the system of equations (1) we take the zero values of the control input:
AUref_ps = 0, AUref_FB = 0, AUa_,dl = 0, open closed loop by power:
APs=APre/ (s),
and solve the system of equations (1), (3), (4) concerning APa(s).
To calculate the TF WOLP(s) parameters, it is necessary to set the initial values of the parameters and coefficients in the equations of the system (1). For a specific implementation of the CDD-LC, the values of the coefficients and parameters for calculating the parameters of the transfer functions of the CDD are summarized in tables 1 to 4.
According to the calculated logarithmic amplitude LOLP(a) = 20lgmod WOLP(s) and phase characteristics (fig. 2), the uncorrected power stabilization loop does not have stability margin, i. e. the loop is unstable.
Current and voltage transients regulated in accordance with the LIA test program should not exceed the limits of the maximum values controlled by the protection system. Therefore, these processes should have the form as close as possible to aperiodic ones with the required rise time tN (the time of the transient change from 10 to 90 %). For an aperiodic transient, the rise time tN is related to the cutoff frequency by an approximate expression [17]:
tN =( 0.3 - 0.6)—.
In accordance with the method of V. V. Solodovnikov [17], for an aperiodic transient process, it is necessary to provide a phase margin.
The analysis shows that in order to provide the required stability margin, it is appropriate to include in the functional circuit of the loop a feedforward compensator with a TF of the following form:
Wa(s) =
T • s +1 T2 • s +1:
(4)
where T1 = 0.0318s and T2 = 133s.
In this case corrected OL TF of power stabilization takes the form:
W
W n,
,(s) = WOL p(s) •Wci(s).
This corresponds to the frequency characteristics of Lcol_p(/), A<Pcol_p/), shown in fig. 2.
It can be seen from fig. 2 that when the power is regulated in a wide range, the required stability margins are provided in the loop.
The voltage stabilization loop at the input of the BTC. The voltage of stabilization UMnT at the input of the BTC is related to the allowed value of the drain-source voltage U&, using transistor switches:
Uln_FB « 0,5U&, = 12 V.
Therefore, in transient modes, the voltage overshoot o2 is limited, and should not exceed the value o2 = 45 %.
For the normal operation of the power stabilization and BTC voltage loops, the condition to2 < to1 must be fulfilled, i. e. the transient time to2 should not be greater than in the power stabilization loop (t0i ~ (3-4) tN). On the basis of the foregoing, we find the frequency /C2 of the cut
cl
in the voltage stabilization loop (VSL) of BTC from condition
fcl < * 4f V2
Fig. 2 shows that the frequency fa is approximately 200 Hz.
Therefore, the cutoff frequency in the VSL of BTC should be fC2 ~ 2500 Hz.
To analyze the stability of the BTC voltage stabilization loop, we find the TF of the open loop:
WoL_u(s) = A Uin_FB(s)/A Uref_FB(s ).
For this, in the system of equations (1) we take the zero values of the control input: AUrejPS = 0, AUrefjFB = 0, AUaJdl = 0, cut off the voltage feedback:
AU = -AU
ref _ FB
(s),
= (T3s +1)-(^4-s +1) c 2W (T5-s +1) • (T6-s +1)
(5)
where T3 = 3.18-10-5s, and T4 = 3.18-10-4s, T5 = 3.18-10-3s and T6 = 3.18-10-6 s .
In this case, the corrected OL TF stabilizing the voltage takes the form:
W„.
r(s) = Wol и (s) -Wc2(s).
and solve the system of equations (1), (3), (4) with respect
to AUinjRB(s).
Analysis of the stabilization loop shows that in order to ensure the required margins of stability and speed, it is appropriate to include in the functional circuit of the loop a feedforward compensator calculated by the method of V. V. Solodovnikov [17], with the TF of the following form:
This expression of the TF corresponds to the frequency characteristics of LcoL_u(f), AymL_u(f), A^coLjUf), given in fig. 3.
It is evident from fig. 3: power control in a wide range in a loop provides necessary margins of stability; when medium and high power are stabilized, the requirements for the cut-off frequency fC2 of the VSL of BTC are fulfilled with a margin, and when the low-power charge/discharge LIA is stabilized, the decrease in the frequency fC2 does not lead to an increase in the voltage overshoot o2 due to the relatively small charge currents of the capacitor at the input of the BTC.
The change in the dynamic properties of the VSL of BTC can lead to a change in the dynamic properties of the PSL of LIA associated with it. To verify compliance with previously established requirements for the stability and speed of the PSL, LCOL P(f), A^>COL_P(f) were recalculated taking into account the correction of both loops and the results are shown in fig. 4.
Fig. 2. Open loop Bode plot of CDD while charging LIA with constant power
Рис. 2. Частотные характеристики разомкнутого контура ЗРУ-НП при заряде ЛИА постоянной мощностью
-200
800W 0.5W
Fig. 3. Open loop Bode plot for WOL U(s) in LIA constant power mode
Рис. 3. Частотные характеристики разомкнутого контура стабилизации напряжения WOL U(s) при стабилизации мощности аккумулятора
Fig. 4. Corrected open loop Bode plot of CDD model for WCOL P(s)
Рис. 4. Частотные характеристики скорректированного разомкнутого контура ЗРУ-НП при заряде ЛИА постоянной мощностью
0
The FC of LAoL_p(f) h 9aol_p(J) (fig. 4) corrected PSL charge/discharge of the LIA when controlling the powers in a wide range have the phase margins A9 > 100° and the cutoff frequency frf in the frequency range of 200 Hz, which meets the requirements.
Absolute stability. In the regime of charge/discharge power stabilization, the current-voltage characteristic (I-V characteristic) of a CDD-LC is non-linear, due to the presence of nonlinear (functional) feedback on the power of the LIA
Pa(t) = Ua(t)-Ia(t).
Since the parameters of the functional feedback vary with time, the CDD-LC in the power stabilization mode is a non-linear non-stationary automatic control system (ACS).
For the stability analysis of such systems, it is appropriate to apply the method developed by B. N. Naumov and Ya. Z. Tsypkin [18-20]. This method requires bringing the ACS to a single-circuit view (fig. 5), containing a stable dynamic linear part (LP) and one static nonlinear element (NE). The criterion allows one to judge the stability of the ACS by the frequency characteristics of the LP system and the differential coefficient kNEmax of the NE transmission.
The equation of a nonlinear element:
Pa da ) = U ш + RQ • IQ ) • I^K^- KycA • K
The linear part of the power stabilization open loop is described by a system of equations:
AUpS (s) = (AU„f _ ps (s) - AUn (s) Wfb _ pS (s)) x x WpS (s) - ZpS (s)- AIa (s), AUn(s) = AUpS(s) -AUa(s) - Za(s) - AT,(s), Ay,(s) = APref (s)-Kcs-Kvsa-Wp (s) - WpwM ,(s), Ay2(s) = (AUn_fb(s)-AUref fb(s))-Kvs-Wu(s) -Wpm2(s),
n fb (s) = U (s) - AIa (s) • R + Ln • s) +
AU,
+ AUin _ fb (sy Y1 + AU, AIa(s) = I* • AYj(s) + AIa (s) • Y* + AI,
/*
'in FB
AY1(s),
+ AUin FB ' s • C1,
in _FB (s) + 2 ' - in FB AY 2 (s)) • П -
(s) • y2 + U*
AUhad (s) = (AU,n_FB -AIl2 (s) • (R2 + L2 •s)
AIin FB (s) = (AIl2 (s) • y2 +1*2 • Ay2 (s)) • n
AIl2(s) = AIload (s) + AUload (s) - s - C2, AUa(s) = AUa a(s) + Za(s) - AI,(s), APa (s) = AU a (s)-1* +AIa (s) - U*, AI load (s) = AU load (s)/Zw (s).
To analyze the absolute stability of the power stabilization loop, we find the TF of the linear part of the open loop
WoLjLp(s) = Ma(s)/APfs),
and LPC Lol lp(s,) 9ol_lp(s) (fig. 6).
According to Naumov-Tsypkin criterion [18-20], for absolute stability of processes in a control system with nonstationary NE it is sufficient that the LP should be stable and the frequency response of the LP should satisfy all frequencies 0 < ra < ro the condition:
Re (Wol_lp (j'®)) + - 1
> 0
Fig. 5. Single-circuit view of the ACS: LP - linear part, NE - non-linear element
Рис. 5. Одноконтурный вид САУ: ЛЧ - линейная часть, НЭ - нелинейный элемент
In the case of a nonstationary system, B. N. Naumov and Ya. Z. Tsypkin showed [18-20] that the processes in the system will be asymptotically stable in general if the criterion of absolute stability is satisfied at the highest value of the differential coefficient kNEmax of NE transmission.
The main output variable of the CDD is the current Ia(t) of the LIA, which when the power is stabilized varies depending on the voltage of the LIA Ua, which according to (1) has the form:
Ua (s) = Ua ш ( s) + Za (s)- Ia (s).
or:
Re ( k
NEmaxWOL LP
Denoting the TF by modified LP (MLP),
WmlP (J®) = kNE max Wol LP (J®) >
we obtain the condition of absolute stability processes in the form:
Re (Wmlp ( j®))>-1
(7)
where the maximum differential transmission coefficient of NE:
(6)
where Kp- coefficient of proportionality.
v dI ,
V a /MAX
In accordance with (6), the coefficient km is a function of three independent variables: the input current Ia, the open circuit voltage UaJdt, the internal resistance Ra of the battery.
Let us study the ranges of kNE coefficient variation depending on these parameters.
k
It follows from fig. 7 that the coefficient kNE reaches its maximum value at the maximum current Ia, voltage UaJdi = 4,2 V and resistance Ra = 20 mOhm, with
kNE_MAX < 12.
Graphical interpretation of condition (7) means that the amplitude-phase characteristic (APC) of the MLD (fig. 8) should lie to the right of the vertical line passing through the point with the coordinates (-1; 0).
Since the frequency characteristics (FC) LP of the CDD-LC (fig. 8) depends on the value of the stabilized power, the analysis of the absolute stability of the processes must be performed for the entire range of power regulation Pa. As a result of APC MLP analysis it was stated that it is sufficient to check the absolute stability with minimum and maximum LIA test power values (fig. 8).
Fig. 6. Bode plot of open-loop linear part (OL_LP) Рис. 6. Частотные характеристики разомкнутого контура линейной части (ЛЧ)
b
Fig. 7. Dependence of the coefficient kNE(Ia) on: a - different voltages Ua idi and resistance Ra = 20 mOhm; b - different resistance Ra and voltage Ua idi = 4,2 V
Рис. 7. Зависимость коэффициента £НЭ(/вх): а - при различных напряжениях иа хх и сопротивлении Ra = 20 мОм; б - при различных сопротивлениях Ra и напряжении иа хх = 4,2 В
a
-S
-2-104
-1 -[ )J -[ £ -C 1.4 ( и I ¡Г ~~~~ 02 04 0i 0J
-2-
4" " - - _
Re
0.5 W
---srnv
Fig. 8. Amplitude phase characteristic WMLD(ja) for kNE = 12: а - for the frequency range 0 < ю < 105, b - in the field of high frequencies (in the vicinity of the point (-1; j '0))
Рис. 8. Амплитудно-фазовая характеристика WMm(jm) при кНЭ = 12: а - для диапазона частот 0 < ю < 105 б - в области высоких частот (в окрестности точки (-1; j'0))
It follows from APC (fig. 8):
1. For the calculated and selected parameters of the MLD WMLD(j'®i) linear part, the condition of absolute processes stability (7) is fulfilled irrespective of the power value Pa of the LIA charge/discharge.
2. The hodographs APC MLD WMLD(j a) at the maximum and minimum input powers differ in the interval of low and medium frequencies and practically coincide in the high-frequency interval, determining the absolute stability of the CDD-LC control system, which indicates the correctness of the synthesis of correcting devices (4) and (5 ).
To prove the adequacy of the developed mathematical models, the experimental sample of the CDD-LC module was investigated.
To obtain transient control processes with power stabilization, the experiment scheme shown in fig. 9 was used. In the tests, instead of the LIA, a test load was used that allowed to estimate the operation in large ranges of currents and voltages of the CDD.
When testing, direction of current when charging the battery is taken for a positive current direction. Fig. 10 shows the process of changing the voltage UinFB at the input of the BTC (upper graph of the oscillogram)
and the current of the battery Ia (lower graph of the oscil-logram) with a linear discharge power surge of the battery from P3 = 3 W to P3 = 640 W. At the same time, the rate of battery power surge is V = 350 A/s. Sweep trace of the voltage channel Uin_FB corresponds to 5V/div (fig. 10) and 80 A/div for channel measurement of current Ia. Time sweep trace - 100 ms/div.
It can be seen from fig. 10 that the current deviation from the linear character differs slightly, and the excessive correction of o2 voltage UBTC does not exceed 42 %, which meets the requirements for the value of o2.
Conclusion. The developed mathematical model of electromagnetic processes of the CDD-LC in the charge/discharge LIA power stabilization mode allows to analyze and synthesize CDD-LC with the required control power stabilization loop quality indicators.
Control system of the CDD-LC is presented in the form of two interrelated control loops: power stabilization loop, and the input voltage stabilization loop of the bridging transformer converter. It is shown that it is appropriate to adjust the power stabilization loop first, and then, taking into account the data obtained, select the parameters of the BTC voltage stabilization loop correcting device.
а
b
Fig. 9. Transient response experiment test structure
Рис. 9. Схема эксперимента для снятия переходных процессов по управлению
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Fig. 10. Transients for linearly increasing power Рис. 10. Переходные процессы при линейном увеличении разрядной мощности
The proposed type of correcting devices allows to ensure absolute stability of processes in the CDD-LC when stabilizing the charge/discharge power of LIA with the required speed and quality of transients.
The experimentally obtained transients meet the necessary requirements, which confirms the adequacy of the CDD-LC mathematical model with the stabilization of the LIA power.
References
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Библиографические ссылки
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