CONTROL PROCESS ABSOLUTE STABILITY ANALYSIS OF CHARGE-DISCHARGE DEVICE WITH LOAD CONVERTER IN CONSTANT POWER MODE

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Nomenclature Ia* Boost converter input current in point of lin- ear decomposition IL2* Reactor L2 current in point of linear decom- position KCS Current sensor transfer ratio KVS Second voltage sensor transfer ratio KVSa Accumulator voltage sensor transfer ratio n Transformer ratio Ua* Accumulator voltage in point of linear de- composition Uin*FB Full-bridge converter input voltage in point of linear decomposition Wfb_PS (s) Feedback loop transfer function of power source WOL_V(s) Open voltage loop transfer function WOL_P(s) Open power loop transfer function WPS (s) Open power source loop transfer function WP(s) Power regulator transfer function WPWM1(s) Power controller PWM transfer function WPWM2(s) Voltage controller PWM transfer function WU(s) Voltage regulator transfer function Za(s) Accumulator impedance Zload (s) Load impedance ZPS (s) Power source impedance ∆Ia Increment of accumulator current ∆Iin_FB Increment of full-bridge converter input cur- rent ∆IL2 Increment of reactor L2 current ∆Iload(s) Increment of load current ∆Ua(s) Increment of accumulator voltage ∆Ua_Idl Increment of accumulator idling voltage ∆Uin Increment of boost converter input voltage ∆Uin_FB Increment of full-bridge converter input voltage ∆Uref FB(s) Increment of full-bridge converter reference voltage ∆Uload(s) Increment of load voltage ∆UPS(s) Increment of power source voltage ∆Uref PS(s) Increment of power source reference voltage ∆P(s) Increment of accumulator power ∆Pref(s) Increment of power controller reference power ∆γ1(s) Increment of boost converter duty cycle ∆γ2(s) Increment of full-bridge converter duty cycle the input voltage stabilization loop of the bridge trans- former converter (BTC). The questions of static and dynamic analysis and syn- thesis 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 calcu- lated 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 investiga- tion 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 equa- tions: ìDUPS (s) = (DUref _ PS (s) - DUin (s) ×Wfb_ PS (s)) ´ ï ï´ WPS (s) - ZPS (s) ×DIa (s), in PS a a a ïDU (s) = DU (s) - DU (s) - Z (s) ×DI (s), γ1* Boost converter duty cycle in point of linear ï Dg (s) = DP (s) × К × К ×W (s) ×W (s), ï 1 e CS VSa P PWM 1 decomposition ïDg (s) = DU (s) × К ×W (s) ×W (s), γ2* Full-bridge converter duty cycle in point of ï 2 e VS U PWM 2 linear decomposition ïDUin _ FB (s) = DUin (s) - DIa (s) × (RL1 + LL1 × s) + ï ï+ DU (s) × g* + DU * × Dg (s), Introduction. Reducing life time testing period ï in _ FB 1 in _ FB 1 of LIA can significantly accelerate and reduce the cost ïDI (s) = I * ×Dg (s) + DI (s) × g* + DI (s) + 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, í a a 1 ï+ DUin _ FB × s × C1, DU (s) = (DU ï ï load in _ FB a 1 in _ FB 2 in _ FB 2 (s) × g* + U * ×Dg (s)) × n - (1) ï- DI (s) × (R + L × s), GOST R IEC 61427-1-2014 [1; 2], in which the LIA life ï time tests are based on the dynamic stress testing (DST) ïDI L 2 2 2 (s) = (DI (s) × g* + I * ×Dg (s)) × n, 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 manufac- turer. To automate the electrical tests of LIA, including life time tests with DST, the authors developed a charge- discharge 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 ad- vantages in comparison with the known ones [8-12]: ï in _ FB L2 2 L2 2 ïDIL2 (s) = DIload (s) + DUload (s) × s × C2 , ï ïDUa (s) = DUa _ Idl (s) + Zа (s) × DIa (s), ïDP (s) = DU (s) × I * + DI (s) ×U *, ï a a a a a îDIload (s) = DUload (s) Zload (s). Considering the power stabilization loop closing equa- tions DPe = DPref (s) - DPa (s) , (2) and the stabilization loop of the input voltage of the BTC - the possibility of providing the required values of the attributes of the DST LIA of a large capacity; DU e = DU in _ FB (s) - DUref _ FB (s) , (3) - 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 inter- connected control loops: the power stabilization loop and 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 ∆Uа_idl 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_а, Ohm RL2, Ohm Rload, Ohm RL1, Ohm 9.3·10-3 2·10-3 3·10-3 0.33 3 5.3·10-3 Table 2 Values of reactive elements of CDD-LC mathematical model LPS, µHY СPS, µF Lа, µHY Lw_а, µHY L2, µHY C2, µF C1, µF Lload, µHY L1, µHY 11 25 1.5 2 60 220 1050 23 31.3 Table 3 TF expressions of CDD-LC mathematical model WPS(s) Wfb_PS (s) WPWM1(s) WPWM2(s) 99 1 + s ×1.59 ×10-4 1 3.7 ×10-3 e-3.3×10-6 ×s 2.8 ×10-4 e-12.5×10-6 ×s Table 4 Expressions of the impedances of CDD-LC mathematical model ZPS(s) Za(s) Zw_а (s) Zload(s) RPS + s × LPS 1 + s × R × C + s2 × L × C PS PS PS PS Ra+ s·La Rw_А+ s·Lw_а 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) = ∆Pa(s)/∆Pref(s). For this reason, in the system of equations (1) we take the zero values of the control input: ∆Uref_PS = 0, ∆Uref_FB = 0, ∆Uа_idl = 0, 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 re- quired stability margin, it is appropriate to include in the functional circuit of the loop a feedforward compensator with a TF of the following form: open closed loop by power: DPe = DPref (s) , WC1 (s) = T1 × s +1 , (4) T2 × s +1 and solve the system of equations (1), (3), (4) concerning ∆Pa(s). To calculate the TF WOL_P(s) parameters, it is neces- sary to set the initial values of the parameters and coeffi- cients in the equations of the system (1). For a specific implementation of the CDD-LC, the values of the coeffi- cients 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 LOL_P(ω) = 20lg·mod WOL_P(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 ωc1 by an approximate expression [17]: N w t = (0.3 - 0.6) 1 . с1 where T1 = 0.0318s and T2 = 133s. In this case corrected OL TF of power stabilization takes the form: WСOL _ P (s) = WOL _ P (s) ×WC1 (s). This corresponds to the frequency characteristics of LСOL_P(f), ∆φСOL_P(f), shown in fig. 2. It can be seen from fig. 2 that when the power is regu- lated 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 UМПТ at the input of the BTC is related to the allowed value of the drain-source voltage Uds, using transistor switches: Uin_FB ≈ 0,5·Uds, = 12 V. Therefore, in transient modes, the voltage overshoot σ2 is limited, and should not exceed the value σ2 = 45 %. For the normal operation of the power stabilization and BTC voltage loops, the condition tσ2 ≤ tσ1 must be fulfilled, i. e. the transient time tσ2 should not be greater than in the power stabilization loop (tσ1 ≈ (3-4) tN). On the basis of the foregoing, we find the frequency fC2 of the cut in the voltage stabilization loop (VSL) of BTC from con- dition WC 2 (s) = (T3s +1) × (T4 × s +1) (T5 × s +1) × (T6 × s +1) , (5) 2 fc 2 £ t » 4pf c1. where T3 = 3.18·10-5s, and T4 = 3.18·10-4s, T5 = 3.18·10-3s -6 s2 and T6 = 3.18·10 s . Fig. 2 shows that the frequency fC1 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 stabiliza- tion loop, we find the TF of the open loop: WOL_U(s) = ∆Uin_FB(s)/∆Uref_FB(s). For this, in the system of equations (1) we take the zero values of the control input: ∆Uref_PS = 0, ∆Uref_FB = 0, ∆Uа_idl = 0, cut off the voltage feedback: DU e = -DUref _ FB (s) , and solve the system of equations (1), (3), (4) with respect to ∆Uin_RB(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: In this case, the corrected OL TF stabilizing the volt- age takes the form: WAOL _U (s) = WOL _U (s) ×WC 2 (s). This expression of the TF corresponds to the fre- quency characteristics of LСOL_U(f), ∆φСOL_U(f), ∆φСOL_U(f), 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 ful- filled 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 σ2 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, LСOL_P(f), ∆φСOL_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 100 80 60 40 20 0.1 1 10 100 1 .103 1 .104 1 .105 20 40 60 80 100 120 140 160 180 - 180 200 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 WСOL_P(s) Рис. 4. Частотные характеристики скорректированного разомкнутого контура ЗРУ-НП при заряде ЛИА постоянной мощностью The FC of LAOL_P(f) и φAOL_P(f) (fig. 4) corrected PSL charge/discharge of the LIA when controlling the powers The linear part of the power stabilization open loop is described by a system of equations: in a wide range have the phase margins ∆φ1 ≥ 100° and the cutoff frequency fс1 in the frequency range of 200 Hz, which meets the requirements. ìDUPS (s) = (DUref _ PS (s) - DUin (s) ×Wfb_ PS ï ï´ WPS (s) - ZPS (s) ×DIa (s), (s)) ´ 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 ïDU (s) = DU (s) - DU (s) - Z (s) ×DI (s), ï in PS a a a ïDg1(s) = DPref (s) × КCS × КVSa ×WP (s) ×WPWM1(s), ïDg (s) = (DU (s) - DU (s)) × К ×W (s) ×W (s), ï the presence of nonlinear (functional) feedback on the ï 2 in _ FB ref _ FB VS U PWM 2 power of the LIA ïDUin _ FB (s) = DUin (s) - DIa (s) × (RL1 + LL1 × s) + ï+ DU (s) × g* + DU * × Dg (s), Pа(t) = Uа(t)·Ia(t). ï in _ FB 1 in _ FB 1 ïDI (s) = I * ×Dg (s) + DI (s) × g* + DI (s) + 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 í a a 1 ï+ DUin _ FB × s × C1, ïDU (s) = (DU a 1 in _ FB (s) × g* + U * ×Dg (s)) × n - ï ï load in _ FB 2 in _ FB 2 (ACS). For the stability analysis of such systems, it is appro- priate to apply the method developed by B. N. Naumov ï- DI (s) × (R + L × s), L 2 2 2 ïDI (s) = (DI (s) × g* + I * ×Dg (s)) × n, ï in _ FB L 2 2 L 2 2 and Ya. Z. Tsypkin [18-20]. This method requires bring- ing the ACS to a single-circuit view (fig. 5), containing a stable dynamic linear part (LP) and one static nonlinear ïDIL 2 (s) = DIload (s) + DUload (s) × s × C2 , ïDU (s) = DU (s) + Z (s) ×DI (s), ï a a _ Idl а a ïDP (s) = DU (s) × I * + DI (s) ×U *, element (NE). The criterion allows one to judge the stabil- ity of the ACS by the frequency characteristics of the LP ï a a a a a ïîDIload (s) = DUload (s) Zload (s). system and the differential coefficient transmission. kNE max of the NE To analyze the absolute stability of the power stabili- zation loop, we find the TF of the linear part of the open loop WOL_LP(s) = ∆Ia(s)/∆Pref(s), and LPC LOL_LP(s,) φOL_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 £ w £ ¥ the condition: Re(WOL_LP ( jw) + 1 > 0 ) k 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 or: NE max Re (kNE maxWOL_LP ( jw)) > -1 . Denoting the TF by modified LP (MLP), WMLP ( jw) = kNE maxWOL_LP ( jw) , the system will be asymptotically stable in general if the criterion of absolute stability is satisfied at the highest we obtain the condition of absolute stability processes in the form: value of the differential coefficient mission. kNE max of NE trans- Re(WMLP ( jw)) > -1, (7) 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 Uа, which according to (1) has the form: where the maximum differential transmission coefficient of NE: è k = æ ¶Pa ö . U (s) = U (s) + Z (s) × I (s). NE max ç ¶I ÷ a øMAX a a _ Idl а a The equation of a nonlinear element: Pa (Ia ) = (Ua _ idl + Ra × Ia ) × Ia × KCS × KVCA × KP , (6) where Кp - coefficient of proportionality. In accordance with (6), the coefficient kNE is a func- tion of three independent variables: the input current Ia, the open circuit voltage Ua_idl, the internal resistance Ra of the battery. Let us study the ranges of kNE coefficient variation de- pending on these parameters. It follows from fig. 7 that the coefficient kNE reaches its maximum value at the maximum current Ia, voltage Ua_idl = 4,2 V and resistance Ra = 20 mOhm, with kNE_МАХ ≤ 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 proc- esses must be performed for the entire range of power regulation Pа. As a result of APC MLP analysis it was stated that it is sufficient to check the absolute sta- bility with minimum and maximum LIA test power val- ues (fig. 8). Fig. 6. Bode plot of open-loop linear part (OL_LP) Рис. 6. Частотные характеристики разомкнутого контура линейной части (ЛЧ) a b Fig. 7. Dependence of the coefficient kNE(Ia) on: а - different voltages Ua_idl and resistance Ra = 20 mОhm; b - different resistance Ra and voltage Ua_idl = 4,2 V Рис. 7. Зависимость коэффициента kНЭ(Iвх): а - при различных напряжениях Ua_хх и сопротивлении Ra = 20 мОм; б - при различных сопротивлениях Ra и напряжении Ua_хх = 4,2 В Fig. 8. Amplitude phase characteristic WМLD(jω) for kNE = 12: а - for the frequency range 0 ≤ ω ≤ 105, b - in the field of high frequencies (in the vicinity of the point (-1; j·0)) Рис. 8. Амплитудно-фазовая характеристика WМЛЧ(jω) при kНЭ = 12: а - для диапазона частот 0 ≤ ω ≤ 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 WМLD(jω) linear part, the condition of absolute processes stability (7) is fulfilled irrespective of the power value Pа of the LIA charge/discharge. 2. The hodographs APC MLD WМLD(jω) at the maximum and minimum input powers differ in the inter- val of low and medium frequencies and practically coin- cide in the high-frequency interval, determining the abso- lute 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 sta- bilization, 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 Uin_FB at the input of the BTC (upper graph of the oscillogram) and the current of the battery Iа (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 VI = 350 A/s. Sweep trace of the voltage channel U in_FB corresponds to 5V/div (fig. 10) and 80 A/div for channel measurement of current Iа. 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 exces- sive correction of σ2 voltage UBTC does not exceed 42 %, which meets the requirements for the value of σ2. 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 con- trol power stabilization loop quality indicators. Control system of the CDD-LC is presented in the form of two interrelated control loops: power stabiliza- tion loop, and the input voltage stabilization loop of the bridging transformer converter. It is shown that it is ap- propriate to adjust the power stabilization loop first, and then, taking into account the data obtained, select the parameters of the BTC voltage stabilization loop cor- recting device. Fig. 9. Transient response experiment test structure Рис. 9. Схема эксперимента для снятия переходных процессов по управлению Fig. 10. Transients for linearly increasing power Рис. 10. Переходные процессы при линейном увеличении разрядной мощности The proposed type of correcting devices allows to en- sure 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 nec- essary requirements, which confirms the adequacy of the CDD-LC mathematical model with the stabiliza- tion of the LIA power.

作者简介

E. Kopylov

Reshetnev Siberian State University of Science and Technology

Email: evgesh72@mail.ru
31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660037, Russian Federation

D. Lobanov

Reshetnev Siberian State University of Science and Technology

31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660037, Russian Federation

E. Mizrakh

Reshetnev Siberian State University of Science and Technology

31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660037, Russian Federation

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