Control process absolute stability analysis of charge-discharge device with load converter in constant power mode

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:

\ U ps ( 5 ) = ( A U ref _ PS ( 5 ) - A U n ( 5 ) ■ W b _ PS ( 5 )) X

X W ps ( 5 ) - Z ps ( 5 ) ■A I a ( 5 ),

A U n ( 5 ) = A U ps ( 5 ) - A U a ( 5 ) - Z a ( 5 ) ■ A I a ( 5 ),

Ay _l ( 5 ) = A P _: ( 5 ) ■ K es ■ K vsa ■ W p ( 5 ) ■ W     1 ( 5 ),

AY 2 ( 5 ) = A U _E ( 5 ) ■ K vs ■ W U ( 5 ) ■ W PWM 2 ( s ),

A U n _ FB ( 5 ) = A U n ( 5 ) - A I a ( 5 ) ■ ( R l 1 + L l 1 ■ 5 ) +

^{+ A} U n _ FB ⁽⁵ ) "Y l ^{+A U}*n _ FB ■^AY ₁ ^{( 5} ),

. A I a ( 5 ) = I*_a ■ AY 1 ( 5 ) + A I a ( 5 ) ■ Y * + A I in _ FB ( 5 ) +

”+A U n _ FB ■ 5 ■ C 1 ,                                         (1)

A U    ( 5 ) = ( A U „ _ FB ( 5 ) ■ y 2 + U n _ FB ■ Ay 2 ( 5 )) ■ П -

-A I l 2 ( 5 ) ■ ( R 2 + L 2 ■ 5 ),

^{A I}n FB ⁽⁵ ) = ^{( A I}L 2 ^{( 5} ) ^2 ⁺ ^2 ■ ^AY 2 ^{( 5} )) ■ n,

A I L 2 ( 5 ) = A I load ( 5 ) + A U load ( 5 ) ■ 5 ■ C 2 ,

A U a ( 5 ) = A U a _ Idi ( 5 ) + Z a ( 5 ) ■A I a ( 5 ),

A P , ( 5 ) =A U a ( 5 ) ■ I* * +A I a ( 5 ) ■ U a * ,

.A I load ( 5 ) = A U_to ad ( 5 )/ Z load ( 5 )-

Considering the power stabilization loop closing equations

A P _e=A P ref ( 5 ) -A P a ( 5 ),                (2)

and the stabilization loop of the input voltage of the BTC

A U _e=A U in _ FB ( 5 ) -A U ref _ FB ( 5 ),           (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 Δ U а_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

Resistance of resistors of CDD–LC mathematical model

Table 1

R PS , Ohm	R a , Ohm	R w а , Ohm	R L 2 , Ohm	R load , Ohm	R L1 , 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

L PS , µHY	С PS , µ F	L а , µ HY	L w а , µHY	L 2 , µHY	C 2 , µ F	C 1 , µ F	L load , µHY	L 1 , µHY
11	25	1.5	2	60	220	1050	23	31.3

TF expressions of CDD–LC mathematical model

Table 3

W PS ( s )	W fb_PS ( s )	W PWM1 ( s )	W PWM 2 ( s )
99 1 + s - 1.59 ■Ю - 4	1	3.7 - 10 - 3 e -^' s	2.8 -10-4 e -|2-5-10-6- s

Expressions of the impedances of CDD–LC mathematical model

Table 4

ZPS ( s )	Z a ( s )	Z w_а ( s )	Z load ( s )
________ R PS + s ■ L PS ________ 1 + s ■ R PS • CPS + s ■ LPS • CPS	R a + s·L a	R w_А + s·L w_а	R lod + s·L load

To analyze the stability of the power stabilization loop, we find the transfer function (TF) of the open loop (OL)

W OL_P ( s ) = Δ P a ( s )/Δ P ref ( s ).

For this reason, in the system of equations (1) we take the zero values of the control input:

Δ U ref_PS = 0, Δ U ref_FB = 0, Δ U а_idl = 0, open closed loop by power:

bP;=A Pref ( s ), and solve the system of equations (1), (3), (4) concerning ΔPa(s).

To calculate the TF W_{OL_P} ( 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 L OL_P (ω) = 20lg ∙ mod W OL_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 t N (the time of the transient change from 10 to 90 %). For an aperiodic transient, the rise time t N is related to the cutoff frequency ω c 1 by an approximate expression [17]:

t N = ( 0.3 - 0.6)— .

^Ю с 1

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:

W 1 ( s ) =

7 1 ■ s + 1

7 2 ■ s + 1,

where T ₁ = 0.0318 s and T ₂ = 133 s .

In this case corrected OL TF of power stabilization takes the form:

W col _ P ( s ) = W ol _ P ( s ) ■ W c 1 ( s ).

This corresponds to the frequency characteristics of L СOL_P ( f ), Δ φ СOL_P ( f ), 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 U МПТ at the input of the BTC is related to the allowed value of the drain-source voltage U_ds , using transistor switches:

U in_FB ≈ 0,5∙ U ds , = 12 V .

Therefore, in transient modes, the voltage overshoot σ 2 is limited, and should not exceed the value σ 2 = 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 condition fc2≤2 ≈4πfc1.

t σ 2

Fig. 2 shows that the frequency f C 1 is approximately 200 Hz.

Therefore, the cutoff frequency in the VSL of BTC should be f C 2 ≈ 2500 Hz.

To analyze the stability of the BTC voltage stabilization loop, we find the TF of the open loop:

W OL_U ( s ) = Δ U in_FB ( s )/Δ U ref_FB ( s ).

For this, in the system of equations (1) we take the zero values of the control input: Δ U ref_PS = 0, Δ U ref_FB = 0, Δ U а_idl = 0, cut off the voltage feedback:

ΔUε = -ΔUref_FB(s), and solve the system of equations (1), (3), (4) with respect to ΔUin_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:

_{W ( s ) =} ( T 3 s + 1) ⋅ ( T 4 ⋅ s + 1) _, ^{C 2}        ( T 5 ⋅ s + 1) ⋅ ( T 6 ⋅ s + 1)

where T ₃ = 3.18∙10^–5 s , and T ₄ = 3.18∙10^–4 s , T ₅ = 3.18∙10^–3 s and T 6 = 3.18∙10^–6 s .

In this case, the corrected OL TF stabilizing the voltage takes the form:

W AOL _ U ( s ) = W OL _ U ( s )⋅ W C 2 ( s ).

This expression of the TF corresponds to the frequency characteristics of L СOL_U ( f ), Δφ СOL_U ( f ), Δφ СOL_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 f C 2 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 f C 2 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, L_{СOL_P} ( f ), Δφ _{СOL_P} ( f ) were recalculated taking into account the correction of both loops and the results are shown in fig. 4.

0.5W

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

Рис. 3. Частотные характеристики разомкнутого контура стабилизации напряжения W_{OL_U} ( s ) при стабилизации мощности аккумулятора

Fig. 4. Corrected open loop Bode plot of CDD model for W СOL_P ( s )

Рис. 4. Частотные характеристики скорректированного разомкнутого контура ЗРУ–НП при заряде ЛИА постоянной мощностью

The FC of L AOL_P ( f ) и φ AOL_P ( f ) (fig. 4) corrected PSL charge/discharge of the LIA when controlling the powers in a wide range have the phase margins Δφ 1 ≥ 100° and the cutoff frequency f с 1 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

P а ( t ) = U а ( t ) ∙I a ( 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 k _{NE max} of the NE transmission.

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 k _{NE max} of NE transmission.

The main output variable of the CDD is the current I a ( t ) of the LIA, which when the power is stabilized varies depending on the voltage of the LIA U а , which according to (1) has the form:

U a ( 5 ) = U a _ Idl ( 5 ) + Z а ( 5 ) ' I a ( 5 ).

The equation of a nonlinear element:

P a ( I a ) = (U a _idl + R a ' I a ) ' I a ' K CS ' K vCA ' K p ^,      ( ⁶ )

where К p – coefficient of proportionality.

The linear part of the power stabilization open loop is described by a system of equations:

A U p_S ( 5 ) = ( A U ref _ PS ( 5 ) -A U n ( 5 ) • W fb _ PS ( 5 )) X

X W ps ( 5 ) - Z ps ( 5 ) •A I a ( 5 ),

A U n ( 5 ) = A U ps ( 5 ) - A U a ( 5 ) - Z_a (5 ) • A I a (5 ),

AY 1 ( 5 ) = A p ef ( 5 ) • K _Cs • K vsa • W p ( 5 ) • W _:W_V , ( 5 ),

AY 2 ( 5 ) = ( A U _M _ fb ( 5 ) — A U ref _ FB ( 5 )) ' K ys ' W u ( 5 ) • W pWM 2 ( s ), A U n _ FB ( 5 ) = A U n ( 5 ) - A I a ( s ") • ( R l 1 + L l 1 • 5 ) +

^{+ A U} n _ FB ^{( 5} ) •Y * ^{+A U}*n _ FB ’^A Y 1^{( 5} X

. A I a ( 5 ) = I * • AY 1 ( 5 ) + A I a ( 5 ) • Y * + A I in _ FB ( 5 ) +

+ A U n _ FB • 5 • C 1 ,

A U oad ( 5 ) = ( A U n _ FB ( 5 ) • Y 2 + U * _ FB • AY 2 ( 5 )) • П -

-A I l 2 ( 5 ) • ( R 2 + L 2 • 5 ),

^{A I} .n FB ^{( 5} ) = ^{( A I} L 2 ^{( 5} ) • Y 2 ^{+ I} L 2 • ^AY 2 ^{( 5} )) • n ,

A I L 2 ( 5 ) = A I _l0a_d ( 5 ) + A U _toad ( 5 ) • 5 • C 2 ,

A U a ( 5 ) =A U a _ Idl ( 5 ) + Z a ( 5 ) •A I a ( 5 ),

A p , ( 5 ) = A U a ( 5 ) • I a +A I a ( 5 ) • U a ,

^A I load ( 5 ) = A U ioad ( 5 )/ Z oad ( 5 )-

To analyze the absolute stability of the power stabilization loop, we find the TF of the linear part of the open loop

WOL_LP(s) = ΔIa(s)/ΔPref(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 < to < » the condition:

Re ( ^wol _ lp ( j ^to ) ) ⁺ T^— >  ⁰ kNE max or:

^Re ^{( k}NE max ^WOL _ Lp ^{( j to ))} > ^-1 ^.

Denoting the TF by modified LP (MLP),

WMLP (jto) kNE maxWOL_Lp (jto) , we obtain the condition of absolute stability processes in the form:

Re ( WMlp (j to))>-1,               (7)

where the maximum differential transmission coefficient of NE:

k NE max

MAX

In accordance with (6), the coefficient k NE is a function of three independent variables: the input current I a , the open circuit voltage U a_idl , the internal resistance Ra of the battery.

Let us study the ranges of k NE coefficient variation depending on these parameters.

It follows from fig. 7 that the coefficient k NE reaches its maximum value at the maximum current I a , voltage U a_idl = 4,2 V and resistance R a = 20 mOhm, with k NE_ 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 processes must be performed for the entire range of power regulation P а . 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)

a

Fig. 7. Dependence of the coefficient k_NE ( I_a ) on: а – different voltages U_aidl and resistance R_a = 20 mОhm; b – different resistance R_a and voltage U_{a_idl} = 4,2 V

------- Ra= SmOhm

----- R„— ImOhm

b

Рис. 7. Зависимость коэффициента k _НЭ( I _вх): а – при различных напряжениях U_{a хх} и сопротивлении R_a = 20 мОм; б – при различных сопротивлениях R_a и напряжении U_{a_хх} = 4,2 В

Fig. 8. Amplitude phase characteristic W МLD ( j ω) for k NE = 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 W _МЛЧ( jω ) при k _НЭ = 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 W МLD ( j ω) linear part, the condition of absolute processes stability (7) is fulfilled irrespective of the power value P а of the LIA charge/discharge.

• 2.    The hodographs APC MLD W МLD ( j ω) 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 U_{in_FB} at the input of the BTC (upper graph of the oscillogram)

and the current of the battery I_а (lower graph of the oscillogram) with a linear discharge power surge of the battery from P 3 = 3 W to P 3 = 640 W. At the same time, the rate of battery power surge is V I = 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 I а . 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 σ 2 voltage U BTC 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 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.

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 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.