A Cell-to-Cell Equalizer Based on Three-Resonant-State ... - MDPI

energies Article

A Cell-to-Cell Equalizer Based on Three-Resonant-State Switched-Capacitor Converters for Series-Connected Battery Strings Yunlong Shang, Qi Zhang, Naxin Cui * and Chenghui Zhang * School of Control Science and Engineering, Shandong University, Jinan 250061, China; [email protected] (Y.S.); [email protected] (Q.Z.) * Correspondence: [email protected] (N.C.); [email protected] (C.Z.); Tel.: +86-531-8839-2907 (N.C.); +86-531-8839-5717 (C.Z.) Academic Editors: Rui Xiong, Hailong Li and Joe (Xuan) Zhou Received: 23 December 2016; Accepted: 6 February 2017; Published: 11 February 2017

Abstract: Due to the low cost, small size, and ease of control, the switched-capacitor (SC) battery equalizers are promising among active balancing methods. However, it is difficult to achieve the full cell equalization for the SC equalizers due to the inevitable voltage drops across Metal-Oxide-Semiconductor Field Effect Transistor (MOSFET) switches. Moreover, when the voltage gap among cells is larger, the balancing efficiency is lower, while the balancing speed becomes slower as the voltage gap gets smaller. In order to soften these downsides, this paper proposes a cell-to-cell battery equalization topology with zero-current switching (ZCS) and zero-voltage gap (ZVG) among cells based on three-resonant-state SC converters. Based on the conventional inductor-capacitor (LC) converter, an additional resonant path is built to release the charge of the capacitor into the inductor in each switching cycle, which lays the foundations for obtaining ZVG among cells, improves the balancing efficiency at a large voltage gap, and increases the balancing speed at a small voltage gap. A four-lithium-ion-cell prototype is applied to validate the theoretical analysis. Experiment results demonstrate that the proposed topology has good equalization performances with fast equalization, ZCS, and ZVG among cells. Keywords: battery equalizers; battery management systems; switched-capacitor (SC) converters; zero-voltage gap (ZVG); modularization; electric vehicles (EVs)

1. Introduction The world is being confronted with unprecedented crises, i.e., the depletion of fossil fuels and the global warming [1]. Energy conservation is becoming of paramount concern to people. In response to the crises, electric vehicles (EVs) have been implemented and are considered to be the inevitable development trend of vehicles for the future [2]. Due to high energy density, long lifetime, and environmental friendliness, lithium-based batteries have been dominating the high power battery packs of EVs [3,4]. However, the terminal voltage of a single lithium battery cell is usually low, e.g., 3.7 V for lithium-ion batteries and 3.2 V for lithium iron phosphate (LiFePO4) batteries [5,6]. In order to meet the demands of the load voltage and power, lithium batteries are usually connected in series and parallel [7]. For example, Tesla Model S uses 7616 lithium-ion 18650 cells connected in series and parallel [8]. Unfortunately, there are slight differences among cells in terms of capacity and internal resistance, which cause the cell voltage imbalance as the battery string is charged and discharged. On the one hand, this imbalance reduces the available capacity of battery packs. On the other hand, it may lead to over-charge or over-discharge for a cell in the battery pack, increasing safety risks. In fact, the most viable solution for this problem might not originate merely from the improvement in the

Energies 2017, 10, 206; doi:10.3390/en10020206

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battery chemistry. also uses suitable Itpower electronics to prevent the cell imbalance, improvement in theItbattery chemistry. also uses suitabletopologies power electronics topologies to prevent which is known as battery equalization. the cell imbalance, which is known as battery equalization. During the the last last few few years, have been been proposed, proposed, which which can can be During years, many many balancing balancing topologies topologies have be classified classified into two categories: the passive balancing methods [7,9] and the active balancing methods [10–32]. into two categories: the passive balancing methods [7,9] and the active balancing methods [10–32]. The The passive equalizers employ a resistor connected in parallel with each cell to drain excess energy passive equalizers employ a resistor connected in parallel with each cell to drain excess energy from from theenergy high energy cellsThese [7,9]. methods These methods the outstanding advantages of small size, the high cells [7,9]. have thehave outstanding advantages of small size, low cost, low cost, and easy implementation. However, their critical disadvantages are energy dissipation and easy implementation. However, their critical disadvantages are energy dissipation and heat and heat management [7]. To overcome these drawbacks, cell balancing topologies management problemsproblems [7]. To overcome these drawbacks, activeactive cell balancing topologies are are proposed, which employ non-dissipative energy-shuttling elements to energy move energy from the proposed, which employ non-dissipative energy-shuttling elements to move from the strong strong cells to the weak ones [7], reducing energy loss. Therefore, active balancing methods have cells to the weak ones [7], reducing energy loss. Therefore, active balancing methods have higher higher balancing and efficiency than the passive equalization can be further balancing capacitycapacity and efficiency than the passive equalization ones. Theyones. can beThey further divided into divided into three groups, which are capacitor based [10–18], inductor based [19–21], and transformer three groups, which are capacitor based [10–18], inductor based [19–21], and transformer based [22–32] based [22–32] methods. Among these active balancing topologies, switched-capacitor (SC) based methods. Among these active balancing topologies, switched-capacitor (SC) based solutions have the solutions advantages have the inherent advantages smaller size, lower cost, simpler higher efficiency. inherent of smaller size, of lower cost, simpler control, and control, higher and efficiency. Ref. [10] Ref. [10] proposes an SC equalizer series battery shown1a, in Figure 1a, oneiscapacitor is proposes an SC equalizer for series for battery packs. As packs. shownAs in Figure one capacitor employed employed to shift charge between adjacent twocapacitor cells. The switched back and forth to shift charge between the adjacentthe two cells. The is capacitor switched is back and forth repeatedly, repeatedly, which the imbalanced until two cell voltages match completely [10]. which diffuses thediffuses imbalanced charge untilcharge the two cellthe voltages match completely [10]. The main The main disadvantage of this structure is the high switching loss. To solve this problem, an automatic disadvantage of this structure is the high switching loss. To solve this problem, an automatic equalization circuit circuit based based on resonant SC converters converters is is proposed proposed in [15]. As As shown shown in Figure 1b, an equalization inductor LL0 0is is added to form a resonant inductor-capacitor (LC) converter, which operates alternatively added to form a resonant inductor-capacitor (LC) converter, which operates between the charging and discharging with zero-current (ZCS) to automatically alternatively between state the charging state andstate discharging state withswitching zero-current switching (ZCS) to balance the cellbalance voltages it is difficult to apply this topology to the with automatically the[15]. cellHowever, voltages [15]. However, it is difficult to apply thissystems topology to low the voltage gap among cells. For example, the voltage difference among lithium-ion battery cells is not systems with low voltage gap among cells. For example, the voltage difference among lithium-ion allowedcells to exceed 0.1 V [15]. This small difference causes the Metal-Oxide-Semiconductor battery is not allowed to exceed 0.1 Vvoltage [15]. This small voltage difference causes the Metal-OxideField Effect Transistor (MOSFETs) of the equalizersoftothe failequalizers to conduct, in the inevitable Semiconductor Field Effect Transistor (MOSFETs) towhich fail toresults conduct, which results residual voltage gap among cells. Moreover, the equalization current becomes smaller as the voltage in the inevitable residual voltage gap among cells. Moreover, the equalization current becomes gap getsassmaller, resulting in a very longresulting balancing smaller the voltage gap gets smaller, in time. a very long balancing time. Q0

Q0

Q0

B0

B0

B0

Q1

Q1

Q1

L0

L0

C0

C0

C0

Q4

Q2 B1

Q3

(a)

Q2

Q2

B1

B1

Q3

(b)

Q3

(c)

Figure equalizers based on switched-capacitor (SC) converters. (a) the classical equalizer Figure 1.1.Battery Battery equalizers based on switched-capacitor (SC) converters. (a) the SC classical SC [10]; (b) the resonant SC equalizer [15]; and (c) the proposed equalizer based on an inductor-capacitorequalizer [10]; (b) the resonant SC equalizer [15]; and (c) the proposed equalizer based on an switch (LCS) converter. (LCS) converter. inductor-capacitor-switch

In order to overcome these problems, a battery equalizer is proposed based on a resonant LC In order to overcome these problems, a battery equalizer is proposed based on a resonant LC converter and boost converter that offers several major advantages, e.g., ZCS and zero-voltage gap converter and boost converter that offers several major advantages, e.g., ZCS and zero-voltage gap (ZVG) among cells, etc. [16]. However, the balancing efficiency of this topology is strongly related to (ZVG) among cells, etc. [16]. However, the balancingefficiency of this topology is strongly related to e  Voutput Vin the voltage conversion ratio, which is expressed as . The lower the conversion ratio (or the voltage conversion ratio, which is expressed as ηe = Voutput /V in . The lower the conversion ratio the larger the voltage difference), the larger the balancing current, balancing (or the larger the voltage difference), the larger the balancing current,but butthe the lower lower the the balancing efficiency. This means that high efficiency cannot be achieved at a large voltage gap. [33] efficiency. This means that high efficiency cannot be achieved at a large voltage gap. Ref. [33] Ref. proposes proposes a high-efficiency SCthat converter thatthe decouples efficiency from the voltage conversion a high-efficiency SC converter decouples efficiencythe from the voltage conversion ratio. Ref. [34] ratio. Ref. [34] applies the switched-capacitor gyrator to photovoltaic systems, demonstrating ultimate improvement in the power harvesting capability under different insolation levels. Based on

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gyrator to photovoltaic systems, demonstrating ultimate improvement 3 of 16 in the power harvesting capability under different insolation levels. Based on these works, the objective of this is to of introduce an adjacent cell-to-cell battery equalization topology based on these works, thepaper objective this paper is to introduce an adjacent cell-to-cell battery equalization three-resonant-state LC converters, with potentialwith of fulfilling the expectations of high current topology based on three-resonant-state LCthe converters, the potential of fulfilling the expectations capability, high efficiency, easy modularization, ZCS, and ZVG among cells. As shown in Figure 1c, of high current capability, high efficiency, easy modularization, ZCS, and ZVG among cells. As except the classical an classical additional switch to be connected with the shown in Figure 1c, design, except the design, anQadditional Q4 is addedintoparallel be connected in 4 is addedswitch LC tank,with which hereinafter to be referred as to thebeinductor-capacitor-switch (LCS) converter.(LCS) This parallel theisLC tank, which is hereinafter referred as the inductor-capacitor-switch structure obtains another resonant current resonant path to release thepath residual energy in the capacitor to converter. This structure obtains another current to release thestored residual energy stored thethe inductor, which the foundations the bi-directional flow and weakens the in capacitor to thelays inductor, which lays to theachieve foundations to achieve thepower bi-directional power flow and couplingsthe of acouplings large voltage with low efficiency small voltage withvoltage slow balancing weakens of a gap large voltage gap with and lowaefficiency and gap a small gap withspeed. slow balancing speed. 2. The Proposed Equalizer

2. The Proposed Equalizer 2.1. Basic Circuit Structure 2.1. Basic Circuitin Structure As shown Figure 2, the proposed equalizer can be easily extended to a long series battery string without limit. The architecture consists of n battery cells connected in series and n − 1 resonant As shown in Figure 2, the proposed equalizer can be easily extended to a long series battery LCS tanks connected in parallel with each two adjacent battery cells, through which energy can be string without limit. The architecture consists of n battery cells connected in series and n − 1 resonant exchanged among all cells. LCS tanks connected in parallel with each two adjacent battery cells, through which energy can be The proposed equalizer has several major advantages per the following: exchanged among all cells. advantages per the (1) The The proposed proposed equalizer equalizer has can several achievemajor ZCS for all MOSFETs, andfollowing: obtain ZVG among cells. (2) (1) (2) (3) (3)

Dueproposed to the other resonant efficiency is improved at a large The equalizer cancurrent achievepath, ZCSthe for balancing all MOSFETs, and obtain ZVG among cells.voltage gap among cells, resonant and the balancing speed increased at a small voltage gap. at a large voltage Due to the other current path, theis balancing efficiency is improved gap among cells, the balancing is increased at a small voltage gap. By changing the and parameters of the speed resonant LCS converter, different balancing speeds can be By changing the parameters of the of resonant converter, achieved to meet the requirements differentLCS energy storagedifferent devices. balancing speeds can be to meet the requirements of different energy devices. (4) achieved The concept is modular [35], and the topology can bestorage extended to any long series-connected (4) The concept is or modular [35],cells andwithout the topology battery strings individual limit. can be extended to any long series-connected battery strings or individual cells without limit.

Q10

Q11

Q14

Q34

Qn-2,4

L10 C10

L30 C30

Ln-2,0 Cn-2,0

Q12

B0

Q13

Q30

B1

Q20

Q31

Q32

Q33

Qn-2,0 Qn-2,1

B2

Q21

Q22 L20 C20 Q24

Bn-3

Q23

Qn-3,0 Qn-3,1

Qn-3,2 Qn-3,3

Ln-3,0 Cn-3,0 Qn-3,4

Qn-2,2 Qn-2,3 Bn-2

Bn-1

Qn-1,0 Qn-1,1

Qn-1,2 Qn-1,3

Ln-1,0 Cn-1,0 Qn-1,4

Figure 2. Schematic diagram of the proposed system for n series-connected battery cells. Figure 2. Schematic diagram of the proposed system for n series-connected battery cells.

2.2. Operation Principles 2.2. Operation Principles In order to simplify the analysis for the operation states, the following assumptions are made: In order to simplify the analysis for the operation states, the following assumptions are made: the the proposed equalizer is applied to two cells connected in series, i.e., B0 and B1, where B0 is overproposed equalizer is applied to two cells connected in series, i.e., B0 and B1 , where B0 is over-charged charged and B1 is undercharged. The operation principles are shown in Figure 3. The switching and B1 is undercharged. The operation principles are shown in Figure 3. The switching sequence is set sequence is set as (Q0, Q2), (Q1, Q3), and Q4, as shown in Figure 4. Three resonant states S1–S3 are as (Q0 , Q2 ), (Q1 , Q3 ), and Q4 , as shown in Figure 4. Three resonant states S1 –S3 are employed to charge, employed to charge, discharge, and release the LC tank, which is connected to a voltage of VB0, VB1, discharge, and release the LC tank, which is connected to a voltage of VB0 , VB1 , or 0 in each switching or 0 in each switching state, respectively. Figure 5 shows the theoretical waveforms of the proposed state, respectively. Figure 5 shows the theoretical waveforms of the proposed equalizer at VB0 > VB1 . equalizer at VB0 > VB1.

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B0 B B00

4 of 16 4 of 16 4 of 16

Q0 Q0 Q0

B0 B B0 0

Q1 iL0 Q1 ii L0 Q1 L0

L0 LL0 0 C0 C C0

Q2

0

Q22 Q

B1

B11 B

Q4 vQQ4

iL0

C0 4

i iL0 BL01

vvC0C0

BB 1 1

Q3

Q0 Q0 Q0 Q1 Q Q1 1

Q2

B0 B B0 0 L0 L L0 0 C0 C0C0

Q2Q2 Q3

Q1 Q1

Q1

Q4 Q4 Q vC04

Q2

vC0vC0

Q3Q3

Q33 Q

Q0 Q0

Q0

B1

Q2 Q2

B1 B1

Q3

L0 Q4 L0 L0 C0 vQC04 Q4 C0 C0vi vC0 L0 C0 iL0 iL0

Q3 Q3

(a) (b) (c) (a) (b) (a) (b) (c)(c) Figure 3. Operating states of the proposed equalizer at VB0 > VB1. (a) charge state S1; (b) discharge state S2; Figure 3. Operating states of the proposed equalizer at VB0 > VB1 . (a) charge state S1 ; (b) discharge (c) release state S3. states Figure 3. Operating Operating statesofofthe theproposed proposedequalizer equalizeratat >B1V.B1(a) . (a) charge state 1; (b) discharge state 3. VB0VB0 >V charge state S1; S(b) discharge state S2; S2; state S(c) (c) release state S3 . 2 ; release state S3. (c) release state S3.

Q0, Q2 QQ00, ,QQ2 2 Q1, Q3 QQ11, ,QQ3 3

Voltage (V) or Current (A)(A) Voltage (V) or Current (A) Voltage (V) or Current

Q4 QQ4 4 Figure 4. Switching sequences of the proposed equalizer at VB0 > VB1. Figure 4. Switching sequences of the proposed equalizer at VB0 > VB1.

7

Figure 4. Switching sequences ofofthe atVVB0B0 Figure 4. Switching sequences theproposed proposed equalizer equalizer at > V>B1V . B1 .

PWM1 Vh1 76 PWM2 7 PWM1 PWM3 Vh1 65 PWM2 i PWM1 V 6 h1 vC0 PWM3 Vc PWM2 54 i PWM3 5 vC0 Vc i 43 vC0 Vc 4 32 iL0 PWM Vr 3 21 iL0 PWM Vr 2 iL0 PWM Q0, Q2 ON Q1, Q3 ON VQr4 ON 10 -Vh2 Q0, Q2 ON Q1, Q3 ON Q4 ON 1 0-1 Q0, Q2 ON Q1, Q3 ON Q4 ON -Vh2 S2 S1 0 S 3 t2 t3 -1-2 t1 t0 -Vh2 2.2 2.4 2.6 2.8 3 S2 3.2 S 3.4 3.6 3.8 4 S1 -1 2 3 t t3 -3 t1 (s) -2 t0 2 Time S1 3 S2 3.2 S33.4 x 104 2 2.2 2.4 2.6 2.8 3.6 3.8 t2 and the t3 resonant current at-3VB0 > VB1. t1 -2 Figure 5. Theoretical waveforms oft0the capacitor voltage 2 2.2 2.4 2.6 2.8 Time3(s) 3.2 3.4 3.6 3.8x 10 4 Figure 5. Theoretical waveforms of the capacitor voltage and the resonant current at VB0 -3> VB1. Time (s) x 10

Charge State S1 [twaveforms 0-t1]:waveforms At t0, switches Q0 andvoltage Q 2 are and turned ON withcurrent ZCS. at The Figure 5. Theoretical of the capacitor voltage and VB0VLC >B0VB1 . VB1is Figure 5. Theoretical of the capacitor the resonant resonant current at >tank . connected B0 Sin1 parallel through Q0 andQQ0 2and , as shown Figure ON 3a. Bwith 0, L0, and formLC a resonant Chargewith State [t0-t1]: At t0, switches Q2 areinturned ZCS.C0The tank is Charge State S1parallel [t0-t1]:Cthrough At 0, switches and Q2 are turned Theaof LC tank is current loop. The capacitor 0 is tcharged by B 02.,0vas C0 increases from −V3a. h2,ON which is aZCS. remnant C0 from connected with B0 in Q0 and QQ shown in Figure B 0, Lwith 0, and C 0 form resonant Charge State S(see [t -t ]: At tL00 ,and switches Q and Qbe are turned ON with ZCS. The LC tank is 1 0 1 0 2 the last period Figure 5). i v C0 in this state can expressed as connected with B 0 in parallel through Q 0 and Q 2 , as shown in Figure 3a. B 0 , L 0 , and C 0 form a resonant current loop. The capacitor C0 is charged by B0. vC0 increases from −Vh2, which is a remnant of C0 from connected with B0The in parallel Q Q , as increases shown infrom Figure 3a. B0 , Lis0 , aand C0 form resonant 0 and current loop. capacitor charged B2state 0. vC0 can −Vas h2, which remnant of Ca0 from the last period (see Figure through 5).Ci0L0isand vC0 in by this be expressed VB 0  Vh 2  n ( t  t0 ) 2   h2 , which is a remnant of C0 current The capacitor . vC0can increases from − V )i  eB sin  1    ( t  t ) 0(tis 0 theloop. last period (see FigureiCL 05). L0charged and vC0 inby this state be expressed as 0 (1)  n  VB 0 1Vh 2 2  n (t t0 ) 2   can r iL 0 (t ) 5).  ZiL0 e in this  sinstate 1  be  (expressed t  t0 )  , as from the last period (see Figure and2 vC0 n (1)   V V  t0 ) 2  iL 0 (t ) Z r  B 01  h 2  e n (t  sin   n  1 q  (t  t0 ),    (1) n ( t  t0 ) 2   e  Z  1   2 V + V  r B0 h2 v ( t )   V  ( V  V )  1  − ρω ( t− t0()t t )  cos n  1    (2t  t0,)   n  (2) C 0 h 2 B 0 h 2  p  t0 ) , · e  e n ·0 2sin ωn · 1 − ρ · (t − i L0 (t) = (1)  vC 0 (t ) Z  rV · h 2 1(V−B 0ρ2Vh 2 )  1  1 (2t t )cos n  1   2  (t  t0 )   , (2) n 0     e1      v (t )  V  (V  V(  cos n  1   2  (t  t0) ,  ) (2) h 2 )  1   parasitic 2 . R  represents   −R 2 Z where Z r  L0 C 0 C ,0 n  1h 2 L0CB0 0 , and the equivalent q S r S ρω ( t − t ) 1   n 0  e ,   2 L0 − CV n(V  1B0 + L0C RS 2Zr . RS· cos where , + represents vC0Zr(tin )= V0h2 ,) ·and1 − p ωn · 1the − ρequivalent · (t − t0 )parasitic , (2) resistance each current path. 0 h2 2

1−ρ TheZcharge ends when (1), the of thisparasitic state resistance in each L0 state Ccurrent npath.  1 Li0L0C0crosses  at  RtS= 2t1Z. rFrom where , andzero . RS Equation represents theduration equivalent r  0 , √ √ The charge state ends when i L0 crosses zero at t = t1. From Equation (1), the duration of this state whereresistance Z = inL each /C ,current ω =path. 1/ L C , and ρ = R /2Z . R represents the equivalent parasitic r

0

0

n

0 0

S

r

S

charge state ends when iL0 crosses zero at t = t1. From Equation (1), the duration of this state resistanceThe in each current path. The charge state ends when iL0 crosses zero at t = t1 . From Equation (1), the duration of this state is determined by

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∆t = t1 − t0 =

ωn ·

π p . 1 − ρ2

(3)

At t1 , vC0 is positively charged to Vh1 , which can be given by e−ρωn ∆t Vh1 = vC0 (t1 ) = (VB0 + Vh2 ) · (1 + p ) − Vh2 . 1 − ρ2

(4)

Discharge State S2 [t1 -t2 ]: At t1 , the switches Q1 and Q3 are turned ON with ZCS, connecting B1 to the resonant LC tank. B1 , L0 , and C0 form a resonant loop. B1 is charged by C0 . iL0 and vC0 in this state are given as i L0 (t) = −

  q Vh1 − VB1 p · e−ρωn (t−t1 ) · sin ωn · 1 − ρ2 · (t − t1 ) , Zr · 1 − ρ2 (

vC0 (t) = Vh1 − (Vh1 − VB1 ) ·

 ) q e−ρωn (t−t1 ) · cos ωn · 1 − ρ2 · (t − t1 ) . 1− p 1 − ρ2

(5)

(6)

At t = t2 , the discharge state ends when iL0 drops to zero. The voltage Vr of C0 at t = t2 is represented by e−ρωn ∆t Vr = Vh1 − (Vh1 − VB1 ) · (1 + p ). (7) 1 − ρ2 Release State S3 [t2 -t3 ]: During this state, the resonant LC tank is short-circuited by turning on the switch Q4 with ZCS. This releases the residual charge of the capacitor into the inductor and even charges reversely the capacitor C0 , so B0 can charge C0 with a large current at the beginning of S1 . This state provides the opportunity to transfer energy from a low voltage cell to a high voltage one, which lays the foundations to achieve ZVG among cells. iL0 and vC0 in this state are given by i L0 (t) = −

  q V pr e−ρωn (t−t2 ) · sin ωn · 1 − ρ2 · (t − t2 ) , Zr · 1 − ρ2

  q e−ρωn (t−t2 ) · cos ωn · 1 − ρ2 · (t − t2 ) . vC0 (t) = Vr · p 1 − ρ2

(8)

(9)

The release state ends when iL0 crosses zero at t = t3 . The voltage Vh2 of C0 at t = t3 can be expressed as

−Vh2 where

  q e−ρωn (t3 −t2 ) 2 · cos ωn · 1 − ρ · (t3 − t2 ) = −λVr , ≡ vC0 (t3 ) = Vr · p 1 − ρ2

√ 2 e−ρωn ∆t e−πρ/ 1−ρ = p . λ= p 1 − ρ2 1 − ρ2

(10)

(11)

By solving Equations (4), (7), and (10), Vh1 , Vr , and Vh2 can be calculated as Vh1 =

VB0 + λ2 VB1 , 1 − λ + λ2

(12)

Vr =

VB1 − λVB1 , 1 − λ + λ2

(13)

λ(VB1 − λVB0 ) . 1 − λ + λ2

(14)

Vh2 =

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Vr 

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VB1  VB1 , 1   2

(13)

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 (VB1  VB 0 ) Vh 2  . (14) The operating period T is composed of three1 resonant    2 states, which can be expressed as

√ The operating period T is composed of three resonant states, which can be expressed as

3π · L0 C0 3π p = p . (15) 2 ωn · 31− ρ 3  L10− C0 ρ2 T  . (15) n  1   2 1  2 The direction of the balancing power flowing can be changed by controlling the switching sequences. above analysis, the switching sequence The According direction of to thethe balancing power flowing can be changed by (Q controlling the switching 0 , Q2 ), (Q 1, Q 3 ), Q4 is to deliver energyAccording from B0 totoBthe the case of energy transferred from B(Q the1, switching sequences. above analysis, the switching sequence , QB20),, (Q Q3), Q4 is tosequence deliver is 1 . In 1 0to energy 1. 0 In the case of energy transferred from B 1 to B 0 , the switching sequence is changed changed tofrom (Q1 , BQ0 3to ), B(Q ,Q ), Q . Figure 6 shows the three consecutive operating states of the proposed 2 4 to (Q 1 , Q 3 ), (Q 0 , Q 2 ), Q 4 . Figure 6 shows the three consecutive operating states of the proposed equalizer: (a) charge state; (b) discharge state; and (c) release state at VB0 < VB1 . Figure 7 shows the equalizer: (a)switching charge state; (b) discharge state; and (c) by release state at the VB0 VB1 . at switching sequences VB0 > V10 B1. Figures 10 and 11 show the three consecutive operating states Q4 and the corresponding sequences at Vsequences The principlesprinciples of this system B0 < VB1 . at without using Q4 and theswitching corresponding switching VB0operation < VB1. The operation of are this similar to those shown in Figures 3–6 and will not be described here in detail. system are similar to those shown in Figures 3–6 and will not be described here in detail. T=

Q0

iL0

B0

Q0

Q0

B0

B0 Q1

Q1

Q1 L0 Q4

Q4

Q4

C0

C0

C0

Q2

Q2

Q2

B1

iL0

L0

L0

iL0

B1

B1 Q3

Q3

Q3

(a)

(b)

(c)

Figure 6. Operating states of the proposed equalizer at VB0 < VB1. (a) charge state; (b) discharge state;

Figure 6. Operating states of the proposed equalizer at VB0 < VB1 . (a) charge state; (b) discharge state; (c) release state. (c) release state. Energies 2017, 10, 206 7 of 16

Q1, Q3

Q0, Q2

Q4 Figure Switchingsequences sequences of of the the proposed < V< B1. Figure 7. 7.Switching proposedequalizer equalizeratatVB0 VB0 VB1 . Q0

Q0 B0 iL0

Q0

B0

B0 Q1

Q1 L0 C0 Q2 Q3

(a)

L0

C0

iL0

C0

Q2

B1

B1

Q1 L0

B1 Q3

(b)

Q2 iL0 Q3

(c)

Q0, Q2 Q0,, Q Q2 Q 0 2 Q4 Q4 Q 4

Energies 2017, 10, 206

7 of 15

Figure 7. Switching sequences of the proposed equalizer at VB0 < VB1. Figure 7. Switching sequences of the proposed equalizer at VB0 < VB1. Figure 7. Switching sequences of the proposed equalizer at VB0 < VB1.

B0 B0 B0 iL0 iL0 iL0

B1 B1 B1

Q0 Q0 Q0 Q1 Q1 Q1

Q2 Q2 Q2

B0 B0 B0

L0 L0 L0 C0 C0 C0

Q3 Q3 Q3

iL0 iL0 iL0 B1 B1 B1

(a) (a) (a)

Q0 Q0 Q0 Q1 Q1 Q1

Q2 Q2 Q2

B0 B B00 L0 L0 L0 C0 C0 C0

B1 B B11

Q3 Q3 Q3

(b) (b) (b)

Q0 Q Q00

Q1 Q Q11

Q2 Q Q22iL0 iL0 iL0 Q3 Q Q33

L0 L L00 C0 C C00

(c) (c) (c)

Figure 8. Operating states of the proposed equalizer without using Q4 at VB0 > VB1. (a) charge state; (b) Figure Operating states states of of the the proposed proposedequalizer equalizerwithout withoutusing usingQQ4 4atatVV (a) charge state; Figure 8. 8. Operating B0B0 > V>B1V. B1 (a). charge state; (b) Figure 8. Operating states of the proposed equalizer without using Q4 at VB0 > VB1. (a) charge state; (b) discharge state; (c) release state. (b) discharge state; (c) release state. discharge state; (c) release state. discharge state; (c) release state.

Q0 Q Q00 Q2 Q Q22

Q1 Q Q11 Q3 Q Q33

Figure 9. Switching sequences of the proposed equalizer without using Q4 at VB0 > VB1. Figure 9. Switchingsequences sequences of theproposed proposed equalizerwithout without usingQQ4atatVVB0 > >V B1. Figure VB1 B1 Figure9.9.Switching Switching sequencesofofthe the proposedequalizer equalizer withoutusing using Q4 4 at VB0 B0 > V ..

B0 B0 B0

B1 B1 B1iL0 iL0 iL0

Q0 Q0 Q0 Q1 Q1 Q1

Q2 Q2 Q2 Q3 Q3 Q3

(a) (a) (a)

iL0 iL0 iL0 B0 B0 B0 L0 L0 L0 C0 C0 C0

B1 B1 B1

Q0 Q0 Q0 Q1 Q1 Q1

Q2 Q2 Q2 Q3 Q3 Q3

(b) (b) (b)

B0 B0 B0 L0 L0 L0 C0 C0 C0

B1 B1 B1

Q0 Q0 Q0

Q1 Q1 Q1

Q2 Q2 Q2iL0 iL0 Q3iL0 Q3 Q3

(c) (c) (c)

L0 L0 L0 C0 C0 C0

Figure 10. Operating states of the proposed equalizer without using Q4 at VB0 < VB1. (a) charge state; Figure 10. Operating states of the proposed equalizer without using Q4 at VB0 < VB1. (a) charge state; Figure 10. Operating Operating states of state. the proposed proposedequalizer equalizerwithout withoutusing usingQQ4atatVVB0