The electric vehicle (EV) is targeted as an efficient method of decreasing CO2 emission and reducing dependence on fossil fuel. Compared with filling up the internal combustion engine (ICE) vehicle, the EV power charging time is usually long. However,to the best of our knowledge, the current charging strategy does not consider the battery state of health (SOH). It is noted that a high charging current rate may damage the battery life. Motivated by this, an optimal charging strategy is proposed in the present paper, providing several optimal charging options taking into account the EV battery health, trying to prevent ‘abused battery utilization’ happening

Conti, Matteo

McDonald, Stephen

Putrus, Ghanim

Tianxiang, Jiang

Zhiwei, Gao

English

Northumbria Research Link

Northumbria Research Link
Citation:  Tianxiang,  Jiang,  Putrus,  Ghanim,  Zhiwei,  Gao,  Conti,  Matteo  and  McDonald,  Stephen 
(2013) Optimal  Charging Strategy for  EVs with  Batteries at  Different  States of  Health.  In:  Fourth 
International Conference on Power Engineering, Energy and Electrical Drives (POWERENG 2013),  
13th - 17th May 2013, Istanbul, Turkey. 
URL:  http://doi.org/10.1109/PowerEng.2013.6635628 
<http://doi.org/10.1109/PowerEng.2013.6635628>
This version was downloaded from Northumbria Research Link: http://nrl.northumbria.ac.uk/21069/
Northumbria University has developed Northumbria Research Link (NRL) to enable users to access 
the University’s research output. Copyright © and moral rights for items on NRL are retained by the 
individual author(s) and/or other copyright owners.  Single copies of full items can be reproduced, 
displayed or performed, and given to third parties in any format or medium for personal research or 
study, educational, or not-for-profit purposes without prior permission or charge, provided the authors, 
title and full bibliographic details are given, as well as a hyperlink and/or URL to the original metadata 
page. The content must not be changed in any way. Full items must not be sold commercially in any  
format or medium without formal permission of the copyright holder.  The full policy is available online: 
http://nrl.northumbria.ac.uk/pol  i cies.html  
This  document  may differ  from the  final,  published version of  the research  and has been made 
available online in accordance with publisher policies. To read and/or cite from the published version 
of the research, please visit the publisher’s website (a subscription may be required.)
                        

Optimal Charging Strategy for EVs with Batteries at 
Different States of Health   
 
Tianxiang Jiang
*
, Ghanim Putrus, Matteo Conti, Zhiwei Gao, Steve McDonald  
1. Northumbria University, UK  
* Email address: tianxiang.jiang@northumbria.ac.uk 
 
  Abstract - The electric vehicle (EV) is targeted as an efficient 
method of decreasing CO2 emission and reducing dependence on 
fossil fuel. Compared with filling up the internal combustion 
engine (ICE) vehicle, the EV power charging time is usually long. 
However, to the best of our knowledge, the current charging 
strategy does not consider the battery state of health (SOH). It is 
noted that a high charging current rate may damage the battery 
life. Motivated by this, an optimal charging strategy is proposed in 
the present paper, providing  several optimal charging options 
taking into account the EV battery health, trying to prevent 
‘abused battery utilization’ happening.   
 
  Index Terms- Fast charging, Battery state of health (SOH), Cycle 
life, Optimization  
I.    INTRODUCTION 
Electric Vehicle (EV) and Plug-in Hybrid Electric Vehicle 
(PHEV) are being developed in a positive effort to deplete 
exhaust emissions, reduce the dependence on fossil fuel. To 
support the environmental and economic benefits which EVs 
bring, each government sets its own EV development plan, for 
example the UK government expects that EVs and PHEVs can 
take 40% of the motor market in 2020 [1]. However the 
development of this plan is not as encouraging as expected.  
Table 1 shows the barriers to EV uptake. Surmising from this 
Table, it is easy to find that the battery technology is the main 
bottleneck. Although, from lead-acid to nickel cadmium 
(NiCd), Nickel-metal hydride (NiMH), as far as lithium-ion (Li-
ion), Li-air and polymer Li-ion, the battery 
technologies have made remarkable achievements; the limited 
energy/power capacity and cycle life (compared with the 
internal combustion engine vehicle) affect the range, life and 
cost of EVs[2, 3].  
Due to the limited running range, EV customers need to 
recharge their EVs frequently. Currently, International 
Electrotechnical Commission (IEC) 62196 set of standards for 
charging of electric defining four modes of charging: 
 Mode 1: normal, slow charging, with a normal household 
socket. 
 Mode 2: same as model 1, but with an in-cable protection, 
the most common charging model today. 
 Mode 3: slow or fast charging with on board charger, 
using dedicated plugs and protection. 
 Mode 4: fast charging using an external charger, as for 
instance CHAdeMO. 
 
Table I. Ranking of barriers to EVs uptake [4] 
 
Fast charging is significant as it can help the users to finish 
charging in relatively short time compared with other charging 
models for the same capacity battery pack. For example, the 
CHAdeMO fast charger can support 50 kW to allow the Nissan 
LEAF to charge to 80% state of charge (SOC) in 20 minutes. 
This charging pattern is quite attractive for customers on a tight 
schedule. 
What is the reason for choosing the fast charger’s power rate 
to be 50 kW rather than 10 kW, or 160 kW?  Figure 1 indicates 
the reason from the manufacturer’s cost and users’ time point of 
view. 
 
Figure 1. Optimal output of fast charger [5] 
From Figure 1, it can be found that the total charging time is 
not linearly dependant on power, also the infrastructure costs 
are increasing with a higher power rate. The 50 kW charger has 
the best performance-price ratio. However, it is not the only 
consideration for choice of charging power rate. As the 
battery’s inherent characteristic, the battery cycle life will 
decrease with the increase in current rate. For instant, a specific 
Li-ion battery can be charged 4000 times charging at mode 1 
but reduces to 100 times under fast charging [6]. So the SOH of 
the EV battery pack should be considered as another factor for 
EV charger development or an additional factor added into the 
existing charger, especially, when EVs are widely used.   
As mentioned, the battery cycle life and charging time 
(current rate) are two inter-constraining factors. In order to 
balance and address this issue, in this paper, an optimized 
charging strategy is proposed. The charging current is 
determined by evaluating battery SOH and charging time. 
These two factors are qualified by cycle life and charging 
current. The mathematical objective function is developed and 
validated here. Generic Algorithm (GA) is utilized to determine 
the optimized point which gives a value for charging current.  
 
II.    OPTIMIZED CHARGING STRATEGY 
 
(i). Battery cycle life and current rate normalization  
Charging time and battery cycle life are the two factors that 
users focus on. Reducing the charging time can make users’ 
lives more flexible but reducing the the charging current 
(extend charging time) can extend battery life and reduce the 
cost [7, 8]. Flexibility is needed, but both these two items 
should be given consideration. Here, a mathematical model is 
built to quantify this abstract concept.  
As material, shape, manufacturing and testing environment 
are different for each manufacturer; there is no generic function 
to describe the aging behaviour. Normally, empirical or semi-
empirical equations based on experimental data are used to 
express batteries’ aging behaviours. Here [6], a specific battery 
                    is chosen as the example. These kinds of 
batteries are cycled with different current rates at room 
temperature and 70% depth of discharge (DOD). The test 
results (cycle numbers against current rate experimental data) 
are shown in Figure 2. 
 
 
Figure 2. battery cycle life at different current rates 
 
 
Figure 3. Normalized battery cycle life at different current rates 
 
 
Figure 4. Normalized battery cycle life at different normalized current rates 
 
From the data above, it can be found that the cycle life 
increases exponentially with decreasing current rate. Then 
normalize the cycle number to obtain a linear function as 
shown in Figure 3. 
By using coefficient normalization and curve fitting, a 
function between cycle life and current rate can be acquired. 
The least square error R
2
 is 0.9893 which proves the function 
fits quite well. Then normalizing the current rate (by dividing 
common base value 1.8), a new function y=-x+1 can be 
obtained as shown in Figure 4. 
 
(ii). Objective function  
Charging time is inversely related to the charging current rate. 
In order to maximize the battery life and minimize charging 
time, the value of charging current rate multiplied by cycle life 
should be maximized by using the objective function: 
 
max      z=x
m
y
n
                                     (1) 
where    ( )  
x is charging current rate; 
y is normalized battery cycle life; 
m, n are the weight factors; 
 
Before applying this objective function in EV smart charging 
control, it is necessary to prove this objective has maximum 
value; the validation is presented in the Appendix. 
Here the SOH of battery is divided into five regions 
according to the battery usable capacity (80% usable capacity is 
considered to be end of life) which is shown in Figure 5.  
 
Figure 5 usable capacity of battery health 
 
For different SOH of the battery, the weighting factors are 
different. The Table II defines weighting factors according to 
the battery SOH. The weighting factors are used to evaluate the 
importance of charging current and battery life cycles. m 
represents the weighting of current rate and n represents cycle 
number. Of course, the manufactures can set different health 
regions and weight factors according to the battery performance 
and user requirements.  
Implement the function (1) into Matlab and plot the curves 
for different weighting factors, as shown in Figure 6. Then 
calling Genetic Algorithm (GA) algorithm to determine the 
optimized value, the optimized normalized current rate values 
can be obtained. The real charging current rate can be 
determined as the normalized current rate multiplied by the 
common base value (1.8). Charging time is the reciprocal of the 
real charging current rate. All this information is shown in 
Table II. From Table II, it can be found that, as expected, the 
charging time will decrease or increase with the weighting of 
current rate or cycle number. For example, when n=1, with m 
increasing from 1 to 3, the charging time reduces from 66 to 43 
mins. Conversely, with weighting towards cycle number, the 
charging time increases from 66 to 150 mins. 
 
Figure 6 Objective function curves 
 
Table II Optimized charging plan 
 
(iii). Impact on the cycle life 
According to the mathematical expression of cycle number 
and current rate which is fitted from Figure 2; 
                 
the corresponding battery cycle life number can be determined 
according to the current rate as shown in Table III 
 
Table III cycle life charged at different current rates 
 
From Table III, it can be found that in principle this specific 
(                   ) battery only can be cycled 17 times at 
high current rate (1.38C) which equates to a charging time of 
43 mins. But the cycle life dramatically increases with 
decreasing charging current rate (charging time extended).  
 
III.    CONCLUSION  
  In order to reduce the adverse effect on the battery cycle life 
due to fast charging, an optimized charging strategy has been 
presented in this paper. Life cycle and charging time have been 
taken as two factors to evaluate the optimal charging current. 
The proposed charging strategy can offer several options 
according to the SOH of battery by using communications with 
the battery management system (BMS), which can relatively 
extend the battery life compared with fast charging technique.  
 
REFERENCES 
 
[1] E. C. Bentley, P. Suwanapingkarl, S. Weerasinghe, T. Jiang, G. A. 
Putrus, and D. Johnston, "The interactive effects of multiple EV 
chargers within a distribution network," in Vehicle Power and 
Propulsion Conference (VPPC), 2010 IEEE, pp. 1-6. 
[2] G. Sarre, P. Blanchard, and M. Broussely, "Aging of lithium-ion 
batteries," Journal of Power Sources, vol. 127, pp. 65-71, 2004. 
[3] V. Marano, S. Onori, Y. Guezennec, G. Rizzoni, and N. Madella, 
"Lithium-ion batteries life estimation for plug-in hybrid electric 
vehicles," in Vehicle Power and Propulsion Conference, 2009. 
VPPC '09. IEEE, 2009, pp. 536-543. 
[4] A. Ltd., "Electric Vehicles: Driving and change," WWF Scotland, 
April 2011. 
[5] Z. E. R. O. (ZERO), "Taking charge - Introducing fast chargers in 
Norway " 2010. 
[6] M. Majima, S. Ujiie, E. Yagasaki, K. Koyama, and S. Inazawa, 
"Development of long life lithium ion battery for power storage," 
Journal of Power Sources, vol. 101, pp. 53-59, 2001. 
[7] X. Wang, Y. Sone, H. Naito, C. Yamada, G. Segami, and K. Kibe, 
"Cycle-life testing of 100-Ah class lithium-ion battery in a simulated 
geosynchronous-Earth-orbit satellite operation," Journal of Power 
Sources, vol. 160, pp. 602-608, 2006. 
[8] R. B. Wright, J. P. Christophersen, C. G. Motloch, J. R. Belt, C. D. 
Ho, V. S. Battaglia, J. A. Barnes, T. Q. Duong, and R. A. Sutula, 
"Power fade and capacity fade resulting from cycle-life testing of 
Advanced Technology Development Program lithium-ion batteries," 
Journal of Power Sources, vol. 119-121, pp. 865-869, 2003. 
[9] P. Ramadass, B. Haran, R. White, and B. N. Popov, "Mathematical 
modeling of the capacity fade of Li-ion cells," Journal of Power 
Sources, vol. 123, pp. 230-240, 2003. 
[10] R. Spotnitz, "Simulation of capacity fade in lithium-ion batteries," 
Journal of Power Sources, vol. 113, pp. 72-80, 2003. 
 
 
APPENDIX 
Objective function validation: 
Here are the constraints: 
a, b are the current rate values 
 
  [   ]       
m, n are weight factors,  
 
      
the function of    ( )  is linear or convex and strictly 
decrease, which can be express as: 
 
                                     {
 ( )   
  ( )   
   ( )   
                                      (2) 
  
Validation: 
If function z has maximum value, using mathematic expression, 
which can be written as: 
 
{
    
     
     [   ]                       (3) 
 
First, prove     . Here reduction to absurdity is used: 
Assume     , then  
 
                                      {
    
  
    
      [   ]                       (4) 
 
If assume     , then function z strictly decrease, then can 
obtain 
: 
 ( )   ( )                                  (5) 
 
But on the other side,  
If current rate a is relatively small, then    ;  
If current rate is very large, then the battery cycle life  ( ) will 
be very limited and approaching to zero ( ( )   ).  
 
Due to           ( ) , 
 ( )     ( )    
 ( )     ( )    
 
So  
 ( )   ( )                                      (6) 
Which is conflict with equation (5). 
 
Similarly ， it can be obtained that if assume     , then 
function z strictly increase,  ( )   ( ), which is also conflict 
with equation (5). 
 
So,  
                          [   ]            
 (  )   ;                 (7) 
which means function z has peak value. 
 
 
Secondly, we need to prove      ; It is easy to obtain: 
 
        [ ( )]     [ ( )]     ( )                       (8) 
 
       (   )    [ ( )]         [ ( )]     ( ) 
      (    )  [ ( )]   [  ( )]     [ ( )]      ( )        
(9) 
 
As previous mentioned that,  
 
     [   ]            
 (  )    
 
So from equation (8), it can be got that: 
 
 (  )
     
   [ (  )]
     
 [ (  )]
     (  )    
 
which can be given as: 
 
 (  )   
  
 
  (  )                               (10) 
 
Substitute equation (10) into (9),  
 
 (  )
   [ (  )]
     
   {
   
 
    
 [  (  )]
  
                            
 [  (  )]
      (   )  
 [  (  )]
  
                        
  
 
  
   (  )   (  )}                                             (11) 
 
Transfer equation (11), it can be obtained that 
 (  )
   [ (  )]
     
   {    
 [  (  )]
 (
 
 
  )  
                       
  
 
  
   (  )   (  )}                                           (12) 
 
Compare equation (12) with zero. From equation (2), it can be 
found that: 
 
      [   ]          ( )    
   
      
      
   
   [ (  )]
       
       (  )     
  (  )    
 {    
 [  (  )]
 (
 
 
  )  
  
 
  
   (  )   (  )}     
  (  )
                                                                              (13) 
 
The equation (7) prove optimized function (1) has peak value, 
and equation (13) validate that this peak value is maximum 
value.  
 
So in summary, this objective function has optimized point 
subject to the constraints.  
 
 
 


Optimal Charging Strategy for EVs with Batteries at

Different States of Health

Citation:  Tianxiang,  Jiang,  Putrus,  Ghanim,  Zhiwei,  Gao,  Conti,  Matteo  and  McDonald, Stephen (2013)  Optimal  Charging Strategy for  EVs  with  Batteries  at  Different  States  of Health. In:  Fourth International Conference on Power Engineering,  Energy and Electrical Drives (POWERENG 2013), 13th - 17th May 2013, Istanbul, Turkey. URL:  http://doi.org/10.1109/PowerEng.2013.6635628 <http://doi.org/10.1109/PowerEng.2013.6635628>This  version  was  downloaded  from  Northumbria  Research  Link: http://nrl.northumbria.ac.uk/21069/Northumbria University has developed Northumbria Research Link (NRL) to enable users to access the University’s research output. Copyright © and moral rights for items on NRL are retained by the individual author(s) and/or other copyright owners.  Single copies of full items can be reproduced,  displayed or  performed,  and given to  third parties in  any format  or medium for personal research or study, educational, or not-for-profit purposes without prior permission or charge, provided the authors, title and full bibliographic details are given, as well  as a hyperlink and/or URL to the original metadata page. The content must  not  be changed in any way. Full items must not be sold commercially in any format or medium without  formal  permission  of  the  copyright  holder.   The  full  policy  is  available  online: http://nrl.northumbria.ac.uk/policies.htmlThis document may differ from the final, published version of the research and has been made available online in accordance with publisher policies. To read and/or cite from the published version of the research, please visit the publisher’s website (a subscription may be required.)Optimal Charging Strategy for EVs with Batteries at Different States of Health    Tianxiang Jiang*, Ghanim Putrus, Matteo Conti, Zhiwei Gao, Steve McDonald  1. Northumbria University, UK  * Email address: tianxiang.jiang@northumbria.ac.uk    Abstract - The electric vehicle (EV) is targeted as an efficient method of decreasing CO2 emission and reducing dependence on fossil fuel. Compared with filling up the internal combustion engine (ICE) vehicle, the EV power charging time is usually long. However, to the best of our knowledge, the current charging strategy does not consider the battery state of health (SOH). It is noted that a high charging current rate may damage the battery life. Motivated by this, an optimal charging strategy is proposed in the present paper, providing  several optimal charging options taking into account the EV battery health, trying to prevent ‘abused battery utilization’ happening.      Index Terms- Fast charging, Battery state of health (SOH), Cycle life, Optimization  I.    INTRODUCTION Electric Vehicle (EV) and Plug-in Hybrid Electric Vehicle (PHEV) are being developed in a positive effort to deplete exhaust emissions, reduce the dependence on fossil fuel. To support the environmental and economic benefits which EVs bring, each government sets its own EV development plan, for example the UK government expects that EVs and PHEVs can take 40% of the motor market in 2020 [1]. However the development of this plan is not as encouraging as expected.  Table 1 shows the barriers to EV uptake. Surmising from this Table, it is easy to find that the battery technology is the main bottleneck. Although, from lead-acid to nickel cadmium (NiCd), Nickel-metal hydride (NiMH), as far as lithium-ion (Li-ion), Li-air and polymer Li-ion, the battery technologies have made remarkable achievements; the limited energy/power capacity and cycle life (compared with the internal combustion engine vehicle) affect the range, life and cost of EVs[2, 3].  Due to the limited running range, EV customers need to recharge their EVs frequently. Currently, International Electrotechnical Commission (IEC) 62196 set of standards for charging of electric defining four modes of charging:  Mode 1: normal, slow charging, with a normal household socket.  Mode 2: same as model 1, but with an in-cable protection, the most common charging model today.  Mode 3: slow or fast charging with on board charger, using dedicated plugs and protection.  Mode 4: fast charging using an external charger, as for instance CHAdeMO.  Table I. Ranking of barriers to EVs uptake [4]  Fast charging is significant as it can help the users to finish charging in relatively short time compared with other charging models for the same capacity battery pack. For example, the CHAdeMO fast charger can support 50 kW to allow the Nissan LEAF to charge to 80% state of charge (SOC) in 20 minutes. This charging pattern is quite attractive for customers on a tight schedule. What is the reason for choosing the fast charger’s power rate to be 50 kW rather than 10 kW, or 160 kW?  Figure 1 indicates the reason from the manufacturer’s cost and users’ time point of view.  Figure 1. Optimal output of fast charger [5] From Figure 1, it can be found that the total charging time is not linearly dependant on power, also the infrastructure costs are increasing with a higher power rate. The 50 kW charger has the best performance-price ratio. However, it is not the only consideration for choice of charging power rate. As the battery’s inherent characteristic, the battery cycle life will decrease with the increase in current rate. For instant, a specific Li-ion battery can be charged 4000 times charging at mode 1 but reduces to 100 times under fast charging [6]. So the SOH of the EV battery pack should be considered as another factor for EV charger development or an additional factor added into the existing charger, especially, when EVs are widely used.   As mentioned, the battery cycle life and charging time (current rate) are two inter-constraining factors. In order to balance and address this issue, in this paper, an optimized charging strategy is proposed. The charging current is determined by evaluating battery SOH and charging time. These two factors are qualified by cycle life and charging current. The mathematical objective function is developed and validated here. Generic Algorithm (GA) is utilized to determine the optimized point which gives a value for charging current.   II.    OPTIMIZED CHARGING STRATEGY  (i). Battery cycle life and current rate normalization  Charging time and battery cycle life are the two factors that users focus on. Reducing the charging time can make users’ lives more flexible but reducing the the charging current (extend charging time) can extend battery life and reduce the cost [7, 8]. Flexibility is needed, but both these two items should be given consideration. Here, a mathematical model is built to quantify this abstract concept.  As material, shape, manufacturing and testing environment are different for each manufacturer; there is no generic function to describe the aging behaviour. Normally, empirical or semi-empirical equations based on experimental data are used to express batteries’ aging behaviours. Here [6], a specific battery                     is chosen as the example. These kinds of batteries are cycled with different current rates at room temperature and 70% depth of discharge (DOD). The test results (cycle numbers against current rate experimental data) are shown in Figure 2.   Figure 2. battery cycle life at different current rates   Figure 3. Normalized battery cycle life at different current rates   Figure 4. Normalized battery cycle life at different normalized current rates  From the data above, it can be found that the cycle life increases exponentially with decreasing current rate. Then normalize the cycle number to obtain a linear function as shown in Figure 3. By using coefficient normalization and curve fitting, a function between cycle life and current rate can be acquired. The least square error R2 is 0.9893 which proves the function fits quite well. Then normalizing the current rate (by dividing common base value 1.8), a new function y=-x+1 can be obtained as shown in Figure 4.  (ii). Objective function  Charging time is inversely related to the charging current rate. In order to maximize the battery life and minimize charging time, the value of charging current rate multiplied by cycle life should be maximized by using the objective function:  max      z=xmyn                                     (1) where    ሺ ሻ  x is charging current rate; y is normalized battery cycle life; m, n are the weight factors;  Before applying this objective function in EV smart charging control, it is necessary to prove this objective has maximum value; the validation is presented in the Appendix. Here the SOH of battery is divided into five regions according to the battery usable capacity (80% usable capacity is considered to be end of life) which is shown in Figure 5.   Figure 5 usable capacity of battery health  For different SOH of the battery, the weighting factors are different. The Table II defines weighting factors according to the battery SOH. The weighting factors are used to evaluate the importance of charging current and battery life cycles. m represents the weighting of current rate and n represents cycle number. Of course, the manufactures can set different health regions and weight factors according to the battery performance and user requirements.  Implement the function (1) into Matlab and plot the curves for different weighting factors, as shown in Figure 6. Then calling Genetic Algorithm (GA) algorithm to determine the optimized value, the optimized normalized current rate values can be obtained. The real charging current rate can be determined as the normalized current rate multiplied by the common base value (1.8). Charging time is the reciprocal of the real charging current rate. All this information is shown in Table II. From Table II, it can be found that, as expected, the charging time will decrease or increase with the weighting of current rate or cycle number. For example, when n=1, with m increasing from 1 to 3, the charging time reduces from 66 to 43 mins. Conversely, with weighting towards cycle number, the charging time increases from 66 to 150 mins.  Figure 6 Objective function curves  Table II Optimized charging plan  (iii). Impact on the cycle life According to the mathematical expression of cycle number and current rate which is fitted from Figure 2;                  the corresponding battery cycle life number can be determined according to the current rate as shown in Table III  Table III cycle life charged at different current rates  From Table III, it can be found that in principle this specific (                   ) battery only can be cycled 17 times at high current rate (1.38C) which equates to a charging time of 43 mins. But the cycle life dramatically increases with decreasing charging current rate (charging time extended).   III.    CONCLUSION    In order to reduce the adverse effect on the battery cycle life due to fast charging, an optimized charging strategy has been presented in this paper. Life cycle and charging time have been taken as two factors to evaluate the optimal charging current. The proposed charging strategy can offer several options according to the SOH of battery by using communications with the battery management system (BMS), which can relatively extend the battery life compared with fast charging technique.   REFERENCES  [1] E. C. Bentley, P. Suwanapingkarl, S. Weerasinghe, T. Jiang, G. A. Putrus, and D. Johnston, "The interactive effects of multiple EV chargers within a distribution network," in Vehicle Power and Propulsion Conference (VPPC), 2010 IEEE, pp. 1-6. [2] G. Sarre, P. Blanchard, and M. Broussely, "Aging of lithium-ion batteries," Journal of Power Sources, vol. 127, pp. 65-71, 2004. [3] V. Marano, S. Onori, Y. Guezennec, G. Rizzoni, and N. Madella, "Lithium-ion batteries life estimation for plug-in hybrid electric vehicles," in Vehicle Power and Propulsion Conference, 2009. VPPC '09. IEEE, 2009, pp. 536-543. [4] A. Ltd., "Electric Vehicles: Driving and change," WWF Scotland, April 2011. [5] Z. E. R. O. (ZERO), "Taking charge - Introducing fast chargers in Norway " 2010. [6] M. Majima, S. Ujiie, E. Yagasaki, K. Koyama, and S. Inazawa, "Development of long life lithium ion battery for power storage," Journal of Power Sources, vol. 101, pp. 53-59, 2001. [7] X. Wang, Y. Sone, H. Naito, C. Yamada, G. Segami, and K. Kibe, "Cycle-life testing of 100-Ah class lithium-ion battery in a simulated geosynchronous-Earth-orbit satellite operation," Journal of Power Sources, vol. 160, pp. 602-608, 2006. [8] R. B. Wright, J. P. Christophersen, C. G. Motloch, J. R. Belt, C. D. Ho, V. S. Battaglia, J. A. Barnes, T. Q. Duong, and R. A. Sutula, "Power fade and capacity fade resulting from cycle-life testing of Advanced Technology Development Program lithium-ion batteries," Journal of Power Sources, vol. 119-121, pp. 865-869, 2003. [9] P. Ramadass, B. Haran, R. White, and B. N. Popov, "Mathematical modeling of the capacity fade of Li-ion cells," Journal of Power Sources, vol. 123, pp. 230-240, 2003. [10] R. Spotnitz, "Simulation of capacity fade in lithium-ion batteries," Journal of Power Sources, vol. 113, pp. 72-80, 2003.   APPENDIX Objective function validation: Here are the constraints: a, b are the current rate values    [   ]       m, n are weight factors,         the function of    ሺ ሻ  is linear or convex and strictly decrease, which can be express as:                                       {  ሺ ሻ     ሺ ሻ      ሺ ሻ                                         (2)   Validation: If function z has maximum value, using mathematic expression, which can be written as:  {              [   ]                       (3)  First, prove     . Here reduction to absurdity is used: Assume     , then                                         {                [   ]                       (4)  If assume     , then function z strictly decrease, then can obtain :  ሺ ሻ   ሺ ሻ                                  (5)  But on the other side,  If current rate a is relatively small, then    ;  If current rate is very large, then the battery cycle life  ሺ ሻ will be very limited and approaching to zero ( ሺ ሻ   ሻ.   Due to           ሺ ሻ ,  ሺ ሻ     ሺ ሻ     ሺ ሻ     ሺ ሻ     So   ሺ ሻ   ሺ ሻ                                      (6) Which is conflict with equation (5).  Similarly， it can be obtained that if assume     , then function z strictly increase,  ሺ ሻ   ሺ ሻ, which is also conflict with equation (5).  So,                            [   ]             ሺ  ሻ   ;                 (7) which means function z has peak value.   Secondly, we need to prove      ; It is easy to obtain:          [ ሺ ሻ]     [ ሺ ሻ]     ሺ ሻ                       (8)         ሺ   ሻ    [ ሺ ሻ]         [ ሺ ሻ]     ሺ ሻ       ሺ    ሻ  [ ሺ ሻ]   [  ሺ ሻ]     [ ሺ ሻ]      ሺ ሻ        (9)  As previous mentioned that,        [   ]             ሺ  ሻ     So from equation (8), it can be got that:   ሺ  ሻ        [ ሺ  ሻ]      [ ሺ  ሻ]     ሺ  ሻ     which can be given as:   ሺ  ሻ        ሺ  ሻ                               (10)  Substitute equation (10) into (9),    ሺ  ሻ   [ ሺ  ሻ]        {         [  ሺ  ሻ]                               [  ሺ  ሻ]      ሺ   ሻ   [  ሺ  ሻ]                                  ሺ  ሻ   ሺ  ሻ}                                             (11)  Transfer equation (11), it can be obtained that  ሺ  ሻ   [ ሺ  ሻ]        {     [  ሺ  ሻ] ቀ    ቁ                                 ሺ  ሻ   ሺ  ሻ}                                           (12)  Compare equation (12) with zero. From equation (2), it can be found that:        [   ]          ሺ ሻ                         [ ሺ  ሻ]              ሺ  ሻ       ሺ  ሻ     {     [  ሺ  ሻ] ሺ    ሻ          ሺ  ሻ   ሺ  ሻ}       ሺ  ሻ                                                                              (13)  The equation (7) prove optimized function (1) has peak value, and equation (13) validate that this peak value is maximum value.   So in summary, this objective function has optimized point subject to the constraints.     

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http://nrl.northumbria.ac.uk/id/eprint/21069/1/Optimal_Charging_Strategy_for_EVs_with_Batteries_at_Different_States_of_Health_.pdf

Optimal Charging Strategy for EVs with Batteries at Different States of Health

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