Vol.1, No.3, 222-228 (2009)
Copyright © 2009 Openly accessible at http://www.scirp.org/journal/NS/
Natural Science
Control strategy optimization using dynamic
programming method for synergic electric system on
hybrid electric vehicle
Yuan-Bin Yu, Qing-Nian Wang, Hai-Tao Min, Peng-Yu Wang, Chun-Guang Hao
The State Key Laboratory of Automobile Dynamic Simulation, JiLin University, Changchun, China
Received 23 August 2009; revised 27 September 2009; accepted 22 October 2009.
Dynamic Programming (DP) algorithm is used
to find the optimal trajectories under Beijing
cycle for the power management of synergic
electric system (SES) which is composed of
battery and super capacitor. Feasible rules are
derived from analyzing the optimal trajectories,
and it has the highest contribution to Hybrid
Electric Vehicle (HEV). The methods of how to
get the best performance is also educed. Using
the new Rule-based power management strat-
egy adopted from the optimal results, it is easy
to demonstrate the effectiveness of the new
strategy in further improvement of the fuel
economy by the synergic hybrid system.
Keywords: Dynamic Programming; Control
Strategy; Optimization; Synergic Electric System;
Hybrid Electric Vehicle (HEV) could improve fuel
economy and exhaust emission using the electric system
to adjust the load of engine and could make it work in
high efficiency. But this depends on the performance of
the electric system on board. Because of the low power
density of battery, it’s hard to be competent for this pur-
pose. When it was combined with super capacitor to
form a new electric system which was called the Syner-
gic Electric System (SES), could exert both merits,
which is energy and power density [1,2].
Rule-based or fuzzy logic control strategy was used to
supervise the power flow between the two parts at pre-
sent [3,4]. For every state of the SES, each control will
come to a new stateat the same time, we can get a loss
of energy. With regard to a given cycle of motor power
requirement, how to get the best control rate has been
put to us which can achieve the optimal performance of
the system. Dynamic Programming is a numerical
methodology developed for solving sequential or multi-
stage decision problems just like the SES. The algorithm
searches for optimal decisions at discrete points in a time
sequence. It has been shown to be a powerful tool for
optimal control in various application areas [4,5,6,7].
The dissertation proposed the dynamic programming
algorithm from optimum control theory which solve the
problem of overall control rate for SES in the whole
cycle and try to answer the question like what can SES
do on energy-saving.
A hybrid city bus was chosen as a researching flat, and
the SES was made up of 280 Nickel-Hydrogen batteries
units (1.2V\15Ah each) and 120 super capacitors (2.5V
\2000F each). A Buck/ Boost DC/DC power converter
whose Peak / rated power were 60/30kw was used in this
system to coordinate the voltage of battery and capacitor,
and to control the current of capacitor actively. Figure 1
was the layout of SES in this research.
Control objective of SES was as follow: ensure the vehi-
cle’s dynamic as the premise and give full play to the
super capacitor’s “push and pull” role; decrease large
current’s impact; extent battery’s service life; regenerate
braking energy as much as possible to improve fuel
Figure 1. Topology of Buck/Boost DC/DC Converter.
Y. B. Yu et al. / Natural Science 1 (2009) 222-228
SciRes Copyright © 2009 Openly accessible at http://www.scirp.org/journal/NS/
economy. The principle of the rule-based control strat-
egy is: During the operation, battery provides the aver-
age power requirement; capacitor will give a comple-
ment. The control rule was as follow:
1) if Pm<0, and super capacity was not over charged,
then Pbat=0
2) or if Pm <Pbat_max, and super capacity was not over
discharged, then Pbat=PfilterPscap=Pm-Pfilter
3) otherwise Pbat=Pm
4) when super capacity run out of energy, charge the
capacitor from battery, Pbat =Pchg+Pbat.
The variables’ meanings above are: Pm——the power
required from motor, Pbat ——batteries’ power needed
from motor, Pscap ——capacitor’s power requirement,
Pfilter——filtering power, which is calculated by For-
mula (1).
filter LP
 
Pe e
In the formula: Pload——road resistance; HLP——filter
function; ττ’——engine and battery’s low-pass filter
time constant; t——acceleration time; Pm——motor
power required; Pchg——charge power from battery
when capacitor’s state of charge (CSOC) is low. It is the
function of battery’s state of charge (SOC) and is equal
to Pbat_max*SOC-SOCmin/ (SOCmax-SOCmin).
The capacitor’s electrical-quantity shortage is deter-
mined by the demand. When the actual voltage of ca-
pacitor is lower than the ideal voltage, it is calculated out
according to vehicle speed as Formula (2). The disserta-
tion provides that the ideal voltage limit is in a reverse
proportion with vehicle speed. The object of setting this
value is to develop a goal of electrical-quantity for ca-
pacitor, which could ensure that it could preserve
enough energy for acceleration when vehicle speed is in
the formula: Vscap——capacitor’s actual voltage of low
and regenerate enough braking energy capacity. When
vehicle speed is high. The speed requirement of the ve-
hicle, the current of single battery, current of SES’s bat-
tery and capacitor according to rule-based control strat-
egy are shown in Figure 2.
max max
scap car
scap car
In the formula: capacitor; max
maximum voltage; vcar——actual vehicle speed;
——maximum vehicle speed; k——capacitor’s energy
utilization rate in cycle, in value equivalent to 0.75
Figure 2. Simulation results under Beijing cycle.
Dynamic programming is a kind of math method solving
the optimal problem in process of multi-stage decision.
It is proposed and established in the Fifties of 20th Cen-
tury by American mathematicians (Bellman) who claim
the famous Optimum Theory and transmit the process of
multi-decision into a serious of single-stage issue. Multi-
stage problem is a kind of activities which could be di-
vided into several interrelated stages. Each stage needs a
decision. The decision not only decides the benefit of
this stage but also the initial state of next stage. A se-
quence of decisions would be created after every stage’s
decision has been made. The multi-stage decision prob-
lem is to get a control strategy that can optimize the sum
of every stage’s benefit [3]. Dynamic programming al-
gorithm can fully utilize the limited resources which is
an important content of investment-determination. As
for multi-stage determination problem, dynamic pro-
gramming method could be used to make sure the sum
of benefit from every stage optimal [4].
The power requirement of motor at every moment is
fulfilled by battery and ultra capacitor. However, be-
cause of the difference between the resistance and the
efficiency of charge and discharge, every decision made
by system at every moment will create impact on the
whole control effect. For the power requirement of mo-
tor is consist of the power of battery and ultra capacitor
at any proportion. So relationship of the power of battery
and ultra capacitor is:
Y. B. Yu et al. / Natural Science 1 (2009) 222-228
SciRes Copyright © 2009 http://www.scirp.org/journal/NS/
at scap m
P +P=P 3
Figure 3. Recursive of system’s state and control object.as
shown in Formula (6).
Every moment, the battery and ultra capacitor in the
SES have the corresponding SOC and CSOC which rep-
resent the state X of battery and ultra capacitor. When
the power flow through battery and ultra capacitor, it can
create an incentive to change the state of the two power
source, at the same time can create power loss J. Ac-
cording to the experiment result of component’s charac-
ter, the resistances of battery and ultra capacitor are
functions of SOC and CSOC and different system loss
would be created under different control rate U. Thus,
conclusions can be drawn that system’s power loss J, at
the same time J is the function of state object X and con-
trol rate U. Recursive equation of X and power loss
equation can be expressed by Formula (4).
= E((1)())E((1)
bat Scap
bat Scap
Lxk uk
bat ScapDC/DC
loss lossloss
=P ()+P()+P
If every control rate in the cycle is given properly, the
sum of power loss at every moment will decrease to the
minimum. Under the same cycle power requirement, it
can be seen as the best cycle power efficiency, the big-
gest regenerate of braking energy and the best perform-
ance of power. Consequently, optimal goal could be set
Under given driving cycle, the power requirement
from motor can be drawn, then the system’s loss on this
moment was only decided by the two parts’ state X and
the chosen control rate U. From this moment on, differ-
ent power requirement Pm and control rate U will create
different state X and confront with the problem of
choosing a new control rate till the end of the cycle. Be-
cause the initial condition of vehicle simulation has been
decided, the problem of optimum control rate on SES
can be attributed to: Free optimal control issues on cer-
tain initial condition x(0)=x, as shown in Figure 3.
Lxk ukG
In cycle, the charge and discharge power and the energy
of batteries and capacitors must subject to the limitation
of Formulas (7) and (8).
It is considered that the changes of battery’s state will
have a certain influence on the power loss effect. So,
when calculating the value of J, battery and capacitor’s
electrical-quantity state G should be fully considered.
Calculation of G showed in Formula (5).
bat min max
min max
bat bat
batbat bat
 T
curre n t
To Workspace3
Table_W e
To Workspace2
Table_F C
To Workspace1
Table_SO C
Power control
model of SES
Input u1
Buck /Boost
Rint model
of super capacitor
Rint model of
B atte ry
Figure 4. Solving model of synergic system.
Openly accessible at
Y. B. Yu et al. / Natural Science 1 (2009) 222-228
SciRes Copyright © 2009 Openly accessible at http://www.scirp.org/journal/NS/
Table 1. Optimal solution of synergic system’s energy loss at the 126th sec of Beijing cycle J1* e+005.
Table 2. Optimal control rate at the 126th sec of Beijing cycle U1.
0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8
0.5 6 6 6 6 6 6 6 6 6
0.55 5 5 5 5 5 5 5 5 5
0.6 4 4 4 4 4 4 4 4 4
0.65 3 3 3 3 3 3 3 3 3
0.7 2 2 2 2 2 2 2 2 2
0.75 1 1 1 1 1 1 1 1 1
0.8 6 6 6 6 6 6 6 6 6
0.85 6 6 6 6 6 6 6 6 6
0.9 6 6 6 6 6 6 6 6 6
0.95 5 5 5 5 5 5 5 5 5
1 5 5 5 5 5 5 5 5 5
Table 3. Optimal solution of synergic system’s energy loss at the 127th sec of Beijing cycle J1* e+005.
0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8
0.5 9.9914 9.9318 9.8686 9.865 9.8617 9.8702 9.8787 9.9383 9.9964
0.55 9.9866 9.9275 9.8645 9.8608 9.8574 9.866 9.8746 9.9342 9.9919
0.6 9.9731 9.914 9.8514 9.8479 9.8447 9.8531 9.8616 9.9208 9.9784
0.65 9.9618 9.9028 9.8403 9.8368 9.8322 9.8393 9.8465 9.906 9.964
0.7 9.9478 9.887 9.8194 9.8135 9.808 9.8152 9.8224 9.8817 9.9395
0.75 9.9361 9.9361 9.8046 9.7987 9.7933 9.8004 9.8076 9.867 9.9248
0.8 9.9124 9.8491 9.782 9.7762 9.7708 9.7779 9.7851 9.8441 9.9015
0.85 9.8888 9.825 9.7576 9.7518 9.7463 9.7535 9.7608 9.82 9.8777
0.9 9.8731 9.8106 9.7444 9.7387 9.7333 9.7404 9.7476 9.8601 9.8629
0.95 9.8572 9.7946 9.7283 9.7226 9.7173 9.7244 9.7315 9.79 9.8469
1 9.8505 9.7882 9.7222 9.7166 9.7113 9.7183 9.7254 9.7838 9.8405
Table 4. Optimal control rate at the 127th sec of Beijing cycle U1.
0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8
0.5 8 8 8 8 8 8 8 8 8
0.55 7 7 9 9 9 9 9 7 7
0.6 7 7 7 7 7 7 7 7 7
0.65 7 7 7 7 7 7 7 7 7
0.7 8 8 8 8 8 8 8 8 8
0.75 8 8 8 8 8 8 8 8 8
0.8 7 7 7 7 7 7 7 7 7
0.85 7 7 7 7 7 7 7 7 7
0.9 6 6 6 6 6 6 6 6 6
0.95 5 5 5 5 5 5 5 5 5
1 5 5 5 5 5 5 5 5 5
0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75
0.5 9.9952 9.936 9.873 9.8693 9.8659 9.8745 9.883 9.942 10
0.55 9.982 9.9229 9.8602 9.8567 9.8535 9.8619 9.870 9.929 9.98
0.6 9.971 9.912 9.8495 9.846 9.8414 9.8485 9.855 9.915 9.97
0.65 9.9574 9.8966 9.829 9.8232 9.8177 9.8248 9.832 9.891 9.94
0.7 9.9464 9.86 9.7929 9.7871 9.7758 9.7829 9.790 9.849 9.90
0.75 9.9233 9.86 9.7929 9.7871 9.7817 9.7888 9.796 9.855 9.91
0.8 9.9175 9.8541 9.787 9.7812 9.7758 9.7829 9.790 9.849 9.90
0.85 9.8935 9.8152 9.7489 9.7432 9.7378 9.7449 9.752 9.810 9.86
0.9 9.8777 9.8152 9.7489 9.7432 9.7378 9.7449 9.752 9.810 9.86
0.95 9.8615 9.7988 9.7325 9.7269 9.7216 9.7286 9.735 9.794 9.85
1 9.8545 9.7922 9.7262 9.7206 9.7153 9.722 9.729 9.787 9.84
Y. B. Yu et al. / Natural Science 1 (2009) 222-228
SciRes Copyright © 2009 Openly accessible at http://www.scirp.org/journal/NS/
scap min max
min max
scap scap
EtE tT
scapscap scap
 
In the Formula (7) and (8), P and E are the power re-
quire from battery and capacitor, the max and min are
the maximum and minimum value respectively.
1) Model for System Solving
Figure 4 shows the solving model of SES under dy-
namic programming. From the initial moment according
to the model established, the algorithm calculates out
every moment’s system loss and stored them in memory.
The code of this program is:
table_x1_n(a,b,c)=Table_CSOC(2); table of CSOC
table_x2_n(a,b,c)=Table_SOC(2); table of SOC
table_u1_n(a,b,c)=u1_Theta_c; control rate table
FC_inst(a,b,c)=Table_FC(2); power loss table
2) Algorithm Solving
Making use of the result in the last section, start from
the terminal, calculate the value-added power loss of the
current and the last moment in sequence and find out the
minimum point according to the reverse deducing
method using the min function of MATLAB language.
Record every moment and state’s minimum value of J
and the corresponding optimal control rate u and store
them in matrix J1 and U1 respectively. The calculating
program is:
_x2_n(:),table_x1_n(:),'nearest'); %sum of the past b =
%sum of the past and current
%FC_inst(:) every moment’s power loss
%find out the optimal solve from the first step to the
current step using min function
[J1(:,:),U1(:,:)] = min(b,[],3); %find the optimal solve
and the corresponding u1
%for J1: x represent SOC and y represent CSOC,
volume is the system’s power loss
%reverse calculation
%U1 store the optimum control rate u1.
Based on the object of optimization, the minimum
power loss of any moment and the moment behind can
(a) (b)
(c) (d)
Figure 5. Optimum solution of synergic system’s dynamic programming under Beijing cyclebatteryesscapacitor
ess2. (a) electrical-quantity state of battery and capacitor; (b) voltage process of battery and capacitor; (c) current distribu-
tion of battery and capacitor; (d) power distribution of battery and capacitor.
Y. B. Yu et al. / Natural Science 1 (2009) 222-228
SciRes Copyright © 2009 Openly accessible at http://www.scirp.org/journal/NS/
be calculated out under the reverse method and stored in
J1. At the same time, the optimum control rate is stored
in U1.Under Beijing cycle, the electrical power require-
ment of the 126th sec and the 127th sec is -16859w and
37467w. Table 1 to 4 list out the optimum solve J1 and
optimum control rate U1, and from which, the trend and
principle of controlling can be induced.
As is shown from the data, if the SOC and CSOC of
battery and super capacitor at some moment are known,
then the optimum solution J1 and optimum control rate
U1 can be checked up from tables above. After known
about the point’s power demand Pm and the optimum
control rate U1, the SOC and CSOC of battery and super
capacitor on the next moment can be calculated out
through every parts model, and then continue to look-up
tables to find out the optimum solution and optimum
control rate. Followed by analogy, on the situation when
the forward calculation mode. Connect every moment’s
optimal control rate, the optimum control rate of SES
under Beijing cycle can be drawn approximately. During
calculation, if the less interval of every component’s
state and control rate were used, the closer the result to
the synergic system’s optimal control rate.
In this way, under Beijing cycle, divide the battery’s
SOC from 0.4 to 0.8 into 9 parts. The interval of each
calculation point is 0.05. Divide the capacitor’s CSOC
from 0.5 to 1 into 11parts to ensure that the same inter-
val of calculation point is chose. After optimized calcu-
lation, the optimal control rate can be counted out shown
in Figure 5. Figure 5a illustrates the SOC of battery and
capacitor resulted from the optimized control under Bei-
jing cycle. Figure 5b shows the voltage process of bat-
tery and capacitor resulted from optimum control. Figure
(c&d) illustrate the current and power distribution re-
sulted from the optimum control rate at every moment
3) Improved Rule-based Control Strategy
The result is based on cycle, so the Dynamic Pro-
gramming algorithm can be not directed applied to en-
gineering control. However, some enlightening can be
drawn used as the guidance of actual control algorithm
programming. For example, based on the result of pre-
vious section, the following four laws can be educed:
a).During power assisting, if the motor required a low
current, the battery would provide the power.
b).During power assisting, the capacitor should
provide more power when the motor is under high
power demand.
The two tips above are all induced by the low efficient
of DC/DC under low load.
c).If the capacitor is in high CSOC, and then it should
share a greater proportion of discharge. However, as the
CSOC drops, the discharge proportion of capacitor
would diminish responsively, while battery’s proportion
would increase gradually till the battery pack power the
motor alone.
d).When low-power braking, capacitor can recovery
all the vehicle braking energy; While, with the increase
of braking power, collaborative work mode of battery
and capacitor would be taken by the optimized control
algorithm to make the two both working on a high effi-
ciency and contribute to the balance of battery. Also the
proportion of discharge from capacitor would increase as
the braking power increase. This is different from the
rule of rule-based control strategy that energy firstly go
to the capacitor and it gives inspiration for the improve-
ment of control strategy.
Based on the analysis above, this section proposed a
modification on rule-based control strategy of SES when
motor power assisting and braking energy regeneration.
Details as follows:
(1) When power assisting, if Pm>0and Pm<Pset
then Pbat=Pm;
(2) Otherwise, if Pm>Pset1then Pscap=K1*Pmbut
Pbat=(1- K1) *Pmin the formula: K1 is the function of
capacitor’s CSOC showed in table 5. Pset1=12.33kw.
(3) When braking: if P
m<0and Pm>Pset2
capacitor is not over recharged, then Pbat=0otherwise
when Pm<Pset2Pscap=K2*Pm, and Pbat=(1- K2) *PmK2
increases as Pm decreases.Pset2=8.7kw.
Figure 6 shows working process manipulate of SES’s
control rate, battery and capacitor resulted from the
simulation of rule-based control strategy, which is modi-
fied by applied optimal algorithm. Table 6 lists com-
parison of SES’s HEV simulation result under different
control algorithm. Conclusions can be drawn from the
various SES’s simulation results listed in the table, that
HEV vehicle have optimum fuel economy under control
of dynamic programming algorithm. Using the principle
of optimized DP algorithm, designer could know the
maximum energy-saving efficient contributed by SES
and also could compile optimal control algorithm of SES,
which can be used in actual engineering. Under this cir-
Table 5. Relationship between Ki and CSOC.
CSOC0.5 0.6 0.7 0.8 0.9 1.0
K1 0 0.2 0.4 0.6 0.8 1.0
Table 6. Simulation result comparison of synergic system.
Control strategyRule-based Dynamic
Fuel economy
L/100km 25.7 23.2 24.5
Fuel economy
improvement 1 2.1% 1.6%
Power effi-
ciency 87% 93% 91%
Brake energy
recovery effi-
8.34% 9.74% 9.35%
Y. B. Yu et al. / Natural Science 1 (2009) 222-228
SciRes Copyright © 2009 http://www.scirp.org/journal/NS/
Openly accessible at
Figure 6. Simulation result based on the optimized
rule-based control strategy under Beijing cycle; (a)
SOC,current and voltage of battery; (b) CSOC, current
and voltage of capacitor.
cumstance,capacitor can keep appropriate power state at
anytime and make vehicle fuel consumption a further
reduction. All the advantages above can prove the sig-
nificance improvement by dynamic programming opti-
mization algorithm
1) Dynamic programming algorithm had been applied to
SES to find out the optimum control rate using off-line
2) Rule-based control algorithm was established based
on the optimum control rate. The simulation results
demonstrated that the method is practical and effective.
3) Based on the control rules of overall optimized al-
gorithm, the rule-based control strategy of the SES had
been improved and the control effect had also been
evaluated. The dissertation explicated the best perform-
ance and most efficient control method of SES which
could make the best contribution to vehicle energy-saving.
4) The overall optimized principle was suitable for the
programming and optimizing of control algorithm for
HEV vehicle with multi-energy power source.
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