Hybrid Vehicle (City Bus) Optimal Power Management for Fuel Economy Benchmarking

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tion.

2.2.1. Criterion

The criterion of optimization also known as the cost

function or the objective function is the function that we

seek to minimize, which is the fuel consumption in this

case

0

Minimize ,

N

i

CTeiwei Ts

(1)

2.2.2. Cons tr a i nts

In a parallel single shaft hybrid powertrain topology, the

sum of engine and motor torque must be instantaneously

equal to the torque demand described in the engine shaft.

and the engine and motor speeds are proportional to the

wheel speed by the final drive and gearbox ratios. Also

we must constrain the engine and motor torque to make

sure that they do not exceed their maximum torques and

finally constrain the battery state of charge to remain be-

tween two limits denoted as SOCmax and SOCmin. Con-

straining the battery SOC in this way helps to prolong its

life, the constraints are described by the equations below:

TdiTe iTm i

ifitiw iwe iwmi

,min ,maxwewe iwe

,min ,maxTeTe iTe

,min ,maxwmwm iwm

,min ,maxTmTm iTm

SOC1 SOCSOC2i

2.2.3. St ate Equation

The state equation gives the variation of the energy sto red

in the battery (X) as a function of the electric power fur-

nished by this battery. In discrete time this variation is

described by

1,

iXiPewmiTmi Ts (2)

2.2.4. Limit Condition

In order to be able to perform the optimization the zone

of acceptable solution must be closed, which leads to

constraining the battery SOC to converge to a known

limit, this limit is described by SOCfinal, in our article a

limit condition u sed is described by:

SOCfinal SOCinitial 80% (3)

3. Principal of the Method of Dynamic

Programming

Dynamic Programming (DP) is a powerful mathematical

technique developed to solve dynamic optimization pro-

blems. The advantage is that it can easily handle the con-

straints and nonlinearity of the problem while obtaining a

globally optimal solution. The DP technique is based on

Bellman’s Principle of Optimality, which states that the

optimal policy can be obtained if we first solve a one

stage sub problem involving only the last stage and then

gradually extend to sub-problems involving the last two

stages, last three… etc. until the entire problem is solved

(backward method). In this manner, the overall dynamic

optimization problem can be decomposed into a sequen ce

of simpler minimization problems [10].

In HEV the sequence of choices represents the power

split between the internal combustion engine and the

el e c t r i c moto r a t s u c c e s s i v e t i me s t e p s . T h e ob j ec t i v e func-

tion can be fuel consumption, emissions, or any other de-

sign objective. The set of choices at each instant is de-

termined by considering the state of each powertrain com-

ponent and the total power requested by the driver. Given

the current vehicle speed and the driver’s demand (ac-

celerator position); the controller determines the total

power that should be delivered to the wheels. Then, using

maps of the components and feedback on their present

state, it also determines the maximum and minimum

power that each energy source can deliver. If the power

demand equals or exceeds the total available power from

both sources, there is no choice to be made: each of them

should be used at the maximum of its capabilities. Oth-

erwise, there are infinite combinations such that the sum

of the power from engine and motor equals the power

demand. In most algorithms, including dynamic program-

ming, instead of considering this continuum of solutions,

a discrete number is selected and evaluated. The number

of solution candidates that can be considered is a com-

promise between the computational capabilities and the

accuracy of the result: in fact, the minimum cost may not

exactly coincide with one of the selected points, but the

closer these are to each other, the better the approxima-

tion of the optimal solution. Once the grid of possible

power splits, or solution candidates, is created associat-

ing a cost to each of the solution candidates, the optimal

cost is calculated for each grid point, and stored in a ma-

trix of costs. When the entire cycle has been examined,

the path with the lowest total cost represents the optimal

solution (Figure 1).

3.1. Torque Demand Calculation

As we said before, this method requires the knowledge of

the whole reference speed in advance to precede the op-

timization; thus; after knowing the speed, we can calcu-

late the power demand and also the torque demand (since

both engine and motor run at the same speed) in each

sample time using the wheel speed and its derivatives as

shown below:

Energy of the power source (Engine and Motor)

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