Page 102 - IJEEE-2022-Vol18-ISSUE-1

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????????h(??????h - ???????? )
G(??, ??) = [???????? (?????? - ????????)]
??????(???? ?)
Where????????, ???????? are the required speed of the vehicle and
the desired value of the SOC respectively, ????????h, ???????? and
?????? are penalty weights. While the fuel flow rate equation
(???? ?) is described in (13).
The cost function of the series-parallel HEV is formulated in
discrete-time form as:-
?? = ?????=??0-1((????????h(??????h(?? + ?? + 1) - ????????)2 +
(????????(?????? (?? + ?? + 1) - ????????)2 + (??????(???? ?(?? + ?? +
1)))2) (28)
Where: - ?? is represented sampling time.
B. Building NMPC control strategy for series-parallel
HEV by using the non-linear MPC control block
A series-parallel HEV model was created using MATLAB
whose controller was built based on the NMPC control
strategy using the non-linear MPC control block provided in
the MPC Toolbox in MATLAB SIMULINK®. This block is
based on calculating the optimal control trajectory over the
prediction horizon ( ???? ) by solving the nonlinear
optimization problem which includes the nonlinear objective
function subject to the nonlinear predictions of the future
behavior of the plant and physical constraints of the plant. To
implement this block, the number of the state variables,
inputs control, and outputs concerning the predictive model
of the plant is defined. In this study, the predictive model of
the plant contains [26]:-
? Three state variables, which are the engine speed ????
(rad/sec), motor speed ???? (rad/sec), and SOC.
? Three inputs, which are the engine torque ???? (N.m),
motor torque????(N.m), and generator torque????(N.m)
? Two outputs, which are Vehicle velocity (km/h), and
SOC.
After defining the number of variables related to the
predictive model of the series-parallel HEV the following
will be specified:-
1. The dynamic states functions (time derivative of state
functions) for a nonlinear prediction model, where the
dynamic state equations in this study are motor speed
??? ?? (rad/sec), engine speed ??? ?? (rad/sec), and the state of
charge ????? ?? of the battery as shown in (9), (10), and
(16) respectively. The computational efficiency of the
controller becomes better when using an analytical
Jacobian for dynamic states functions, and when not
using an analytical Jacobian, the controller calculates
the Jacobian by numerical perturbation. The Jacobian of
the state functions are[26]:-
????1 ? ????1
????1 ????????
???? ?? = ???? = ?
???? ? ? ? , and
???????? ????????
[ ????1 ???????? ]
????1 ? ????1
????1 ????????
?????? = ???? = ? (29)
???? ? ? ?
???????? ????????
[ ????1 ???????? ]
Al-Flehawee & Al-Mayyahi
Where:-
??1 to ??????are dynamic states functions of the model,
??1to ?????? are states variables of the model,
??1to ?????? are manipulated variables of the model.
2. The outputs functions for a nonlinear prediction model
are vehicle velocity (km/h) is defined by (4), and SOC
is represented by the third state variable. An analytical
Jacobian was also used to improve the efficiency of the
computational.
3. The non-linear MPC controller block needs to define a
cost function that represents the objectives for which
the controller is built or to be achieved by the controller.
The cost function of any study when using a nonlinear
MPC controller block can be represented by either
using the standard cost function that consists of four
terms where each term describes an aspect of the
controller performance as shown in the following
equation:-
??(????) = ????(????) + ????(????) + ??????(????) + ????(????) (30)
Where the standard cost function terms are output
reference tracking ????(????) , manipulated variable
tracking ????(????) , manipulated variable move
suppression ??????(????) , and constraint violation ????(????)
respectively. If the standard cost function cannot
represent the cost function of a particular study, a
function called the custom cost function can be built so
that it contains terms that are not present in the standard
cost function. Sometimes it is required to combine the
standard cost function and a custom cost function to
represent the control cost function. In this study, both
the standard cost function and the assigned cost
function are used to express the cost function of the
series-parallel HEV shown in (28) is expressed by
dividing it into two parts. The first part of the cost
function illustrates making the vehicle move at the
desired speed while maintaining the state of the charge
of the battery at the desired value, which is represented
by the first term of the standard cost function, which is
called output reference tracking. The rest of the terms
of the standard cost function are eliminated by making
the penalty weight zero for each of them. While the
second part of the cost function explains minimizing
fuel consumption as it is represented by the custom cost
function, which is the fuel flow rate equation (17).
4. Define the physical constraints of the system, which
consist of the standard bounds on states, inputs, and
outputs as shown in (27). While the equality and
inequality of custom constraints were not used in this
study.
After providing the previously mentioned requirements
for the design or construction of a nonlinear MPC control
block for the series-parallel HEV, this block calculates at
each sampling step the optimal control inputs (engine
torque????, motor torque ????, generator torque????). The optimal
control input is the solution of the nonlinear optimization
problem represented by the cost function (28) subject to (26)
and (26) but after formulating both (26) and (27) in the form
of a discrete time as shown in (21) and (22) respectively.
Where the nonlinear MPC controllers solve the nonlinear