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????+1 = ???? + ?????????? - ???? ???????????????? (38) respectively, and also need to determine their Jacobian
?????????? ??????????????
Where
???? = ????+1 - ????
???? = (???(????+1) + ????=? 1 ???? · ?????(????+1)) - (???(????) +
????=? 1 ???? · ?????(????))
Since the solution of a nonlinear optimization problem
has many solutions, it is difficult to find a solution to this
problem unless we start with guess points that fall within the
feasible solution regions that enable the SQP algorithm to
make the first iteration to solve the optimization problem. In
subsequent iterations, the predicted state variables and the
optimal control inputs (control interval) from the previous
step act as initial guesses for solving the optimization
problem at that iteration. This is why feedback is built for
this block, as shown in Fig.2. So when starting to solve a
series-parallel HEV optimization problem, the guess torque
values of the vehicle's energy converters are chosen and
these values should be determined within the possible
solution regions.
VII. SIMULATION RESULTS
The model of the series-parallel HEV with the
specifications mentioned in Table I was built and simulated
using the MATLAB / Simulink (2019b) environment. As
mentioned previously, this vehicle model consists of three
blocks. The first block values represent the reference values
or the values to be achieved by the outputs of this vehicle
model, where the New European Driving Cycle (NEDC) was
chosen to represent the required speed of the vehicle during
the vehicle's trip. As for the second reference value, it is the
SOC value, which represents the value required for the state
of charge of the batteries during this trip, as this value was
chosen to be 65% of the full value of the charging state of the
batteries. Where this SOC value allows the battery to store
the vehicle's kinetic energy captured from the vehicle's
deceleration, as well as the possibility to equip the motor
with electrical energy[32].
The second block, which represents the control unit of the
series-parallel HEV model, was created by MATLAB Model
Predictive Control toolbox using a nonlinear MPC controller
block. This block needs to specify the number of the state
variables, the manipulated variables, and the outputs where
in this study are 3, 2, and 3 respectively, it also needs to
specify the controller sample time, prediction horizon, and
control horizon are determined by the following values 0.1
sec, 10, and 5 respectively. When creating the Nonlinear
MPC controller block, a mathematical model to represent a
series-parallel HEV is required to predict the vehicle's future
behavior, where this model includes the engine speed ????
(rad/sec), motor speed ???? (rad/sec), and SOC are designated
as the state variables. While the engine torque ???? (N.m),
motor torque ????(N.m), and generator torque ????(N.m) are set
as manipulated variables (MVs), and each the vehicle
velocity (km/h) and SOC are the plant outputs.
To implement the Nonlinear MPC controller block needs
to define the state equations of the nonlinear plant model
( ??? ?? , ??? ?? , and ????? ?? ) which are (9), (10), and (16)
Al-Flehawee & Al-Mayyahi
equations using (29). Also, this block needs the output
equations of the nonlinear plant model (the vehicle velocity
and ??????) and their Jacobian equations are determined by the
following (29). In addition to setting the constraints for each
(39) of the state variables, the manipulated variables (MVs), and
the output variables, as in (27). The second block also needs
the values of the manipulated variables from the previous
sampling step and the values of the state variables received
(40) by the third block for the purpose of finding the optimal
torques values for each of the engine, motor, and generator
to apply them to the third block, which represents the vehicle
model (the powertrain and vehicle dynamic equations).
The aim of this simulation is to study the possibility of
making the vehicle drive the required speed and also make
the state of charge of the battery at the desired value, and that
this is all done with minimal fuel consumption To
accomplish this, we construct the objective function as
shown in (28), since the objective function is composed of
two parts where the first part of the objective function is
represented by the first part of the standard cost
function(30), which is called output reference tracking. In
order to implement this part, the reference values for each of
the outputs variables (vehicle speed and SOC) are received
from the first block and the weight coefficients of the output
variables are determined where the values 100 and 200 are
chosen for each of the weight coefficients for the output
variables (vehicle speed and SOC) respectively. As for the
second part of the objective function, it is the custom cost
function shown in (17).
After completing the simulation of the series-parallel HEV
model, the results obtained were good because the vehicle
control unit was able to achieve the desired objectives. Fig.6
shows the state variables are the engine speed ???? (rad/sec),
motor speed ???? (rad/sec), and SOC. where it is observed
that the state variable SOC changes between two values
(65.2%) and (64%) during the driving cycle and this is a good
result in tracking the desired value (65%), and this change is
considered a small, insignificant change. While the outputs
of the control unit are the optimal values of the manipulated
variables (MVs) (engine torque (Nm), motor torque (Nm),
and generator torque (Nm)) which are obtained by solving
the optimization problem (minimizing the cost function
subject to the nonlinear prediction model and physical
constraints of the vehicle). Where Fig.7 represents these
Fig.6: The state variables trajectory
these optimal values that are applied to the engine, motor,
and generator to drive the vehicle in order to achieve
tracking of each of the reference values of the vehicle speed