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Al-Flehawee & Al-Mayyahi | 97
[24].Where the standard NMPC algorithm procedures are as V. NMPC CONTROL STRATEGY FOR SERIES-
PARALLEL HEV
follows [20]:-
? At each sampling step, use the current values of optimal In this study, the NMPC control strategy, which is one of
the energy management strategies for hybrid electric
control inputs (MVs) and measure or estimate the current vehicles, was chosen, as this strategy works to accomplish
the tasks of the series-parallel HEV controller, and this
values of the system state variables to use all these values to strategy is built by formulating the optimization problem and
solving it in one of the methods of mathematical
predict the future behavior of the plant across the prediction optimization. The optimization problem includes the cost
function subject to the nonlinear prediction model and
horizon. Calculate the open-loop optimal control from physical constraints of the vehicle, where the cost function
represents the objectives to be achieved by this strategy. In
solving the optimization problem that is subject to dynamics this study, the non-linear MPC control block was chosen to
build and implement this strategy on the series-parallel HEV.
of the system and constraints of the input and state over the The following is explained how to build the cost function and
prediction horizon????. the non-linear MPC control
? Calculate the optimal control inputs trajectory by solving A. Formulation of the Optimization Problem for a Series-
Parallel HEV
the optimization problem, where the optimization problem is
the cost function subject to predictions of the future behavior
of the plant in addition to the physical constraints of the
plant.
? Implement the first part of the optimal control inputs
trajectory until the next sampling instant.
? Continue with step (1) when the next sampling step is
reached. The optimization problem includes the cost function that
In the MPC algorithm, the prediction trajectories for the represents the objectives to be achieved by the vehicle, and
state variables of the plant and the plant output are a linear the cost function is subject to the predictions of the plant
function of both the current state variable and the optimal model that represents the equality constraints of the cost
control input used in the current sampling step. Therefore, function. These predictions are obtained through the
the solution of the optimization problem deals with the application of the mathematical model that describes the
solvers that are efficient and high-performance. While the work of the vehicle based on the current values of the state
NMPC algorithm, the prediction trajectories for the state variable and optimal control inputs (MVs) of the vehicle. The
variables and outputs of the plant are a nonlinear function for cost function is also subject to the physical constraints of the
the current state variable and the optimal control input used vehicle that represents the inequality constraints of the cost
at the current sampling step. Thus the optimization problem function. In this study, the cost function of the series-parallel
becomes a nonlinear optimization problem and also known HEV is formulated to minimize fuel consumption while
as nonlinear programming (NLP) problems, which needs a ensuring that the vehicle can move at the speed required by
different approach (solvers) than in the MPC algorithm and the driver and maintain the state of the charge of the battery
where it is more computationally complex [25]. at the desired value. The optimization problem of the series-
In NMPC the optimization problem is solved at every parallel HEV is shown in the following equations:-
sampling step, which is represented by the cost function and ?????? ?? = ????(??, ??)?2????
??(??)
the inequality and equality constraints as shown in below:- (25)
?????? ?? = F[??(??), ??(??)] (18) Subject to
??(??)
Subject to the following inequality constraints: 1. Equality constraints:
???????? > ??(?? + ?? |??) > ????????, 0 > ?? > ???? - 1 (19) {????? = ??(??, ??) (26)
= ??(??, ??)
???????? > ??(?? + ?? |??) > ????????, 1 > ?? > ???? (20)
Where:-
In addition, the equality constraints:
????
??(?? + ?? + 1|??) = ??[??(?? + ?? |??), ??(?? + ??|??)], ???? ?? = [????], ?? = [???????????h?]
?? = [ ???? ],
0 > ?? > ???? - 1, (21) ????
??????
??(?? + ?? |??) = h[??(?? + ??|??)] , 1 > ?? > ???? (22)
Where:- are the vectors of state, control inputs and tracking outputs
? ??(?? + 1|??), ??(?? + 1|??) are the state variable and respectively.
optimal control input predicted at time ?? + 1 from 2. inequality constraints:
???????????? = ?????? = ????????????
measurements of the process model at time ??
?????????? = ???? = ??????????
respectively. ?????????? = ???? = ??????????
? ??(??) ?????? ??(??) are the optimal control inputs and (27)
outputs predicted from the process measurements ?????????? = ???? = ??????????
?????????? = ???? = ??????????
of the model at time ?? respectively. ????? ????? = ???? = ????? ?????
??(??) = ????????h???? = ??????h = ????????h????
[??(??|??),?? ??(?? + 1|??)??, … . , ??(?? + ???? - 1|??)??]
(23) Where the superscripts "min" and "max" denote the lower
??(??) = (24)
[??(??|??),?? ??(?? + 1|??)??, … . , ??(?? + ???? |??)??] and upper bounds of the parameters.
In (21) the integrand G(??, ??) is defined as:-
