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Electric car optimization finds the optimal control strategy for driving an electric car 100 meters in 10 seconds while minimizing energy consumption. The mathematical model uses ordinary differential equations (ODEs) to describe the car behavior. The manipulated variable is the voltage input to the motor, which can accelerate or recharge the battery with regenerative braking. The manipulated variable {`p(t)`} is a dimensionless variable bounded between -1 and 1. It scales the battery voltage {`V_{b}`} to regulate the electric current {`i(t)`} in the motor:

The electric car optimization problem involves finding the optimal control strategy for driving an electric car 100 meters in 10 seconds while minimizing energy consumption. The mathematical model uses ordinary differential equations (ODEs) to describe the car behavior. The manipulated variable is the voltage input to the motor, which can accelerate or recharge the battery with regenerative braking.

The manipulated variable {`p(t)`} is a dimensionless variable bounded between -1 and 1. It scales the battery voltage {`V_{b}`} to regulate the electric current {`i(t)`} in the motor:

* {`p = 1`} applies maximum positive voltage to accelerate the car.
* {`p = -1`} enables regenerative braking and recharging of the battery.
* {`p = 0`} disconnects the voltage supply, allowing the car to coast.

Starting with this simulation model, the objective is to minimize the total energy consumed while driving the electric car 100 meters in exactly 10 seconds. The manipulated variable {`p`} should be optimized to achieve this goal while adhering to critical constraints. Specifically, the electric current {`i(t)`} must remain between -150 and +150 A to avoid damaging the battery or overheating the electric cables. The control strategy should leverage acceleration, coasting, and regenerative braking efficiently. The final solution should ensure that the position reaches precisely 100 meters at the end of the simulation, with minimal energy consumption as the primary performance metric.

{$ \begin{array}{llcl} \min_{p} & e(T) \\ \mbox{s.t.} & \frac{di}{dt} &=& \frac{w V_{b} - R_m i - K_m \omega}{L_m} \\ & \frac{d\omega}{dt} &=& \frac{K_r^2}{M r^2} \left( K_m i - \frac{r}{K_r} \left( M g K_f + 0.5 \rho S C_x \omega^2 \frac{r^2}{K_r^2} \right) \right) \\ & \frac{dx}{dt} &=& \frac{\omega r}{K_r} \\ & \frac{de}{dt} &=& w i V_{b} + R_{bot} i^2 \\ & x(T) &=& 100 \\ & -150 &\leq& i(t) \leq 150 \\ & -1 &\leq& w(t) \leq 1 \end{array} $}

The electric car model uses several physical and design parameters to accurately represent the system dynamics. Each parameter plays a specific role in the vehicle's performance and control strategy.

* {`R_{bot} = 0.05`} Ω (Battery Resistance): Represents the internal resistance of the battery, which affects energy losses as heat during current flow.

* {`V_{alim} = 150`} V (Battery Voltage): The maximum supply voltage available from the battery to the motor, influencing the potential acceleration and braking forces.

* {`C_x = 0.4`} (Aerodynamic Coefficient): The drag coefficient that quantifies the aerodynamic efficiency of the car's shape, affecting resistance to motion at higher speeds.

These parameters are used in the system's differential equations to model the behavior of the electric car accurately. Understanding their influence helps in designing an optimal control strategy that minimizes energy consumption while meeting performance constraints.

R_bot, V_b, R_m, K_m = 0.05, 150.0, 0.03, 0.27

   energy.dt() == (p * i * V_b + R_bot * i**2)/1000

* {`p = 0`} from 0-1 sec
* {`p = 100/150`} from 1-1.9 sec
* {`p = 0`} from 1.9-10 sec

The manipulated variable {`p(t)`} is a dimensionless variable bounded between -1 and 1. It scales the battery voltage {`V_{alim}`} to regulate the electric current {`i(t)`} in the motor:

{$ \begin{array}{llcl} \min_{p} & e(T) \\ \mbox{s.t.} & \frac{di}{dt} &=& \frac{w V_{alim} - R_m i - K_m \omega}{L_m} \\ & \frac{d\omega}{dt} &=& \frac{K_r^2}{M r^2} \left( K_m i - \frac{r}{K_r} \left( M g K_f + 0.5 \rho S C_x \omega^2 \frac{r^2}{K_r^2} \right) \right) \\ & \frac{dx}{dt} &=& \frac{\omega r}{K_r} \\ & \frac{de}{dt} &=& w i V_{alim} + R_{bot} i^2 \\ & x(T) &=& 100 \\ & -150 &\leq& i(t) \leq 150 \\ & -1 &\leq& w(t) \leq 1 \end{array} $}

R_bot, V_alim, R_m, K_m = 0.05, 150.0, 0.03, 0.27

p_values[(m.time >= 1) & (m.time < 2)] = 100 / V_alim

   i.dt() == (p * V_alim - R_m * i - K_m * omega) / L_m,

   energy.dt() == (p * i * V_alim + R_bot * i**2)/1000

Starting with this simulation model, minimize the energy to travel 100m in 10 sec. Limit the current to between -150 and +150 A to prevent damage to the battery.

The manipulated variable {`w(t)`} is a dimensionless variable bounded between -1 and 1. It scales the battery voltage {`V_{alim}`} to regulate the electric current {`i(t)`} in the motor:
* {`w = 1`} applies maximum positive voltage to accelerate the car.
* {`w = -1`} enables regenerative braking and recharging of the battery.
* {`w = 0`} disconnects the voltage supply, allowing the car to coast.

The optimal control strategy involves switching {`w`} strategically to minimize energy consumption while meeting constraints on position and current. This often results in a "bang-bang" control pattern, using full throttle and full braking as needed to achieve the objective efficiently.

{$ \begin{array}{llcl}
\min_{e(t)} & e(T) \\
\mbox{s.t.} & \frac{di}{dt} &=& \frac{w V_{alim} - R_m i - K_m \omega}{L_m} \\
& \frac{d\omega}{dt} &=& \frac{K_r^2}{M r^2} \left( K_m i - \frac{r}{K_r} \left( M g K_f +
0.5 \rho S C_x \omega^2 \frac{r^2}{K_r^2} \right) \right) \\
& \frac{dx}{dt} &=& \frac{\omega r}{K_r} \\
& \frac{de}{dt} &=& w i V_{alim} + R_{bot} i^2 \\
& x(T) &=& 100 \\
& -150 &\leq& i(t) \leq 150 \\
& -1 &\leq& w(t) \leq 1
\end{array} $}

where {`i(t)`} is the electric current (A), {`\omega(t)`} is the angular velocity (rad/s), {`x(t)`} is the position of the car (m), and {`e(t)`} is the consumed energy (J). The manipulated variable {`w(t)`} regulates the current.

The initial simulation is a pulse in the voltage:

* {`w = 0`} from 0-1 sec
* {`w = 100/150`} from 1-1.9 sec
* {`w = 0`} from 1.9-10 sec

# Manipulated variable (w) as a step input
w = m.MV(value=0)
w.STATUS = 0

w_values = np.zeros(nt)
w_values[(m.time >= 1) & (m.time < 2)] = 100 / V_alim
w.VALUE = w_values

   i.dt() == (w * V_alim - R_m * i - K_m * omega) / L_m,

   energy.dt() == w * i * V_alim + R_bot * i**2

plt.plot(m.time, w.value, 'b-', label='Voltage MV (w)')

plt.plot(m.time, energy.value, 'k:', label='Energy Consumed')

The initial solution is a step-change in current with a sequence of 1-second pulses:
- {`i = 0`} A initially
- {`i = 100`} A for 1 second
- {`i = -50`} A for 1 second
- Back to {`i = 0`} A