Difference between revisions of "Electric Car"

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|nx        = 4
 
|nx        = 4
 
|nw        = 1
 
|nw        = 1
 +
|nc        = 2
 
|nri        = 2
 
|nri        = 2
 
|nre      = 5
 
|nre      = 5
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  \displaystyle \min_{x, u} & x_3(t_f)  \\[1.5ex]
 
  \displaystyle \min_{x, u} & x_3(t_f)  \\[1.5ex]
 
  \mbox{s.t.}  
 
  \mbox{s.t.}  
−
& \dot{x}_0 & = & (V_{alim} u – R_m x_0 – K_m x_1) / L_m, \\
+
& \dot{x}_0 & = & (V_{alim} u - R_m x_0 - K_m x_1) / L_m, \\
−
& \dot{x}_1 & = & \frac{K_r^2}{Mr^2} (K_m x_0 - \frac{r}{K_r} ( M g K_f + \frac{1}{2} \rho S C_x \frac{r^2}{K_r^2} x_1^2)),  \\
+
& \dot{x}_1 & = & \frac{K_r^2}{Mr^2} (K_m x_0 - \frac{r}{K_r} ( M g K_f + \frac{1}{2} \rho S C_x \frac{r^2}{K_r^2} x_1^2)),  \\
−
& \dot{x}_2 & = & \frac{r}{K_r} x_1,  \\
+
& \dot{x}_2 & = & \frac{r}{K_r} x_1,  \\
 
& \dot{x}_3 & = & V_{alim} u x_0 + R_{bat} x_0^2, \\[1.5ex]
 
& \dot{x}_3 & = & V_{alim} u x_0 + R_{bat} x_0^2, \\[1.5ex]
−
& x(t_0) &=& (0,0,0,0)^T, \\
+
& x(t_0) &=& (0,0,0,0)^T, \\
 
& x(t_f) & \in & \mathcal{T} \subseteq \mathbb{R}^4,\\
 
& x(t_f) & \in & \mathcal{T} \subseteq \mathbb{R}^4,\\
 
& x_0(t) & \in & [-i_{max}, i_{max}], \\
 
& x_0(t) & \in & [-i_{max}, i_{max}], \\
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</math>
 
</math>
 
</p>
 
</p>
−
Here the four differential states stand for the electrical current (<math> x_0 </math>), the angular velocity (<math> x_1 </math>), the position of the car (<math> x_2 </math>), and the consumed energy (<math> x_3 </math>). The objective function <math> x_3(t_f) </math> is just a reformulation of the Lagrange-type objective function tracking the used energy over time.
+
Here the four differential states stand for the electrical current (<math> x_0 </math>), the angular velocity (<math> x_1 </math>), the position of the car (<math> x_2 </math>), and the consumed energy (<math> x_3 </math>). The objective function <math> x_3(t_f) </math>is just a reformulation of the Lagrange-type objective function tracking the used energy over time.
−
 
+
  
 
== Parameters ==
 
== Parameters ==
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|-
 
|-
 
|Resistance of the battery
 
|Resistance of the battery
−
|<math>R_[bat}</math>
+
|<math>R_{bat}</math>
 
|0.05
 
|0.05
 
|<math> Ohms </math>
 
|<math> Ohms </math>
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== Reference Solutions ==
 
== Reference Solutions ==
−
We look at the particular instance of our problem with <math> t_f = 10s </math> and target set <math> \mathcal{T} = = \mathbb{R} \times \mathbb{R}  \times \{100\} \times \mathbb{R} </math>, in which the car needs to cover 100m in 10s.
+
We look at the particular instance of our problem with <math> t_f = 10s </math>and target set <math> \mathcal{T} = \mathbb{R} \times \mathbb{R}  \times \{100\} \times \mathbb{R} </math>, in which the car needs to cover 100m in 10s.
−
Figure 1 shows a plot of the differential states of the optimal trajectory of the relaxed problem (i.e. <math> u \in [-1,1] </math> instead of <math> u \in \{-1,1\} </math>) for <math> N = 1000, N </math> being the number of time discretization points. The current <math>  x_0 </math> increases to its maximal value of 150A, stays there for a certain time, decreases on its minimal value of -150A, stays on this value and eventually increases slightly. This behavior corresponds to the different arcs bang, path-constrained, singular, path-constrained, bang and can be observed also in Figure 2. It shows the corresponding switching function and the optimal control. Note that the plots show data from the solution with the indirect approach.
+
Figure 1 shows a plot of the differential states of the optimal trajectory of the relaxed problem (i.e. <math> u \in [-1,1] </math>instead of <math> u \in \{-1,1\} </math>) for <math> N = 1000, N </math>being the number of time discretization points. The current <math>  x_0 </math>increases to its maximal value of 150A, stays there for a certain time, decreases on its minimal value of -150A, stays on this value and eventually increases slightly. This behavior corresponds to the different arcs bang, path-constrained, singular, path-constrained, bang and can be observed also in Figure 2. It shows the corresponding switching function and the optimal control. Note that the plots show data from the solution with the indirect approach.
  
 
Applying the sum up rounding strategy results in an integer-feasible chattering solution.
 
Applying the sum up rounding strategy results in an integer-feasible chattering solution.
−
The resulting primal states are shown in Figure 3. One observes the high-frequency zig-zagging of the current  <math> x_0 </math> that results from the fast switches in the control.
+
The resulting primal states are shown in Figure 3. One observes the high-frequency zig-zagging of the current  <math> x_0 </math>that results from the fast switches in the control.
  
 
The direct and indirect approaches are local optimization techniques and only provide upper bounds for the relaxed problem and hence for the original problem. Here the indirect solution of the relaxed problem gives us a bound of <math> x_3(t_f) = 22777.2 </math>.
 
The direct and indirect approaches are local optimization techniques and only provide upper bounds for the relaxed problem and hence for the original problem. Here the indirect solution of the relaxed problem gives us a bound of <math> x_3(t_f) = 22777.2 </math>.
 +
 +
 +
<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="3">
 +
Image:ElectricCar PrimalStates Plot.PNG| Primal states of an optimal trajectory for the relaxed problem on a control discretization grid with <math> N = 1,000 </math>.
 +
Image:ElectricCar Control PathConstraints SwitchingFunction Plot.PNG| The optimal control and switching function. The dotted vertical lines show the switching times <math> \tau_i </math> where transitions between different kinds of arcs occur.
 +
Image:ElectricCar PrimalStates SumUpRounding.PNG| Primal states with Sum up rounding on grid with <math> \Delta t = 10^{-2} </math>.
 +
</gallery>
  
 
== Source Code ==
 
== Source Code ==
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There are several alternative formulations and variants of the above problem, in particular
 
There are several alternative formulations and variants of the above problem, in particular
−
* fixed final velocity, <math> \mathcal{T} = \mathbb{R} \times \{ 50 \frac{K_r}{3.6r} \} \times \{100\} \times \mathbb{R}, r_f = 10s </math>
+
* fixed final velocity, <math> \mathcal{T} = \mathbb{R} \times \{ 50 \frac{K_r}{3.6r} \} \times \{100\} \times \mathbb{R}, t_f = 10s </math>
−
* bounded velocity, <math> x_2(t) \leq 45 \frac{K_r}{3.6r} \forall t, mathcal{T} = \mathbb{R} \times \mathbb{R} \times \{100\} \times \mathbb{R}, r_f = 10s </math>
+
* bounded velocity, <math> \mathcal{T} = \mathbb{R} \times \mathbb{R} \times \{100\} \times \mathbb{R}, t_f = 10s, x_2(t) \leq 45 \frac{K_r}{3.6r} \forall t </math>
−
* fixed final velocity, bounded velocity, longer time horizon, <math> x_2(t) \leq 30 \frac{K_r}{3.6r} \forall t, \mathcal{T} = \mathbb{R} \times \{ 30 \frac{K_r}{3.6r} \} \times \{100\} \times \mathbb{R}, r_f = 15s </math>.
+
* fixed final velocity, bounded velocity, longer time horizon, <math> x_2(t) \leq 30 \frac{K_r}{3.6r} \forall t, \mathcal{T} = \mathbb{R} \times \{ 30 \frac{K_r}{3.6r} \} \times \{100\} \times \mathbb{R}, t_f = 15s </math>.
  
 
== References ==
 
== References ==

Latest revision as of 09:29, 27 July 2016

Electric Car
State dimension: 1
Differential states: 4
Discrete control functions: 1
Path constraints: 2
Interior point inequalities: 2
Interior point equalities: 5

The Electric car problem tries to find an optimal driving policy for an electric car. The goal is to use minimal energy to finish a given distance. As the car can be driven in two discrete modes, which either cause acceleration (and thereby consumption of energy) or the recharging of the battery, the control variable u(t) is supposed to be integer. Additionally the model for the electric car itself contains nonlinearities.

The problem is discussed in detail in [Sager2015]Author: Sager, S.; M. Claeys; F. Messine
Journal: Journal of Global Optimization
Number: 4
Pages: 721--743
Title: Efficient upper and lower bounds for global mixed-integer optimal control
Volume: 61
Year: 2015
Link to Google Scholar.

Mathematical formulation

The mixed-integer optimal control problem is given by

\begin{array}{llclr} \displaystyle \min_{x, u} & x_3(t_f) \\[1.5ex] \mbox{s.t.} & \dot{x}_0 & = & (V_{alim} u - R_m x_0 - K_m x_1) / L_m, \\ & \dot{x}_1 & = & \frac{K_r^2}{Mr^2} (K_m x_0 - \frac{r}{K_r} ( M g K_f + \frac{1}{2} \rho S C_x \frac{r^2}{K_r^2} x_1^2)), \\ & \dot{x}_2 & = & \frac{r}{K_r} x_1, \\ & \dot{x}_3 & = & V_{alim} u x_0 + R_{bat} x_0^2, \\[1.5ex] & x(t_0) &=& (0,0,0,0)^T, \\ & x(t_f) & \in & \mathcal{T} \subseteq \mathbb{R}^4,\\ & x_0(t) & \in & [-i_{max}, i_{max}], \\ & u(t) &\in& \{-1, 1\}. \end{array}

Here the four differential states stand for the electrical current ( x_0 ), the angular velocity ( x_1 ), the position of the car ( x_2 ), and the consumed energy ( x_3 ). The objective function x_3(t_f) is just a reformulation of the Lagrange-type objective function tracking the used energy over time.

Parameters

These fixed values are used within the model.

Parameters
Name Symbol Value Unit
Coefficient of reduction K_r 10 [-]
Air density \rho 1.293 kg/m^3
Aerodynamic coefficient C_x 0.4 [-]
Area in the front of the vehicle S 2 m^2
Radius of the wheel r 0.33 m
Constant representing the friction of the wheels on the road K_f 0.03 [-]
Coefficient of the motor torque K_m 0.27 [-]
Inductor resistance R_m 0.03 Ohms
inductance of the rotor L_m 0.05 [-]
Mass M 250 kg
Gravity constant g 9,81 [-]
Battery voltage V_{alim} 150 V
Resistance of the battery R_{bat} 0.05 Ohms

Reference Solutions

We look at the particular instance of our problem with t_f = 10s and target set \mathcal{T} = \mathbb{R} \times \mathbb{R} \times \{100\} \times \mathbb{R} , in which the car needs to cover 100m in 10s. Figure 1 shows a plot of the differential states of the optimal trajectory of the relaxed problem (i.e. u \in [-1,1] instead of u \in \{-1,1\} ) for N = 1000, N being the number of time discretization points. The current x_0 increases to its maximal value of 150A, stays there for a certain time, decreases on its minimal value of -150A, stays on this value and eventually increases slightly. This behavior corresponds to the different arcs bang, path-constrained, singular, path-constrained, bang and can be observed also in Figure 2. It shows the corresponding switching function and the optimal control. Note that the plots show data from the solution with the indirect approach.

Applying the sum up rounding strategy results in an integer-feasible chattering solution. The resulting primal states are shown in Figure 3. One observes the high-frequency zig-zagging of the current x_0 that results from the fast switches in the control.

The direct and indirect approaches are local optimization techniques and only provide upper bounds for the relaxed problem and hence for the original problem. Here the indirect solution of the relaxed problem gives us a bound of x_3(t_f) = 22777.2 .


  • Reference solution plots

  • Primal states of an optimal trajectory for the relaxed problem on a control discretization grid with N = 1,000 .

  • The optimal control and switching function. The dotted vertical lines show the switching times \tau_i where transitions between different kinds of arcs occur.

  • Primal states with Sum up rounding on grid with \Delta t = 10^{-2} .

Source Code

Model descriptions are available in

Variants

There are several alternative formulations and variants of the above problem, in particular

  • fixed final velocity, \mathcal{T} = \mathbb{R} \times \{ 50 \frac{K_r}{3.6r} \} \times \{100\} \times \mathbb{R}, t_f = 10s
  • bounded velocity, \mathcal{T} = \mathbb{R} \times \mathbb{R} \times \{100\} \times \mathbb{R}, t_f = 10s, x_2(t) \leq 45 \frac{K_r}{3.6r} \forall t
  • fixed final velocity, bounded velocity, longer time horizon, x_2(t) \leq 30 \frac{K_r}{3.6r} \forall t, \mathcal{T} = \mathbb{R} \times \{ 30 \frac{K_r}{3.6r} \} \times \{100\} \times \mathbb{R}, t_f = 15s .

References

[Sager2015]Sager, S.; M. Claeys; F. Messine (2015): Efficient upper and lower bounds for global mixed-integer optimal control. Journal of Global Optimization, 61, 721--743Link to Google Scholar
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