Critical speed control of a solar car
The following optimal control problem is presented: The World Solar Challenge is a 3000 km race for solar powered cars across the Australian continent from Darwin to Adelaide. Each car is powered by a panel of photovoltaic cells which convert sunlight into electrical power. The power can be used directly to drive the car or stored in a battery for later use. Previous papers [\textit{P. G. Howlett, P. J. Pudney, T. Tarnopolskaya}, and \textit{D. Gates}, IMA J. Math. Appl. Bus. Ind. 8, 59--81 (1997; Zbl 0893.49025), \textit{P.G. Howlett} and \textit{P.J. Pudney}, Dyn. Contin. Discrete Impulsive Syst. 4, 553--567 (1998; Zbl 0914.49024)] using a simplified model of the battery, have shown that the optimal strategy is essentially a speedholding strategy. In this paper, with a more realistic model of the battery, they show that the optimal driving strategy is a critical speed strategy. For an optimal journey with no beginning and no ending the solar car must always travel at the critical speed. For an optimal journey of finite length the speed must be close to the critical speed for most of the journey. The critical speed depends on the solar power and will normally vary slowly with time. The mathematical model for this problem is described as follows. Assume that the force generated by the wheels of the driving vehicle is \(F = \frac{p}{v}\) where \(p\) is the power applied to the drive system and \(v\) the velocity of the car. Let \(R(v)\) be the resistive force increasing in \(v\) and suppose that \(\varphi (v) := v R(v)\) is convex. Then the equation of motion of the car is modeled by \[ \frac{d x}{d t} = v,\quad \frac{d v}{d t} = \frac{1}{m} \Big[\frac{p}{v}-R(v)\Big] \] with mass \(m\) and initial-boundary conditions \[ x(0) = x_0, \quad v(0) = v_0, \quad v(T) = v_T . \] Set \(p = b + s\) where \(s\) is the power coming from the solar array (\(b\) battery power). Additionally, the energy storage equations are supposed to be \[ \frac{d q}{d t} = - I(b), \quad q(0)=q_0, \quad q(T)=q_T \] where \(I(b)\) is the battery current required to supply the power \(b\), presuming perfect charge efficiency of the used battery. For silver-zinc batteries one may take \[ I(b) = c_1 b + c_2 b^2 \] where \([b] = \)Watts, \([I] =\) Ampères. The coefficients \(c_1\) and \(c_2\) can be estimated by methods such as least square fit. The authors derive necessary conditions for an optimal journey which maximizes the distance travelled in a day. For this purpose, they form the Hamiltonian \[ H = \pi_1 v + \frac{\pi_2}{m} \Big[\frac{p}{v} - R(v)\Big] - \pi_3 I(b) \] with adjoint equations for \(\pi_i (i=1,2,3)\). Then a correspondingly modified Hamiltonian is maximized. This leads to an optimal control \(b^*\) on \(b\) and optimal strategies by studying the level curves of maximized Hamiltonian \(H^*\). Now, it turns out that critical speed strategies (i.e., holding the speed close to a critical speed) are rather optimal compared to those suggested in previous papers. However, when the solar power is constant, the critical speed strategy coincides with the earlier known speedholding strategy. All in all, the new model and its properties seem to be more realistic than the previous ones.
Bellman principle of dynamic programming
scientific article; zbMATH DE number 1880490