# Engine Speed Control System for Improving the Fuel Efficiency of Agricultural Tractors for
## 2 Controller Design
### Tractor Dynamic Simulation Model  
A schematic representation of the dynamic simulation model of the tractor used for calculating its performance parameters is shown in Figure 2. The simulation model solves the mathematical equations using the previously calculated engine speed, throttle command, and workload to determine the newly calculated engine speed. The engine model was designed using the look-up table module in Matlab/Simulink, and engine torque was defined as a function of rotational speed of engine and throttle opening, given in Equation (1). The tractor simulation model was based on a 48 kW MFWD diesel tractor ([PERSON], Seoul, South Korea). Specifications of the tractor are given in Table 1[7]. \[T_{e}=f\left(N_{e},\theta\right) \tag{1}\]  
where \(T_{e}\) is engine torque in N-m; \(N_{e}\) is engine rotational speed in RPM; \(\theta\) is throttle opening in %. The engine torque for the simulation model was based on the PTO performance data of the T623 tractor tested by the OECD test code considering a power transmission efficiency of 90%. The full-load torque with full throttle was expressed as a second order polynomial of engine speed, \begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **Specification** \\ \hline Unballasted weight, kN & 24.4 \\ Weight distribution (Front: Rear), \% & 47.6: 52.4 \\ Transmission (main/sub/creep) & Forward 24, Backward 24 (4/3/2) \\ \hline Engine & Manufacturer and Model & Caterpillar C3.3T 2232/2200 \\  & Rated power, kW & 48 kW @ 2200 rpm \\ \hline PTO & Rated power, kW & 47.61 \\  & Rated speed, rpm & 540, 733 @ rated engine speed \\ \hline Tire & Front & 11.2-20 SPR (drive tires) \\  & Rear & 14.9-30 SPR (drive tires) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specifications of MFWD tractor used for modelling in this study [7]. Figure 1: Block diagram of engine speed controller. Figure 2: Block diagram of tractor dynamic model. and the partial-load torque with both full and reduced throttles (\(\theta\)) by a linear function of engine speed (\(N_{e}\)) [8; 9; 10; 11; 12; 13]. The engine torque from Equation (1) was used to calculate the tractive force while considering the applied engine load torque, the selected transmission gear ratio, overall transmission efficiency, and draft load using Equations (2) and (3) [9; 10; 11; 12; 13]. \[TF=\frac{(T_{e}\times\eta_{t}\times G_{r})}{r_{d}}-F_{L} \tag{2}\]  
\[T_{L}=\frac{F_{L}\times r_{d}}{G_{r}\times\eta_{t}} \tag{3}\]  
where \(TF\) is tractive force in N; \(T_{L}\) is applied engine load torque; \(\eta_{t}\) is transmission efficiency; \(G_{r}\) is transmission gear ratio; \(r_{d}\) is dynamic radius of driving wheel in m; \(F_{L}\) is draft load in N.  
Finally, Equation (2) can be modified to the equation shown below using Equations (2) and (3). \[TF=\frac{(T_{e}-T_{L})\times\eta_{t}\times G_{r}}{r_{d}} \tag{4}\]  
The actual velocity of the tractor was calculated from the following equations using the draft force and the equivalent mass of the tractor. \(m_{eq}\) was calculated using Equation (6). It is the summation of the mass of the tractor consists and moment of inertia of the rotating part of transmission components, which is divided by the dynamic radius of driving wheel square. The tractor mass (\(m_{t}\)) and dynamic radius of rear wheel (\(r_{d}\)) for the simulation model was 2485 kg and 0.665 m based on the T623 tractor OECD test report. In order to determine the moment of inertia of the rotating part of transmission (\(I_{r}\)), engine speed rising time from idle to a certain speed was measured with the gear engaged and engine torque was also measured during engine speed acceleration with full throttle open. Measured engine speed rising time and engine torque were used to calculate engine speed angular acceleration (\(\alpha\)). For excluding the effect of load torque and tractor mass, the rear wheel of the tractor was suspended in air. Finally, Equation (7) was used for calculating the moment of inertia of the rotating part of transmission (\(I_{r}\)).