## 1 Introduction  
Today, vehicle manufacturers are forced to reduce the fuel consumption of their products because of enhanced environmental regulations. A promising solution is the development of plug-in hybrid electric vehicles (PHEV), as they combine an extended electric cruising range and the possibility of propelling the vehicle by an internal combustion engine (ICE) when the battery is depleted or when high performance is required. Common powertrain configurations of PHEVs are the series, parallel, and power split configurations. The efficiencies of these configurations vary depending on the distance and the power demand of the intended trip [1]. Increased efficiency can be obtained by using so-called multimode or dedicated hybrid transmissions (DHTs). This type of transmission enables switching between different powertrain configurations during vehicle operation and combining the individual advantages of these configurations. A general property of a DHT is that the electric motors are an integral and indispensable part of the transmission [2], which is the case when only the electric motor is able to propel the vehicle in a certain speed range. Since the part of the transmission for the ICE can be designed for the remaining and, in general, smaller speed range, a DHT requires fewer speeds and is less complex in comparison with a conventional automatic transmission. This saves a part of the additional weight and installation space caused by the electrification of the vehicle. Examples of available PHEVs equipped with a DHT can be found in [3; 4; 5; 6; 7]. Figure 1 shows the basic concept of a hybrid electric powertrain with a DHT, which is abstractly considered a configuration of two clutches (C), a transmission (T) with a constant gear ratio \(i_{\text{ED}}\), a planetary gear (PG) with a stationary gear ratio \(i_{0}\), and a two-speed transmission (2 ST) with the gear ratios \(i_{1/2}\). The concept requires only one electric drive and enables a continuous variable transmission mode (CVT), a parallel hybrid mode (PAR), and an electric driving mode (EM), where each mode can be driven in two speeds due to the two-speed transmission at the output. Clutch C\({}_{1}\) connects the ICE to the DHT, and clutch C\({}_{2}\) blocks the planetary gear (i.e., all shafts of the planetary gear rotate with the same speed). In this case, the speed ratios between the transmission's output and the two input shafts are constant and depend on the state of C\({}_{1}\) when either the parallel hybrid or the electric driving mode are active. Due to the chosen transmission ratios, the parallel hybrid mode can only be driven at higher vehicle speeds. Otherwise, the rotational speed of the ICE will fall below its lower limit, and the engine will stall. To operate the vehicle at lower speeds, the electric driving mode or, if the battery is discharged, the CVT mode must be activated. In this mode, the rotational speeds of the ICE and electric motor are superimposed by a planetary gear, whereby the speed of the ICE can be adjusted continuously as a function of the speed of the electric motor and the final drive. The design of the transmission ratios ensures that the electric motor operates as a generator in CVT mode until an appropriate vehicle speed is reached. A more detailed description can be found in a previous work [8], where the basic concept was used for the development of a new DHT including a specific configuration of gears and clutches and the corresponding transmission ratios. The operation of the hybrid electric powertrain requires an appropriate operating strategy, which determines the operation mode of the DHT and the torque distribution between the ICE and electric motor with the aim of maximal fuel economy. According to [9; 10; 11], numerous approaches for implementing such strategies are already known and classified into heuristic and optimization-based methods. Figure 2 shows an overview of the commonly applied methods. Figure 1: Basic concept of a hybrid electric powertrain with a DHT.