*Article* **Linear Quadratic Regulator and Fuzzy Control for Grid-Connected Photovoltaic Systems**

**Azamat Mukhatov 1, Nguyen Gia Minh Thao 2,\* and Ton Duc Do 1,\***


**Abstract:** This work presents a control scheme to control a grid-connected single-phase photovoltaic (PV) system. The considered system has four 250 W solar panels, a non-inverting buck-boost DC-DC converter, and a DC-AC inverter with an inductor-capacitor-inductor (LCL) filter. The control system aims to track and operate at the maximum power point (MPP) of the PV panels, regulate the voltage of the DC link, and supply the grid with a unity power factor. To achieve these goals, the proposed control system consists of three parts: an MPP tracking controller module with a fuzzybased modified incremental conductance (INC) algorithm, a DC-link voltage regulator with a hybrid fuzzy proportional-integral (PI) controller, and a current controller module using a linear quadratic regulator (LQR) for grid-connected power. Based on fuzzy control and an LQR, this work introduces a full control solution for grid-connected single-phase PV systems. The key novelty of this research is to analyze and prove that the newly proposed method is more successful in numerous aspects by comparing and evaluating previous and present control methods. The designed control system settles quickly, which is critical for output stability. In addition, as compared to the backstepping approach used in our past study, the LQR technique is more resistant to sudden changes and disturbances. Furthermore, the backstepping method produces a larger overshoot, which has a detrimental impact on efficiency. Simulation findings under various weather conditions were compared to theoretical ones to indicate that the system can deal with variations in weather parameters.

**Keywords:** fuzzy control; grid-connected PV system; incremental conductance algorithm; linear quadratic regulator; maximum power point tracking; unity power factor

### **1. Introduction**

Renewable energy is emerging as one of the main sources of energy for the future. The key reason for this is the depletion and pollution of fossil fuels. Renewable energy sources are available, clean, eco-friendly, and cost-effective. There are various types of renewable energy sources, of which solar and wind energy systems have become more and more popular in many countries. According to [1,2], harmonic resonances, which often occur in grid-connected wind power farms, cause negative effects on the power quality of the grid.

Nowadays, solar energy is widely used around the world and demonstrates impressive results. To effectively obtain electricity from solar energy, photovoltaic (PV) systems should be installed. The system efficiency is strongly affected by two major factors, as follow [3,4]:


While the prior factor is uncontrollable, the second one depends on the designer, system operator and electric grid. To improve the efficiency of the power electronic parts, appropriate converter topologies together with efficient control schemes are required. From [5–8], there are two modes of operation for the PV systems, which are:

• Stand-alone mode;

**Citation:** Mukhatov, A.; Thao, N.G.M.; Do, T.D. Linear Quadratic Regulator and Fuzzy Control for Grid-Connected Photovoltaic Systems. *Energies* **2022**, *15*, 1286. https://doi.org/10.3390/en15041286

Academic Editors: Marco Pasetti and Wojciech Cieslik

Received: 2 December 2021 Accepted: 1 February 2022 Published: 10 February 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

• Grid-connected mode.

Between these two modes of operation, the grid-connected mode is preferable, as it can avoid the issues of storage systems in the stand-alone mode. For grid-connected systems, the following two problems need to be solved simultaneously [7–9]:


Solving the second problem often requires the following tasks:


One of the most important parts of this research is the MPP tracking part, which is mainly used to find and keep the output power of a PV panel at its maximum value [11–13]. The MPP tracking (MPPT) technique can be divided into two main categories: the perturb and observe (PO) technique and the incremental conductance (INC) algorithm. The negative side of the PO algorithm is its high computational complexity, but it leads to high efficiency. It measures voltage and current values to periodically estimate the power of the solar panel and compares it with the previous power. If the power of the PV module has increased *(dP/dV* > 0), the system will start adjustments in that direction; otherwise, it will adjust in the opposite way. These operations continue until system finds the MPP. In fact, the technique depends on perturbations of the voltage, so if the perturbations are high, the speed of the technique is fast. The advantages of this method are simplicity without interest in the previous PV characteristics; however, the main drawback is that oscillations happening near the MPP, which may lead to power losses in varying weather conditions [10]. The INC type is more advantageous in terms of accuracy in finding and tracking the MPP compared to the second type; therefore, in this paper, the INC algorithm is improved by fuzzy control and then implemented in the grid-connected PV system.

Considering current controller strategies, generally, they can be divided into two main categories: on/off controllers and pulse width modulation (PWM)-based control techniques [14]. The first group has two subdivisions, which are hysteresis control and predictive control. Hysteresis control has high dynamics and fast response; however, its major drawbacks are variations of the switching frequency and high complexity of the system. Predictive control has positive sides such as less computation time, better regulation, and a decrease in offset error. On the other hand, it requires identifying a proper model for the system and the installation cost of the system is high. The second group can be divided into linear and non-linear control [14]. Proportional-integral (PI) controller and its different update versions such as multiple generalized integral (MGI) [15] and Manta Ray Foraging Optimization [16] control are the well-known classical control techniques, along with their improved versions, which can be easily designed to control the current. However, the key disadvantage of this controller is its poor compensation of lower-order harmonics and presence of steady-state errors [17]. The proportional-resonant (PR) controlled can compensate for the harmonics. Moreover, this type of controller has high dynamics, is less complex, and can reach a high gain at the resonance frequency. However, this controller has problems reaching the power factor control, which means that the system is not able to control the losses in the system [18].

Generally, the power factor is a ratio between working power and apparent power. Thus, if there is no control/maintenance of a high-power factor, the system efficiency is consequently low. On the other hand, predictive deadbeat control has a high level of harmonic compensation and rapid fast-tracking performance. The disadvantage of this controller is that it requires a lot of computation effort [19]. Harmonic compensation and steady-state error can also be done by repetitive controllers; their slow tracking response is their main drawback [20]. In various dynamic conditions, the effectiveness of the third order

complex filter (TOCF) control algorithm may be demonstrated. This controller serves as a distribution static compensator, enhances grid power quality by reducing harmonics, and balances grid currents while maintaining a unity power factor. However, the controller's biggest disadvantage is its enormous computational load [21]. Another method which possibly could be proposed for mitigation of the issue is the model predictive control (MPC) method [22] and its improved versions [23,24]. However, these methods present oscillations when the load varies. In addition, the THD of MPC is higher compared to other methods, which makes the technique unsuitable for implementation in this case. There exist different kinds of adaptive filtering techniques, such as leaky least mean mixed norm (LLMMN), least mean mixed norm (LMMN), and least mean square (LMS). However, all of the above-mentioned adaptive approaches have the drawback of being unable to maintain acceptable performance in the presence of DC-offset [25]. The combined affine projection sign algorithm (CAPSA) has high stability and tracking performance, but output has the same problem as the previous mentioned method, which is the high THD [26]. Artificial neural networks (ANN) also can be implemented as controllers in this system, but the main disadvantage are their low output power compared to fuzzy logic [27]. Fuzzy logic controllers (FLC) [14] are one of the popular intelligent control techniques. They are extensively used in renewable energy systems due to their efficiency and ease of use. They are also robust and applicable to a wide range of the dynamics systems, from linear to nonlinear systems. Moreover, this type of controller can perform complex estimations, which are not possible with conventional methods [11]. The linear quadratic regulator (LQR) is an effective control method that is applicable for both linear and nonlinear systems. In this method, the control gain is designed to minimize a quadratic cost function by selection of appropriate weighting matrices. In our study case, the cost function is the quadratic function of the tracking error between the current and its reference and the control efforts. This technique was chosen to be implemented due to its properties such as stability, robustness, and ease in application. Moreover, the computational complexity of the LQR controller is not high, which means that it is fairly simple to implement.

This paper proposes a complete control solution for grid-connected single-phase PV systems based on fuzzy control and an LQR. Our past related research on this topic was conducted with a different type of controller, namely the backstepping approach. The present study is a significant extension and improvement of our former research in terms of enhancing the quality of the control method. The proposed technique is the LQR in appropriate combination with fuzzy control and improved INC algorithms for grid-connected photovoltaic systems; furthermore, detailed explanations on developing the fuzzy association rules of the designed fuzzy logic controllers are newly presented in this study. The main originality of this paper is to show and prove by comparison of our former and present control methods that the newly suggested method is more effective in various aspects. Specific details of the PV system and controllers can be found in our past work in [28], which was used as the basis for this paper. The major advantage of the LQR method is its ability to react in a rapid manner to changes of the system, namely, changes in the module temperature or solar irradiation. In other words, the system can reach its settling time faster, which is important to stabilize the behavior of output. Moreover, it can be said that the LQR technique is robust when faced with different disturbances and changes compared to the backstepping technique and its improved fault tolerant version [29]. In addition, the backstepping method has a higher overshoot, which significantly impacts efficiency in a negative way. As was mentioned above, the speed of the LQR is faster, which makes this kind of controller preferable. These are the key contributions of this study compared to our past research in [28]. Simulation results under different weather conditions show that the proposed control system can cope with changes in weather parameters effectively, and were compared with theoretical ones. Moreover, it was shown that variations in weather parameters do not significantly affect the performance of the proposed control system.

The remains of the paper are organized as follows. Section 2 shows the modeling of the grid-connected PV system, which includes the system description, PV panel model, and modeling of power converters. The control system design is depicted in Section 3, which consists of the MPPT control module, DC-link voltage regulator module, and current controller module. In addition, Section 4 provides simulation results in MATLAB, in which the first test case is with a fixed module temperature, and the second test case is with an unchanged solar irradiation. A detailed comparison and assessment of efficacy between the LQR control method in this study and the backstepping approach in our past work [28] is presented in Section 5; brief comparisons between this research and other related works are also shown in this section. The conclusions are described in the last section.

#### **2. PV Grid-Connected System Modeling**

#### *2.1. System Description*

This paper considers a grid-connected PV system consisting of two stages of power conversion. The nominal power of the system is 1 kW. Figure 1 shows the circuitry of the system; the power generated from the PV array is directed to the non-inverting buck-boost DC-DC converter. After that, to supply the grid, the obtained result is converted to AC via the single-phase DC-AC inverter. To remove unwanted noises and disturbances injected to the grid, the LCL output filter was used [30,31].

**Figure 1.** PV single-phase grid-connected system.

#### *2.2. PV Panel Model*

The PV panels used in this paper have characteristics as presented in [28]. The provided data is applicable when the module temperature is 25 ◦C and the solar radiation is 1000 W/m2. In total, the PV array consists of four panels, where the nominal power of each panel is 250 W. Figure 2 shows the impacts of module temperature and solar radiation on the power and voltage of the PV panel, respectively. Table 1 represents MPPs of the PV panel and array in terms of power and voltage.

**Figure 2.** Power obtained from PV panels for different (**a**) solar radiation values and (**b**) module temperature values.


**Table 1.** Power and Voltage at Maximum Power Points.

#### *2.3. Modeling of Converters*

Figure 1 illustrates all components of the system including the single-phase inverter and the non-inverting buck-boost converter [7]. The input control signals of the noninverting buck-boost converter and the single-phase inverter are *α*<sup>p</sup> and *β*p, respectively.

$$\boldsymbol{a}\_{p} = \begin{cases} \text{ 0; } \text{S}\_{1} \text{ and } \text{S}\_{2} \text{ are OFF} \\ \text{ 1; } \text{S}\_{1} \text{ and } \text{S}\_{2} \text{ are ON} \end{cases} \\ \boldsymbol{\beta}\_{p} = \begin{cases} \text{ 1; } \text{S}\_{3} \text{ and } \text{S}\_{5} \text{ are ON, } \text{S}\_{4} \text{ and } \text{S}\_{6} \text{ are OFF} \\\ \text{ 0; } \text{S}\_{3}, \text{S}\_{4}, \text{S}\_{5} \text{ and } \text{S}\_{6} \text{ are OFF} \\\ \text{ -1; } \text{S}\_{3} \text{ and } \text{S}\_{5} \text{ are OFF, } \text{S}\_{4} \text{ and } \text{S}\_{6} \text{ are ON} \end{cases}$$

The modeling technique, specifically averaging, and Kirchhoff's laws were used to estimate a mathematical model for the two converters. Equation (1) and Table 2 demonstrate details of the previously mentioned procedure

$$\begin{cases} \dot{\mathbf{x}}\_{1} = \frac{1}{\mathbf{C}\_{i}} \bar{I}\_{P} - \alpha \frac{1}{\mathbf{C}\_{i}} \mathbf{x}\_{2} \\ \dot{\mathbf{x}}\_{2} = \alpha \frac{1}{L\_{1}} \mathbf{x}\_{1} - \frac{R\_{1}}{L\_{1}} \mathbf{x}\_{2} + (\alpha - 1) \frac{1}{L\_{1}} \mathbf{x}\_{3} \\ \dot{\mathbf{x}}\_{3} = (1 - \alpha) \frac{1}{\mathbf{C}\_{\text{DC}}} \mathbf{x}\_{2} - \beta \frac{1}{\mathbf{C}\_{\text{DC}}} \mathbf{x}\_{4} \\ \dot{\mathbf{x}}\_{4} = \beta \frac{1}{L\_{f}} \mathbf{x}\_{3} - \frac{R\_{f}}{L\_{f}} \mathbf{x}\_{4} - \frac{1}{L\_{f}} \mathbf{x}\_{5} \\ \dot{\mathbf{x}}\_{5} = \frac{1}{\mathbf{C}\_{f}} \mathbf{x}\_{4} - \frac{1}{\mathbf{C}\_{f}} \mathbf{x}\_{6} \\ \dot{\mathbf{x}}\_{6} = \frac{1}{L\_{\chi}} \mathbf{x}\_{5} - \frac{R\_{\mathcal{S}}}{L\_{\chi}} \mathbf{x}\_{6} - \frac{1}{L\_{\chi}} V\_{\mathcal{S}} \end{cases} (1)$$

**Table 2.** Variables.


#### **3. Control System Design**

The design of the control system considered in this study is shown in Figure 3 and includes three main parts: the MPPT controller, the DC link voltage regulator, and the current controller. In this paper, detailed explanations on developing the fuzzy rules of the two designed fuzzy logic controllers are presented, which are useful as references for designing other fuzzy controllers.

**Figure 3.** System schematics in detail.

#### *3.1. The MPPT Controller Module*

The PV array produces its optimal power despite varying weather with the help of the designed MPPT controller. According to Figure 4, this controller has two parts: the first fuzzy logic controller (FLC-1) and the proportional-integral (PI-1) controller.

**Figure 4.** MPPT controller module schematics.

#### 3.1.1. FLC-1

The main idea of this sub-controller is to improve the conventional INC-MPPT algorithm in terms of response time and efficiency by combining it with a fuzzy logic controller (FLC-1). According to Figure 5, the FLC-1 has two inputs and one output. The first input can be one of the following two kinds:


$$A\_P(k) = \mathcal{G}\_P(k)[\mathcal{S}\_P(k)] = \mathcal{G}\_P(k)\left[I\_p(k) + V\_p(k)\frac{dI\_P(k)}{dV\_P(k)}\right] \tag{2}$$

$$G\_P(k) = \frac{1}{1 + g\_1 \left[\frac{P\_P(k)}{P\_{P,total}^{max}}\right]} \tag{3}$$

where *Pmax <sup>P</sup>*,*total* = 1000 W is the maximum power of PV panels and *g*<sup>1</sup> is a positive coefficient. The second input is the INC algorithm's prior step-size Δ*V*(*k*−1). Figure 6 shows the

detailed flowchart of the proposed method. In addition, the aforementioned scaling module *Gp(k)* is used to suitably increase the sensitivity of slope *Sp*(*k*) as given in Figure 7.

$$\lim\_{P p(k)\to 0} G\_P(k) = 1 \\ \lim\_{P p(k)\to P^{\text{max}}\_{P,total}} G\_P(k) = \frac{1}{1+g1}$$

$$P\_P(k) = V\_P(k) \times I\_P(k) \tag{4}$$

$$dI\_P(k) = I\_P(k) - I\_P(k-1) \tag{5}$$

$$dV\_P(k) = V\_P(k) - V\_P(k-1) \tag{6}$$

To avoid significant changes in the step-size and instability of the PV output power, a switching module is implemented as described in Figure 5. According to the first input, namely, |*Ap(k)*| or |*dIp(k)|,* the system will use the appropriate output coefficient *g*<sup>2</sup> as shown in Table 3.

**Table 3.** Switching module operation.


As is known from previous parts of this paper, the inputs are in the range of [0,1]. It should be noted that all the inputs have the same number of linguistic variables, specifically five linguistic variables: VS—Very Small, SM—Small, ME—Medium, LA—Large, VL—Very Large. The output has nine linguistic variables in a range of [−1;1]; in detail, NL—Negative Large, NM—Negative Medium, NS—Negative Small, NZ—Negative Zero, ZE—Zero, PZ—Positive Zero, PS—Positive Small, PM—Positive Medium, PL—Positive Large. As a result, there are 49 fuzzy rules associated in the FLC-1.

**Figure 5.** FLC-1 structure.

**Figure 6.** INC-MPPT algorithm with fuzzy logic.

**Figure 7.** PV panel power–voltage (P-V) curve.

All the association rules of the FLC-1 are shown in Table 4, while the membership functions of the inputs and output can be referred to in [28]. To explain the fuzzy rules in Table 4, we can analyze several sample cases as follows. In the first case, when the two inputs Δ*V*(*k*−1) and |*dIp(k)*| are VS, that means that the PV system is close to the MPP and the step voltage is also very small; thus, the output of the FLC-1 as the additional voltage *Vadd*(*k*) should be ZE to avoid fluctuations in the PV voltage at the steady state. Whereas, in another case when Δ*V*(*k*−1) is LA and |*dIp(k)*| is ME, the additional voltage *Vadd(k)* will be NZ because the tendency of the PV system is automatically approaching the MPP. On the other hand, when Δ*V*(*k*−1) is vs. and |*dIp(k)|* is VL, it means that the PV system is far from the MPP; therefore, the output Δ*V*(*k*−1) should be PL to force the PV system to

quickly move to the MPP. Furthermore, when |*dIp(k)|* is VL and Δ*V*(*k*−1) is VL, the output Δ*V*(*k*−1) can be chosen as either PZ or ZE for the PV system to automatically move to the MPP; in this study, we want to increase the speed for searching the MPP, so the output Δ*V*(*k*−1) is set as PZ in this case. In general, the other fuzzy rules in Table 4 can be suitably interpreted with the same deductive method.


**Table 4.** Fuzzy association rules for FLC-1.

#### 3.1.2. PI-1 Controller

The PI-1 controller with an anti-windup block (refer to [32]) serves as the second sub-controller of the system. Figures 4 and 8 show detailed schematics of the controller.

**Figure 8.** PI-1 controller in detail.

#### *3.2. DC Link Voltage Regulator Module*

According to Figure 3, the objective of the DC link voltage regulator is to determine an appropriate value of the reference grid current *I ref <sup>g</sup>* used for the current controller module. We note that the ultimate goal here is to make the DC link voltage *VDC* reach its desired value *Vref DC* once the actual grid current *Ig* is well regulated to its reference *I ref <sup>g</sup>* [7,8] by the proposed current controller module using the LQR technique, which will be shown in detail in Section 3.3. In existing studies, a conventional PI controller has often been used to generate the reference grid current *I ref <sup>g</sup>* from the DC link voltage difference *eVdc* <sup>=</sup> *<sup>V</sup>ref DC* − *VDC*, as shown in the upper left part of Figure 9. Nevertheless, it is difficult to manually choose and tune optimal values for the coefficients of the PI-2 controller due to the high nonlinearity of the grid-connected PV system, including a LCL output filter. Furthermore, the response of the conventional PI-2 controller usually has fairly large overshoot in the transient state and achieves the steady state in a relatively slow manner. Thus, this paper proposes a novel hybrid control scheme for the DC link voltage regulator module using another FLC (named FLC-2) as depicted in Figure 9 to overcome the above-mentioned drawbacks of the traditional PI-2 controller and remarkably enhance the response time.

**Figure 9.** Proposed PI-fuzzy hybrid control scheme, where *g*3, *g*<sup>4</sup> and *g*<sup>5</sup> are design coefficients.

Firstly, the theoretical relation between the output power of the DC-AC inverter supplied to the grid *PAC* and the output power of the PV array *PP* can be expressed as follows:

$$P\_{A\C} = \eta\_{\text{DC}-A\C} \times P\_{\text{DC}-link} = \eta\_{\text{DC}-A\C} \times \left(\eta\_{\text{DC}-D\C} \times P\_P\right) = \eta\_{\text{Exp}} \times P\_P \tag{7}$$

where:

*ηDC*−*DC*—the efficiency of the buck-boost DC-DC converter can be estimated theoretically as a ratio value of the DC link power *PDC*−*link* and the output power of PV array *PP*; *ηDC*−*AC*—the efficiency of the DC-AC inverter can be estimated theoretically as a ratio value of the output power of the inverter *PAC* and the DC link power *PDC*−*link*;

*ηExp* = *ηDC*−*DC* × *ηDC*−*AC*—the overall efficiency of the grid-connected PV system.

Equation (7) can be written as:

$$V\_{\mathcal{S}} I\_{\mathcal{S}} \left(\cos \theta\_{\mathcal{S}}\right) = \eta\_{\text{Exp}} P\_P \tag{8}$$

here cos*θ<sup>g</sup>* is the PF of the PV system, *Vg* is the rms value of the grid voltage, and *Ig* is the rms value of the grid current. In the normal operation of the grid, the rms value of the grid voltage is often larger than zero, meaning that *Vg* > 0 V.

Hence,

$$I\_{\mathcal{S}} = \eta\_{\text{Exp}} \frac{P\_{\mathcal{P}}}{V\_{\mathcal{S}}(\cos \theta\_{\mathcal{S}})} \tag{9}$$

When the PF = 1, the grid current will reach the following value.

$$I\_{\mathcal{S}} = \eta\_{\text{Exp}} P\_{\text{P}} / V\_{\mathcal{S}} \tag{10}$$

However, the actual overall efficiency *ηExp* depends not only on the PF, but also on other component parameters of the DC-DC buck-boost converter and the DC-AC inverter, as well as the operating conditions of the PV system (e.g., the PV power, temperature, input voltage, and so forth); as a result, it is difficult to accurately estimate a particular value for *ηExp*. We note that Equations (7)–(10) are only explanations of the theoretical relations among the parameters *ηExp*, *PP*, *PAC*, *Ig*, and *Vg*, which are used as the reference basis for introducing and calculating a new "virtual efficiency" *ηvir exp* in our proposed hybrid control scheme for the DC link voltage regulator module, as depicted in Figure 9. Hence, the calculated value of the "virtual efficiency" *ηvir exp* in this figure and (11) is not the actual value of the overall efficiency of the PV system *ηExp* in (7). In this study, we calculate the "virtual efficiency" value *ηvir exp* instead of estimating the actual overall efficiency *ηExp*.

To fulfill the above goal, the FLC-2 is designed to frequently update a suitable value for the "virtual efficiency" *ηvir exp*(*k*) in real-time, as described in (11) and Figure 9. Then, from (10), an additional value *IFLC <sup>g</sup>* for adjusting the reference grid current *I ref <sup>g</sup>* can be computed as *IFLC <sup>g</sup>* (*k*) = *ηvir exp*(*k*) × *PP*(*k*)/*Vg*(*k*), as shown in the right part of Figure 9. Finally, this computed additional value *IFLC <sup>g</sup>* (*k*) is used to effectively compensate for the output of the conventional PI-2 controller *IPI <sup>g</sup>* (*k*) to appropriately determine the reference grid current *I ref <sup>g</sup>* (*k*), as presented in the upper right part of Figure 9; in detail, *I ref <sup>g</sup>* (*k*) = *IPI <sup>g</sup>* (*k*) + *IFLC <sup>g</sup>* (*k*). The key aims of *IFLC <sup>g</sup>* (*k*) generated by our proposed control scheme using the FLC-2 are to significantly improve the response time for updating a suitable value for the reference current *I ref <sup>g</sup>* (*k*) and to elevate the effectiveness of the conventional PI-2 controller against effects caused by the high nonlinearity of the PV system.

In fact, using the proposed current controller module (refer to Figure 3 and Section 3.3), when the grid current *Ig*(*k*) is regulated to its reference *I ref <sup>g</sup>* (*k*) suitably generated by the designed PI-Fuzzy hybrid control scheme (see Figure 9), the DC link voltage *VDC* achieves its desired value *Vref DC* [7,8]. This means that both the error values of the DC link voltage (*eVdc* in Figure 9) and the grid current in (14) and (15) are considered and regulated by the proposed complete control system, as given in the lower part of Figure 3.

The designed FLC-2 has two inputs and one output.

The two inputs are:

*eVdc*(*k*)—error between desired and present DC link voltage; d*eVdc*(*k*)—change in error.

The output is:

Δ*η*(*k*)—step in efficiency, added to the "virtual efficiency" *ηvir exp* to reach the desired value:

$$
\eta\_{\exp}^{vir}(k) = \eta\_{\exp}^{vir}(k-1) + \Delta\eta(k) \tag{11}
$$

The two inputs are:

$$e\_{Vdc}(k) = V\_{DC}^{ref} - V\_{DC}(k) \tag{12}$$

$$\text{d}e\_{Vdc}(k) = e\_{Vdc}(k) - e\_{Vdc}(k-1) \tag{13}$$

All the inputs have the same number of linguistic variables, specifically seven; the range is [−20;20]: NL—Negative Large, NM—Negative Medium, NS—Negative Small, ZE—Zero, PS—Positive Small, PM—Positive Medium, PL—Positive Large

The output Δ*η*(*k*) has nine linguistic variables, and they range from [−1;1]: NL—Negative Large, NM—Negative Medium, NS—Negative Small, NZ—Negative Zero, ZE—Zero, PZ—Positive Zero, PS—Positive Small, PM—Positive Medium, PL—Positive Large. As a result, there are 49 fuzzy rules formed in the FLC-2.

All the association rules of the FLC-2 are presented in Table 5, while the membership functions of the inputs and output can be referred to in [28]. To interpret the fuzzy rules in Table 5, we can analyze and evaluate some sample cases as follows. Firstly, when d*eVdc*(*k*) is NL and *eVdc*(*k*) is PL, the output of the fuzzy controller Δ*η*(*k*) should be ZE since the tendency of the DC-link voltage *Vdc(k)* is automatically approaching its reference value. On the other hand, when d*eVdc*(*k*) is ZE and *eVdc*(*k*) is NL, it means that *Vdc(k)* is much smaller than its reference value; thus, the output Δ*η*(*k*) should be PL to force *Vdc(k)* to rapidly move to the desired value. Furthermore, when d*eVdc*(*k*) is ZE and *eVdc*(*k*) is PS, it means that *Vdc(k)* is marginally larger than its reference value; hence, the output Δ*η*(*k*) should be NS to slightly decrease *Vdc(k)* to its desired value without oscillation at the steady state. In general, the other fuzzy rules in Table 5 can be appropriately explained with a similar deductive technique.


**Table 5.** Fuzzy association rules for FLC-2.

*3.3. Current Controller Module*

In this section, the current controller is designed by an optimal control method. Firstly, from (2), we have the following dynamic model,

$$\begin{cases}
\dot{\mathbf{x}}\_4 = -\frac{R}{L}\mathbf{x}\_4 - \frac{1}{L}\mathbf{x}\_5 + \frac{\mu}{L} \\
\dot{\mathbf{x}}\_5 = \frac{1}{\mathcal{C}}\mathbf{x}\_4 - \frac{1}{\mathcal{F}}\mathbf{x}\_6 \\
\dot{\mathbf{x}}\_6 = \frac{1}{L\_{\mathcal{S}}}\mathbf{x}\_5 - \frac{R\_{\mathcal{S}}}{L\_{\mathcal{S}}}\mathbf{x}\_6 - \frac{1}{L\_{\mathcal{S}}}V\_{\mathcal{S}}
\end{cases} \tag{14}$$

The main purpose of the current controller is to make the grid current *ig* (i.e., *x*6) converge to its reference *x*6*ref* . Then, from the third equation of (14), the error dynamics of *x*<sup>6</sup> and the reference for *x*<sup>5</sup> (i.e., *x*5*ref*) can be derived as,

$$\dot{\mathbf{x}}\_{6} - \dot{\mathbf{x}}\_{6ref} = \frac{1}{L\_{\mathcal{g}}} \left( \left( \mathbf{x}\_{5} - \mathbf{x}\_{5ref} \right) + \mathbf{x}\_{5ref} \right) - \frac{R\_{\mathcal{g}}}{L\_{\mathcal{g}}} \left( \left( \mathbf{x}\_{6} - \mathbf{x}\_{6ref} \right) + \mathbf{x}\_{6ref} \right) - \frac{1}{L\_{\mathcal{g}}} V\_{\mathcal{g}} \tag{15}$$

Thus, we have .

$$
\dot{\widetilde{\mathbf{x}}}\_{6} = \frac{1}{L\_{\mathcal{S}}} \widetilde{\mathbf{x}}\_{5} - \frac{R\_{\mathcal{S}}}{L\_{\mathcal{S}}} \widetilde{\mathbf{x}}\_{6} - \frac{1}{L\_{\mathcal{S}}} \mathbf{x}\_{5ref} \tag{16}
$$

where *x*5*ref* is determined by

$$\mathbf{x}\_{5ref} = \mathbf{R}\_{\mathcal{S}} \mathbf{x}\_{6ref} + \upsilon\_{\mathcal{S}} + \dot{\mathbf{x}}\_{6ref} L\_{\mathcal{S}} \tag{17}$$

Similarly, with *x*5*ref* achieved from (17), combined with the second equation of (14),

$$
\dot{\mathbf{x}}\_5 - \dot{\mathbf{x}}\_{5ref} = \frac{1}{\mathbf{C}} \left( \left( \mathbf{x}\_4 - \mathbf{x}\_{4ref} \right) + \mathbf{x}\_{4ref} \right) - \frac{1}{\mathbf{C}} \left( \left( \mathbf{x}\_6 - \mathbf{x}\_{6ref} \right) + \mathbf{x}\_{6ref} \right) \tag{18}
$$

then .

$$
\dot{\tilde{\mathfrak{X}}}\_5 = \frac{1}{\bar{C}} \tilde{\mathfrak{X}}\_4 - \frac{1}{\bar{C}} \tilde{\mathfrak{X}}\_6 \tag{19}
$$

where

$$
\dot{\mathbf{x}}\_{4ref} = \mathbf{x}\_{6ref} + \dot{\mathbf{x}}\_{5ref} \mathbf{C}\_f \tag{20}
$$

From *x*5*ref* and *x*4*ref* , obtained in (17) and (20), respectively, the first equation of (14) can be rewritten as

$$\dot{\tilde{\mathbf{x}}}\_{4} = -\frac{R}{L}\tilde{\mathbf{x}}\_{4} - \frac{R}{L}\mathbf{x}\_{4ref} - \frac{1}{L}\tilde{\mathbf{x}}\_{5} - \frac{1}{L}\mathbf{x}\_{5ref} - \dot{\mathbf{x}}\_{4ref} + \frac{1}{L}\mu\_{1} + \frac{1}{L}\mu\_{2} \tag{21}$$

Hence, we have

$$\dot{\mathfrak{X}}\_4 = -\frac{R}{L}\widetilde{\mathfrak{X}}\_4 - \frac{1}{L}\widetilde{\mathfrak{X}}\_5 + \frac{1}{L}u\_1\tag{22}$$

where

$$
\mu\_2 = R\mathbf{x}\_{4ref} + \mathbf{x}\_{5ref} + L\dot{\mathbf{x}}\_{4ref} \tag{23}
$$

Here, we decompose the control input u into two terms: *u*<sup>1</sup> and *u*2; in detail, *u*<sup>1</sup> is used for feedback control to stabilize the error dynamics, whereas *u*<sup>2</sup> is the compensating term used to compensate for the offset in the reference tracking problem. Finally, the error dynamics of (15) are achieved by combining (22), (19), and (16), as follows:

$$
\begin{bmatrix} \mathbf{x}\_4 \\ \mathbf{x}\_5 \\ \mathbf{x}\_6 \end{bmatrix} = \begin{bmatrix} -\frac{R\_\mathcal{S}}{L\_\mathcal{S}} & \frac{1}{L\_\mathcal{S}} & 0 \\ -\frac{1}{L} & 0 & \frac{1}{L} \\ 0 & -\frac{1}{L} & -\frac{R}{L} \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ \frac{1}{L} \end{bmatrix} u\_1 \tag{24}
$$

Equation (24) is rewritten in the following form:

$$
\dot{\mathfrak{x}} = A\mathfrak{x} + Bu\_1 \tag{25}
$$

Consider the following cost function:

$$J(\mathbf{x}, \boldsymbol{\mu}) = \int\_0^\infty \mathbf{x}^T Q \mathbf{x} + \boldsymbol{\mu}\_1^T Q \mathbf{u}\_1 \tag{26}$$

where *Q* ≥ 0 and *R* > 0 are the weighting matrices with appropriate dimensions; that is, 3 × 3 and a scalar, respectively. After that, this cost function is minimized by the following control law:

$$\mu\_1 = -\mathbf{K}\mathbf{x} = -\mathbf{R}^{-1}\mathbf{B}^T\mathbf{P}\mathbf{x} \tag{27}$$

where *K* is the controller gain matrix, and *P* is the positive definite solution of the algebraic Riccati equation as follows

$$PA + A^T P - PBR^{-1}B^T P + Q = 0\tag{28}$$

Typically, *Q* is chosen to be diagonal:

$$Q = \begin{bmatrix} q\_1 & 0 & 0 \\ 0 & q\_2 & 0 \\ 0 & 0 & q\_3 \end{bmatrix} \tag{29}$$

where its elements and *R* can be selected by the following criteria,

$$q\_{\rm i} = \frac{1}{t\_{\rm si} \left(\chi\_{1\rm max}\right)^2}, \; R = \frac{1}{\left(u\_{1\rm max}\right)^2}, \; p > 0 \tag{30}$$

In (30), *ximax* is the |*xi*| constraint, *uimax* is the |*ui*| constraint, and *tsi* is the required settling time of *xi*.

#### **4. Simulation Results**

The simulation performed in MATLAB/Simulink and all related parameters of the considered PV system can be referred to in [28]. The results with the designed LQR control are illustrated in Figures 10–13, in which the time unit in the horizontal axis is second.

**Figure 10.** MPPT module performance for constant temperature: (**a**) PV voltage; (**b**) PV output power.

**Figure 11.** DC link voltage regulator module and current controller module performances with the LQR method for constant temperature case: (**a**) DC link voltage; (**b**) grid current magnitude; (**c**) grid voltage waveform *Vg* (V) and current waveform where Gain is 10 × *Ig* (*A*).

**Figure 13.** DC link voltage regulator module and current controller module performances with the LQR for constant solar irradiation case: (**a**) DC link voltage; (**b**) grid current magnitude; (**c**) grid voltage waveform *Vg* (V) and current waveform where gain is 10 × *Ig* (*A*).

#### *4.1. Simulation 1: Constant Module Temperature*

This case considers when the PV module temperature is constant at 25 ◦C. Irradiation starts from 850 W/m2 at a time from 0 s to 0.3 s, then it becomes 1000 W/m<sup>2</sup> from 0.3 s to 0.6 s and finally becomes 400 W/m<sup>2</sup> from 0.6 s to 0.9 s. Figure 10 shows that the results of the voltage *Vp* of the PV array are close to the reference values for the MPPT. The obtained output powers of the PV array are 847 W, 998 W, and 385 W, which match to the reference data provided in Table 1. Thus, this means that the power loss is small in this test.

The DC-link voltages correspond to each other in Figure 11. Furthermore, the grid current is equal to the reference values. Finally, it was shown and proven that the voltage and current of the grid are in phase, which means that the power factor of the gridconnected PV system is nearly unity.

#### *4.2. Simulation 2: Constant Solar Irradiation*

In the second case, solar irradiation is constant at 1000 W/m2, but the module temperature is varying. From *t* = 0 s to 0.3 s temperature is 25 ◦C, next, at 0.3 s the temperature is 45 ◦C, and lastly, at 0.3 s the temperature is 30 ◦C. According to Figure 12, the performance of the panel is 30.38 V/1000 W, 27.92 V/912.1 W, and 29.76 V/978.8 W, which is highly close to the values represented in Table 1. Despite the module temperature change, *VDC* matched its reference at all times. In addition, the RMS value of *Ig* is maintained according to the reference trend, as presented in Figure 13. The phases of the grid voltage and current match, which means that the system's power factor is unity.

#### **5. Comparison between LQR and Backstepping Approaches**

This research suggests the suitable combination of an LQR and fuzzy control for gridconnected PV systems. To show the effectiveness of the provided technique, it is important to make comparisons between some other methods, such as photovoltaic grid-connected systems using fuzzy logic and backstepping approaches [28] (see this reference paper for the specific details of simulations). Figures 14–17 present the simulation results of fuzzy control and the backstepping approach for a grid-connected photovoltaic system with the module temperature (Figures 14 and 15), and then with constant solar irradiation (Figures 16 and 17), in which the time unit in the horizontal axis is seconds. The obtained simulation results should be compared to those of the above-mentioned method. Specifically, the results in Figures 11 and 13 should be compared with those in Figures 15 and 17, respectively. We can see that both the control methods have good results.

**Figure 14.** MPPT module performance in case with the backstepping approach for constant temperature: (**a**) PV voltage; (**b**) PV output power.

**Figure 15.** DC link voltage regulator module and current controller module performances with backstepping approach for constant temperature case: (**a**) DC link voltage; (**b**) grid current magnitude; (**c**) grid voltage waveform *Vg* (V) and current waveform where gain is 10 × *Ig* (*A*).

**Figure 16.** MPPT module performance in case with the backstepping approach for constant solar irradiation: (**a**) PV voltage; (**b**) PV output power.

**Figure 17.** DC link voltage regulator module and current controller module performances with backstepping approach for constant solar irradiation case: (**a**) DC-link voltage; (**b**) grid current magnitude; (**c**) grid voltage waveform *Vg* (V) and current waveform where gain is 10 × *Ig* (*A*).

#### *5.1. Simulation 1: Constant Module Temperature*

It is obvious that the simulation results of the MPPT parts in both the cases are the same since the main changes were not related to the MPPT controller, but to the LQR. Thus, the behaviors shown in Figures 10 and 14 are same; the performances presented in Figures 12 and 16 are also similar. Comparing the DC link voltage regulator module and the current controller module in both the cases, it can be clearly seen that the LQR case reacts to the changes in the module temperature and irradiation faster; in other words, the settling time of the LQR technique is lower compared to the backstepping method. Moreover, the LQR is robust when faced with different temperature and irradiation changes, which makes this technique preferable. In addition, in the case of the backstepping method, overshooting of the signal was observed, which significantly degrades the output and overall efficiency of the considered PV system. Furthermore, the response speed of the designed LQR is faster; consequently the rise time and peak time of the LQR are lower than those of the backstepping approach.

#### *5.2. Simulation 2: Constant Solar Irradiation*

In addition, the fuzzy-based INC-MPPT controller (see Figures 3–7 and Table 4) in this paper is improved from our past method [33], with significant modifications in the pre-scaling module *Gp(k)* for increasing the sensitivity of the slope *Sp(k),* in the fuzzy association rules for boosting the speed of searching the MPP, and in the control parameter values, which are designed and tuned to be more suitable for grid-connected PV systems with the LCL output filter. The detailed analysis and evaluation of the effectiveness of the prior related MPPT method for a small stand-alone PV system in various simulations and experiments can be referred to in Sections 4 and 5 of our past work [33], in which cases of partial shadow on the PV array were also investigated in experiments. Moreover, the influence of the module temperature on the performance of the stand-alone PV system and MPPT controller can be found in Sections 2.2 and 4.3 of the prior study [33]. In fact, our present paper proposes a complete control solution for grid-connected single-phase PV systems including the LCL output filter based on fuzzy control and LQR techniques with multiple objectives as described in Sections 1 and 3 above; thus, it is noted that the separate assessment of the MPPT controller compared to other MPPT systems is out the scope of this study.

Due to substantial differences in configurations of considered PV systems and control objectives between this study and other existing works, it seems inappropriate and difficult to directly compare the detailed effectiveness of the proposed control scheme in this research to that of other studies. Therefore, we have only performed the detailed comparison of the current controller module using the LQR method proposed in the present paper with our past work using the backstepping approach [28] for evaluation, as described in Section 5 above. Moreover, for further reference, the current controller module using the LQR technique in this paper is briefly compared with the PID-fuzzy hybrid controller introduced in our other study [34], as represented in Table 6.


**Table 6.** Comparison between the proposed current controller module and the previously introduced PID-fuzzy controller in [34].

In this grid-connected PV system using a DC-AC inverter and LCL output filter, due to parasitic capacitance and grounding resistance, the issues of common mode (CM) voltage and leakage current may become significant if the design and control of the inverter and LCL filter are not appropriate [35,36]. As presented in other existing studies [37,38], the issues of CM voltage and leakage current in grid-connected PV systems can be effectively investigated and reduced using various techniques such as improved PWM methods, modified topologies of PV inverters with complementary switches [37], design and implementation of CM filters [38,39], active damping control approaches [40], and so forth. On the other hand, our paper focuses on developing a complete control scheme for gridconnected single-phase PV systems including an LCL filter based on fuzzy logic and an

LQR method with three key goals comprising the MPPT, DC link voltage regulation, and injection of the PV power into the grid with a unity PF. Hence, it should be noted that the assessment and reduction of the CM voltage and leakage current are beyond the scope of the present paper; these issues will be thoroughly considered in our future work.

#### **6. Conclusions**

Based on fuzzy control and an LQR, this study provides a comprehensive control solution for grid-connected single-phase PV systems. In terms of improving the quality of controller methods, this work represents a substantial extension and enhancement of our past research. For grid-connected solar systems, the suggested approach is an LQR suitably combined with fuzzy control, in which the design procedures of all the controllers are also described in detail. The major novelty of this study is to demonstrate and verify that the newly proposed approach is more successful in different aspects by comparing our past and present control methods. The LQR technique's major benefit is its ability to react quickly to unexpected changes in the system, such as changes in module temperature and solar irradiation. In other words, the systems achieve their settling period sooner, which is necessary to steady the output behavior.

Furthermore, as compared to the backstepping approach, the LQR method is more resistant to various changes in weather conditions. The backstepping approach also has the greater overrun, which has a detrimental effect on the efficiency of the investigated PV system. As previously stated, the LQR has the quicker response speed, making this type of controller more desirable. These are the major contributions of our present work as compared to the earlier research. Moreover, the results of simulations under different weather circumstances were compared to theoretical ones, indicating that the proposed system can cope with variations in weather parameters well. It was also demonstrated that abrupt changes in weather factors had no significant effects on the proposed control system's performance.

In future work, intelligent models based on fuzzy control for effectively predicting PV power and load demand will be thoroughly studied and implemented to improve the effectiveness and quality of grid-connected solar energy systems, especially under adverse conditions such shaded solar PV modules.

**Author Contributions:** A.M. worked on all tasks; T.D.D. and N.G.M.T. worked on the methodology and supervised. All the authors participated in writing, editing, and review. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by Nazarbayev University under the Faculty Development Competitive Research Grant Program (FDCRGP), Grant No. 11022021FD2924.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Kinga Skobiej \* and Jacek Pielecha**

Faculty of Civil and Transport Engineering, Poznan University of Technology, pl. M. Sklodowskiej-Curie 5, 60-965 Poznan, Poland; jacek.pielecha@put.poznan.pl

**\*** Correspondence: kinga.d.skobiej@doctorate.put.poznan.pl; Tel.: +48-61-665-2239

**Abstract:** Transportation, as one of the most growing industries, is problematic due to environmental pollution. A solution to reduce the environmental burden is stricter emission standards and homologation tests that correspond to the actual conditions of vehicle use. Another solution is the widespread introduction of hybrid vehicles—especially the plug-in type. Due to exhaust emission tests in RDE (real driving emissions) tests, it is possible to determine the real ecological aspects of these vehicles. The authors of this paper used RDE testing of the exhaust emissions of plug-in hybrid vehicles and on this basis evaluated various hybrid vehicles from an ecological point of view. An innovative solution proposed by the authors is to define classes of plug-in hybrid vehicles (classes from A to C) due to exhaust emissions. An innovative way is to determine the extreme results of exhaust gas emission within the range of acceptable scatter of the obtained results. By valuating vehicles, it will be possible in the future to determine the guidelines useful in designing more environmentally friendly power units in plug-in hybrid vehicles.

**Keywords:** exhaust emission; energy consumption; real driving emissions test

#### **Citation:** Skobiej, K.; Pielecha, J. Plug-in Hybrid Ecological Category in Real Driving Emissions. *Energies* **2021**, *14*, 2340. https://doi.org/ 10.3390/en14082340

Academic Editor: Constantine D. Rakopoulos

Received: 1 March 2021 Accepted: 17 April 2021 Published: 20 April 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

#### **1. Introduction**

The main problem of the ever-growing industry is its negative impact on the natural environment. One of the most dynamically changing sectors of industry is transport, which significantly contributes to air pollution. In order to reduce the impact of vehicles on the environment, increasingly stringent emission standards are introduced and solutions are sought that could allow minimization of the exhaust emissions from vehicles. One of the solutions proposed by the carmakers aiming at a global reduction of exhaust emissions is to replace as many conventional vehicles as possible with electric ones. Yet, due to the high cost of batteries, low vehicle range, and the lack of infrastructure, fully electric vehicles cannot fully replace the conventional ones. Manufacturers of conventional vehicles still struggle with the increasingly stringent exhaust emission standards.

The emission standards are set forth worldwide to control the pollutants emitted from vehicles to the atmosphere. The exhaust emissions are measured under the conditions of a predefined type of approval test. This part of the vehicle certification is responsible for its ecological properties and is the same for all passenger cars. The course of the test reflects the most probable road conditions and its performance, identical for all vehicles, entitles a comparison of the emission results among all the tested vehicles. These days, however, increased attention has been drawn to the performance of road tests, i.e., performed under actual traffic conditions. Currently, these tests are specified in the EU regulations as RDE (real driving emissions) [1–6]. They are carried out to most accurately reproduce the actual traffic conditions in the environmental aspect. Such tests must be performed in compliance with certain precisely defined requirements while at the same time allowing a relative arbitrariness, which significantly spreads the obtained exhaust emission results despite meeting the RDE requirements. The performed qualitative and quantitative analyses of the exhaust emissions in different tests was the subject of [7]. The authors proved that the values of relative road emissions depend more on the distance covered during the test than

203

the duration of the test. The exhaust road emission values determined in different tests depend mainly on the type of test and are greater in shorter tests compared to the RDE test. The analysis of the investigations has confirmed that it is possible to shorten the tests by approximately 20% without a significant change in the exhaust emission results. This is confirmation of the increasing importance being placed on road test results, which must be verified for not only static parameters, but also, to a large extent, for dynamic parameters.

The detailed requirements of the RDE tests and the possibilities of optimization of a combustion engine were investigated by Pielecha and Skobiej [8]. The analysis of the emission level of individual exhaust components allowed the authors to show that the exhaust emission level may be lower by 26% to 81% compared to the road test performed under actual traffic conditions. The performed tests indicate that already at the stage of design, the engines can be optimized in terms of their exhaust emissions. Roadside emission tests have also been shown to be the most reliable vehicle performance information vehicle. The literature presented in this paper intends to indicate the actions and research aimed at increasing the environmental aspects of hybrid vehicles. These are both solutions involving better management of the energy stored in the batteries and technical solutions affecting the development of this type of vehicle.

A partial solution to the problem of the limited range of electric vehicles as well as its dependence on the traffic conditions is the introduction of a plug-in hybrid [9,10]. A plug-in hybrid combines the advantages of hybrid and electric vehicles. Compared to typical hybrids, the batteries used in plug-in hybrid electric vehicles (PHEVs) have greater capacity and range and can be charged from an external power supply. Such vehicles are more economical in terms of fuel consumption and more ecological in terms of exhaust emissions. Due to the fact that approximately 30–50 km [11] can be covered using electrical energy, PHEVs are more ecological than the conventional ones.

Cieslik et al. [12] investigated an electric vehicle under varied vehicle operating conditions, particularly the influence of the weather on the energy consumption by the vehicle. Under actual traffic conditions in sub-zero temperatures, the energy consumption in such a vehicle is greater by 14% compared to warm weather. Similar investigations were carried out by Yi et al. [13], indicating that energy consumption in the electric vehicle (EV) may vary drastically depending on the driving conditions, which is extremely impactful on the vehicle range.

Li et al. in [14], observing the growing popularity of electric and plug-in hybrid vehicles, proposed a methodology of assessment of the energy distribution within a vehicle. Plug-in hybrids allow charging of the batteries while driving, which reduces the demand for energy from the external sources.

Pielecha et al. [15] compared vehicles of different powertrains: conventional, plug-in hybrid, and electric, in tests under actual traffic conditions. They showed that the plug-in hybrid consumes 20% less energy than the conventional vehicle.

As per the IEA, the Global EV Outlook 2020 reported [16] that since 2010 (last 10 years), the number of electric vehicles has increased and today their share in the market amounts to 2.5% (1 in 40 new vehicles is fitted with an electric powertrain), including 74% fully electric ones and 26% plug-in hybrids. The increase in the number of plug-in hybrid vehicles is presented in Figure 1, where the 2010–2019 phase is compared. In 2012, when the sales of plug-in hybrids were initiated, already 100,000 vehicles were located in the US. In 2017, the number of these vehicles exceeded 1 million, for which the greatest share was in Europe. The sales of plug-in hybrid vehicles reached a level of 2.4 million worldwide in 2019 and the highest number of these vehicles was recorded in Europe.

**Figure 1.** Increasing the number of plug-in hybrid vehicles between 2010 and 2019 globally (based on [16]).

Due to the fact that this type of vehicle is equipped with two sources of propulsion—a conventional internal combustion engine and an electric motor—it is necessary to optimally develop an energy management strategy and calibrate its parameters; the new system is designed to reduce emissions on the one hand and minimize energy consumption on the other.

#### **2. Aim of the Paper**

The addressing of the topic discussed in the paper is a result of social expectations, related to fair provision of vehicle emission-related information. Accusations were made that emission tests (particularly the type of approval tests) do not reflect the actual vehicle emissions, and, at the same time, the provided fuel economy information significantly diverges from the values obtained in daily vehicle operation. A similar topic, but in terms of a very broad scope of research, was addressed by the Initial Green Vehicle Index Roadmap project [17], where the assumption was to develop a basis for the comprehensive assessment of vehicles in terms of exhaust emissions, fuel consumption, and energy. The project, divided into phases between 2019 and 2030, assumed the study of exhaust emissions in type approval tests (NEDC, WLTC, and RDE) and, on this basis, the classification of vehicles on a 10-point scale of two indices: the clean air index and energy efficiency index. The clean air index compared on-road exhaust emissions with the permissible values—the lower the exhaust emissions, the higher the index value. The second index, the energy efficiency index, assessed the energy consumption of vehicles represented in kWh/100 km. Determination of the two indices and their respective values made it possible to assign a rating from 0 to 5 stars to the vehicle.

Therefore, the authors of this paper see the need for an objective approach to the assessment of the fuel consumption and exhaust emissions under actual operation of a vehicle. Such a particularization is necessary in the times of valuation of much data influencing, not only the vehicle operation, but also the choices of the end-users. A very good example is the valuation of the energy consumption of electric vehicles (though, conditioning of this parameter on external conditions should be required). The categorization of exhaust emission assessments conditional on the probability distribution of the results obtained in road tests is a novel approach. A novel method of determining the above will be the estimation of the extreme values of the exhaust emissions for a given vehicle within the admissible spreads of the road test results. Such a scenario, in the first stage, forced the application of a mathematical optimization apparatus in order to determine the minimum and the maximum values of the exhaust emissions and fuel consumption and, in the second stage, the need to practically validate the possibilities of obtaining such results

from vehicles fitted with given powertrains (plug-in hybrid vehicles). The realization of such a research plan resulted in the development of a new tool that may be used in the environment-related assessment of motor vehicles under actual traffic conditions. In relation to this, the discipline of transport would gain another solution, serving the purpose of protecting the natural environment. Currently, the certificate of type approval provides the final values related to the conformity of a vehicle to a given ecological category without energy- and emission-related valuations in the said category. Currently, sold vehicles in the emission category of Euro 6 are not classified in terms of adequacy of the actual emission and fuel consumption with the values stated in the certificate of type approval. The aim of the paper is the development of a procedure, according to which one could assess whether the road emission test performed on a plug-in hybrid is reliable and, at the same time, indicate an interval of probability of meeting the requirements for tests carried out in the actual traffic. Such a task remains in line with the optimization tasks, in which one needs to determine the extreme values and indexes, to which the obtained emission result is to be compared. In the paper, the authors introduced valuation of vehicles in road tests (classes from A to C), which facilitates the assessment of the vehicle emission category of a PHEV (not only in the type approval but also in road exhaust emission tests).

The effect of the paper will be the possibility of assessing the adequacy of the exhaust emissions under actual traffic conditions based on the vehicle emission category and determining the index (valuation), directly translating the results of a type approval test to the road tests. By evaluating plug-in hybrids in environmental tests, it will potentially be possible not only to assess their actual environmental impact, but also to establish guidelines that will contribute to the design of more environmentally friendly powertrains in the future. The research presented in this paper is an alternative approach to the issues of environmental assessment of vehicles, although limiting oneself only to the emission of exhaust compounds—in the case of PHEVs—is not a complete solution. The next challenge, which is a continuation of the research, will be to develop a methodology for evaluating the energy consumption of such vehicles. The consequence of these actions will be the determination of the ecological assessment of conventional vehicles (evaluation of exhaust emissions), hybrid vehicles (evaluation of exhaust emissions and energy consumption), and electric vehicles (only energy consumption).

This procedure is motivated by the following reasons:

	- Electrified vehicles (ZEV and PHEV) will have a significantly large share (about 90%) of the European market in 2030, of which more than 70% of such vehicles will be equipped with internal combustion engines [18].
	- The environmental impact of hybrid combustion vehicles powered, e.g., by hydrogen is significantly lower than that of cars powered only with electric motors [19].

4. Increase public awareness of the environmental aspects of vehicles, which have a local and direct impact on human health (exhaust emissions), as opposed to energy consumption, which can be produced in a very distant area or from renewable sources.

#### **3. Methodology**

#### *3.1. Requirements for the Tests under Actual Traffic Conditions*

From 2017 onwards, the approval process for a new type of passenger car in the European Union includes a procedure for measuring emissions under real traffic conditions. The European Union Regulation (715/2007/EC [1] and 692/2008 [2]) on RDE tests is a response to the results of tests concerning the increased emission of nitrogen oxides from cars equipped with compression ignition engines, despite the fact that such vehicles met the acceptable standards in laboratory conditions. According to the RDE rules (Package 1–4), for all new approvals from September 2020, the emission of nitrogen oxides measured under road conditions will not exceed 1.43 times (CF—conformity factors) the maximum limit (for Euro 6d-Temp is 60 mg/km), i.e., 86 mg/km (Table 1). The parameters of the road tests are not arbitrary, and the moving average windows method (MAV) (also referred to in the literature as EMROAD, developed by the JRC) is used to determine emissions.

**Table 1.** Requirements for RDE testing in Europe [3–6].


The route shall be chosen in such a way that the test is carried out without interruption, the data is recorded continuously, and the duration of the test is between 90 and 120 min. The electrical energy for the PEMS (portable emission measurement system) shall be supplied by an external power supply and not by a source that draws its energy directly or indirectly from the engine of the vehicle under test. The installation of the PEMS system shall be carried out in such a way that it affects vehicle emissions, performance, or both as little as possible. Care shall be taken to minimize the mass of the installed equipment and potential aerodynamic changes in the test vehicle. RDE tests shall be conducted on working days, and on paved roads and streets (e.g., off-road driving is not permitted). Prolonged idling after the first ignition of the combustion engine at the start of the emission test shall be avoided (Table 2).




#### *3.2. Research Equipment*

In order to measure the concentration of toxic compounds in the engine exhaust gas, mobile exhaust gas analyzers were used in stationary tests. The concentration of gaseous compounds was measured with the use of a Semtech DS analyser by Sensors. It enables measurement of the concentration of carbon monoxide, hydrocarbons, nitrogen oxides, and carbon dioxide, and, on the basis of oxygen concentration, the coefficient of excess air is determined.

The main purpose of the Semtech DS analyser is to measure the concentration of gaseous compounds from automotive vehicles. In this version, it can be used to test engines powered by different fuels, whose composition should be taken into account in the final data treatment (post processing). It is a representative of a group of PEMS-type measuring devices. It therefore meets the ISO 1065 standard for testing exhaust emissions with mobile systems. In addition to the possibility of using the analyser for in-service vehicle tests, it can be used as a measuring device for stationary tests, e.g., on an engine dynamometer. The Semtech DS analyser consists of the following measurement modules:


The Semtech DS analyser, in cooperation with a suitable flow meter, enables measurement of the exhaust mass flow rate. An important aspect is the appropriate thermal condition of the equipment, which is necessary to ensure stable indications. The time

needed to obtain the appropriate temperature of the analyzer is 60 min. The measurement starts with the introduction of the exhaust sample into the analyser through a measuring probe that maintains a temperature of 191 ◦C. The exhaust gas sample is then filtered from the particulate matter. The filtered sample is then subjected to a hydrocarbon concentration measurement. The next step is to cool the exhaust sample to 4◦C and start measuring the concentration of nitrogen oxides, carbon monoxide, carbon dioxide, and oxygen. Table 3 presents the values of measurement uncertainty and the measurement range of particular modules of the Semtech DS analyser.

**Table 3.** Uncertainty of indications of individual measurement modules of the Semtech DS analyzer.


For the measurement of particle diameters, a TSI Incorporated analyser—EEPS 3090 (Engine Exhaust Particle Sizer™ Spectrometer) was used. It allowed measurement of the discrete range of particle diameters (from 5.6 nm to 560 nm) based on their different electrical mobility. Exhaust fumes are directed to the device through a dilution system and a system that maintains the required temperature. A pre-filter stops particles larger than 1 μm in diameter that are outside the measuring range of the device. After passing through the neutralizer, the particles receive a positive electrical charge depending on their diameter. The particles deflected by the high-voltage inner electrode enter the ring slot. In the space between the inner electrode (having a positive electrical charge) and the outer cylinder (built as a stack of isolated electrodes arranged in rings), the electric charge of the collected particles (on the outer electrodes) is read by the processing system. Technical data of the TSI 3090 EEPS analyser are presented in Table 4.

**Table 4.** Spectrometer characteristics of EEPS TSI 3090.


#### *3.3. Characteristics of the Research Objects*

The characteristic features of the research objects are presented in Table 5. All vehicles were plug-in hybrids. They varied in terms of the displacement of the fitted combustion engine, maximum power output, and torque. The vehicles were selected so as to most efficiently diversify and compare vehicles of different environmental performances. Vehicle A is characterized by the highest power output of the combustion engine and the electric motor and the highest battery capacity (13.6 kW·h). Vehicle B is distinguished by a continuously variable transmission, the lowest curb weight, and an average battery capacity (8.8 kW·h). Vehicle C is characterized by the highest engine capacity but has the lowest battery capacity (3.3 kW·h). Depending on the battery capacity, the vehicles were referred to as large battery (vehicle A), medium battery (vehicle B), and small battery (Vehicle C).


**Table 5.** Technical parameters of plug-in vehicles.

#### *3.4. Adopted Method to Search for the Function Minimum*

In search for the lowest road (specific) emission in the RDE test, a task was applied consisting in seeking the lowest value of the function of many variables while fulfilling the imposed conditions. Such a task was formulated in the form:

$$\min \mathbf{f}(\mathbf{x}) \tag{1}$$

fulfilling the limitations:

$$\mathbf{h}(\mathbf{x}) = 0,\tag{2}$$

$$\mathbf{g}(\mathbf{x}) \le 0\tag{3}$$

where: **f**—objective function of the optimization task, **h**(**x**)—equality limitations vector, and **g**(**x**)—inequality limitation vector.

The above-presented general form of the objective function and the function of limitations requires an introduction of the modification of the objective function and limitations to enable an application of the general reduced gradient method. The general reduced gradient method belongs to a group of methods searching for the function minimums of many variables without limitations. The equality and inequality limitations can be considered by including them in the objective function:

$$\mathbf{K(x)} = f(\mathbf{x}) \sum\_{i} \mathbf{h}\_{i}^{2}(\mathbf{x}) + 1/(2\mu) \sum\_{i} \mathbf{W(g)} + 1/(2\mu),\tag{4}$$

where: W(**g**) = g(**x**)−g(min) for g ≤ <sup>g</sup>(min) and <sup>μ</sup>—par parameter modified in the optimization process.

The generalized reduced gradient (GNG) algorithm implemented in the Excel add-in Solver was used to perform the analysis. The principle of operation of the GNG algorithm in a shortened version is presented on the basis by Lasdon et al. [20]. Such an algorithm works very well for solving nonlinear problems; however, it is sensitive to the choice of initial data.

#### **4. Results**

#### *4.1. Validation of the Tests for Compliance with the Requirements*

In order to be able to compare the exhaust emissions in the first place, the nature of the test drives had to be compared. The performed tests were validated for their compliance with the RDE procedure, which requires three phases: urban, rural, and motorway. First, a formal check of the test drives was performed, and the detailed data are shown in Table A1 in Appendix A. All tests were repeated 5 times; tests in which extreme values were reached were rejected. For the purposes of this paper, the test that did not deviate by more than 10% from the average value in terms of the emissions of each exhaust constituent was assumed to be representative.

Despite meeting all the formal requirements for the tests, the most vital parameters were also compared that could have impact on the different emission results of the investigated vehicles. The course of the test route clearly indicates the three phases of the test (Figure 2a). We can distinguish the urban phase 20–25 km, the rural phase—25 km to 50 km, and the motorway phase—60 to 90 km. All three phases were rather similar in terms of their average speeds (Figure 2b). The spread of the average speed for the urban phase did not exceed 1 km/h; for the rural phase, 2 km/h; and for the motorway phase (the greatest), 5 km/h. The average value of the speed in the entire test was similar in each drive and amounted to 52, 56, and 54 km/h for vehicles A, B, and C, respectively.

**Figure 2.** The tracings of the speed in the tests performed on the plug-in hybrids of different battery capacities (**a**) and average values of the speed in each phase of the test and in the entire RDE test (**b**).

In a second step, the authors compared the contribution of each phase of the RDE test for individual vehicles. The formal requirement is that the urban phase fills from 29% to 44%, the rural phase 23% to 43%, and the motorway phase 23% to 43% of the entire RDE test. The obtained results for the urban phase were 36.7%, 33.1%, and 34.1% for vehicles A, B, and C, respectively (Figure 3). This means that the data variation described with the coefficient of variation CoV (the ratio of the standard deviation to the mean) is 1.5%. This is a very small value and indicates a very high probability of the obtained results. For the rural phase of the test, the obtained results are 27.5%, 31.6%, and 32.7%, which gives a CoV = 7.4%. This value is several times higher than that from the urban phase. This phase of the test is characterized by a greater variability of vehicle speed and a greater share of unpredictable conditions. For the motorway phase, the shares in the entire test were 35.5%, 35.3%, and 33.2%, which gives a value of CoV = 3.3%. Such a comparison clearly confirms the similarity of the phase shares in the RDE test. It is noteworthy that, despite the allowable high variability in phase shares, the authors tried to keep the test drives very similar throughout the study, so that the emission results were only affected by traffic parameters (which the authors had no control over).

**Figure 3.** Comparison of the share of individual phases of the RDE test for the plug-in hybrids with a large (**a**), medium (**b**), and small (**c**) battery capacity.

The third and last stage of the comparison was determining the parameters characterizing the dynamic states of the vehicle operation. The first parameter was the 95% centile of the product of speed and positive acceleration (Figure 4a), which, in each phase of the test, should be lower than the predefined maximum. The requirement is to make sure that the test drive is not excessively dynamic, and the vehicle accelerations do not drastically increase the dynamics, resulting in increased exhaust emissions. The values of this parameter for the urban part, shown in Figure 4a, are in the range 9–12 m2/s3 and are similar for all test drives. For the rural phase, the obtained values of this parameter in individual drives have a greater spread (14 m2/s3 to 18 m2/s3). For the motorway phase, these values are the most similar and amount to approximately 14 m2/s3. It is noteworthy that in each phase of the test, the admissible value is not exceeded (marked with a continuous line). This suggests the correctness of the test realization in terms of driving dynamics. At the same time, it indicates a significant similarity, which will constitute a basis for the comparison of the exhaust emissions from the investigated vehicles.

**Figure 4.** Comparison of the 95% centile of the product of vehicle speed and positive acceleration (**a**) and the relative positive acceleration (**b**) in each phase of the test performed on the plug-in hybrids.

The other parameter describing the dynamic conditions was relative positive acceleration (Figure 4b). This parameter described the minimum dynamic conditions for the test drive not to be a steady state and to eliminate driving with the use of cruise control. Unfortunately, this parameter is determined on such a level that the performed tests are characterized by only slightly higher values than the minimum ones. For the urban part of

the test, for all the vehicles, the relative positive acceleration fell in the range 0.14 m/s2 to 0.16 m/s2 and was approximately 10% greater than the minimum (13 m/s2) defined for the obtained average speed in this phase (approximately 30 km/h). In the rural phase of the test, the values were even closer to one another, and fell in the range of approximately 0.6–0.7 m/s<sup>2</sup> and were greater than the minimum by approximately 5% (for the average speed of 75 km/h). In the motorway phase of the test, the relative positive acceleration was the lowest. It amounted to 0.3 ± 0.04 m/s2 and was slightly higher than the minimum defined on the level of 0.25 m/s2.

The presented characteristics of the tests as well as the steady state and dynamic properties of the test drives prove that the formal requirements (presented in a concise form) for each of the test drives and each phase of the test were met. Such a situation implies the possibility of moving on to the next stage of the investigations, consisting in determining the exhaust emission values. Similar test conditions and similar dynamic parameters obtained for all the vehicles tested provide grounds for concluding that the exhaust gas emission results are not burdened with inaccuracy resulting from differences in the test course. At the same time, the similarity of the test runs indicate differences resulting only from individual vehicle characteristics, such as the engine used or battery capacity.

#### *4.2. Exhaust Emission Results*

Due to the fact that the engine was not in the initial phase of the RDE test, the characteristic aspect of the emission of individual exhaust components is the initial flat period of the relation. It results from the fact that the initial phase of the RDE test is performed only with the electric motor activated and the cold phase of the test is moved from the urban phase to the rural one (Figure 5a). This is particularly visible when we analyze the emission of carbon dioxide (shown in increments), from which we conclude that the vehicle of the lowest battery capacity drove approximately 40 km on the electric motor. The other vehicles used the electric motor for the distance of over 50 km. The advantage of this situation is that the engine warm-up time is reduced. This is due to the fact that there is a higher engine speed and load during the rural phase of the test. This results in much faster catalytic converter firing and a reduced cold start time. The downside, however, is the fact that the cold engine operation at higher loads and speeds results in higher emissions of individual exhaust components. This is particularly visible in the analysis of the emission of carbon monoxide (Figure 5b), where the start of a cold engine (vehicle A) in a very short time resulted in emission of almost 50% of the entire emission of this component (approximately 30 mg/km for the distance of approximately 50 km and for the entire test this value was approximately 60 mg/km for the distance of approximately 100 km).

A separate comment is required related to the nature of the changes in the emission of carbon monoxide for the medium-battery plug-in hybrid, for which this emission drops as the test continues (from 70 km onwards). This is caused by the lowest increment of the mass of carbon monoxide compared to the distance covered by the vehicle. In the outstanding two cases, the increase in the mass of carbon monoxide was greater than the increment in the covered distance. For each investigated vehicle, the emission of carbon monoxide was lower than the admissible one (1000 mg/km Euro 6d-Temp). For individual vehicles, the obtained values of the emission of this component, defined as the total mass of the emitted component against the total covered distance, are: 63, 74, and 109 mg/km, for vehicle C, B, and A, respectively.

**Figure 5.** Dependence of the emission of carbon dioxide (**a**), carbon monoxide (**b**), nitrogen oxides (**c**), and particle number (**d**) on the distance covered during the road tests for individual investigated vehicles; the curves were made using the emission intensity results obtained in each second of the test.

The nature of the changes in the emission of nitrogen oxides (Figure 5c) was heavily dependent on the engine displacement and the vehicle weight. Vehicle C (of the highest curb weight and engine displacement) was characterized by the highest emission of nitrogen oxides and, at the same time, had a constant growing trend in the rural and motorway cycles. The final value of the emission of nitrogen oxides obtained using all the data related to the emission intensity in the entire RDE test was 9.5 mg/km, which is an approximately 50% higher value compared to vehicle B (6.4 mg/km) and more than 3 times higher than the results obtained for vehicle A (approximately 3 mg/km). In the final described case, vehicle A had the engine of the lowest displacement, but the engine was turbocharged. The catalytic converter significantly reduces the concentration of nitrogen oxides in the exhaust system, which is particularly visible in modern turbocharged engines. The obtained values of the emission of nitrogen oxides do not exceed the limits prescribed in the Euro 6d-Temp standard, which amounts to 60 mg/km.

A different tracing from the previous one was obtained for particle number (Figure 5d). The smallest increase was observed for vehicle C, for which the greatest emission of nitrogen oxides was recorded, which confirms the inversely proportional relation between these exhaust components. When comparing the obtained results of all the investigated vehicles, we know that the particle number is lower than the prescribed limit, which amounts to 6·10<sup>11</sup> 1/km for the direct-injected engines. Determining the total exhaust emissions in the tests requires an application of an averaging algorithm of the emissions in the measurement windows that are determined based on the emission of carbon dioxide in individual phases of the WLTC test (red dots in Figure 5).

The value of the road emission of carbon dioxide in the RDE test, determined individually for each of the vehicles, in the majority of cases falls between the lower (Tollow = 0) and the upper (Tolhigh = +45%) limit of tolerance. In all the investigations of the plug-in hybrids, one can see a similarity of the distribution of the average emission of carbon dioxide in the measurement windows (Figure 6): for the speed in the range up to 75 km/h, the observed values do not exceed 100 g/km. In the range from 75 km/h to 125 km/h, the emission values of carbon dioxide are from 100 g/km to 200 g/km for vehicle A (Figure 6a); for vehicle B, up to 220 g/km (Figure 6b); and for vehicle C, up to 150 g/km (Figure 6c).

**Figure 6.** Characteristic curves determining the relation between the emissions of carbon dioxide in the measurement windows that are the basis for the determination of the emission of individual exhaust components in the RDE test for: Plug-in, Large Battery (**a**), Plug-in, Medium Battery (**b**), Plug-in, Small Battery (**c**).

Analyzing the carbon dioxide emissions for the different phases of the RDE test reveals zero emissions of this component in the urban phase for all plug-in hybrid vehicles tested (Figure 7a). The differences are visible only in the subsequent phases of the test: in the rural phase, the greatest emission occurs for the vehicle with the smallest battery capacity (49 g/km) and the lowest emission occurs for vehicle B with the medium battery capacity (32 g/km). The greatest values of the emission of carbon dioxide were recorded in the motorway phase of the test: 3–4 times higher compared to the rural phase. The greatest emission of carbon dioxide (159 g/km) is for the vehicle with the lowest engine displacement, which suggests the load of the powertrain. At the same time, it is confirmation of the fact that downsized engines fitted in vehicles of relatively high curb weight do not ensure the expected environmental results. The final result of the road emission of carbon dioxide for all the vehicles is 63 ± 2 g/km, which means that none of the values differed from one another by more than 3.2%. This is a very similar result confirmed by the gas mileage of 2.6, 2.8, and 3.0 dm3/100 km for vehicles A, B, and C, respectively. This is caused by the fact that the electric mode was used in each phase of the test by each vehicle and its share was 60%, 53%, and 48% for vehicles A, B, and C, respectively.

**Figure 7.** Road emission of carbon dioxide (**a**), carbon monoxide (**b**), nitrogen oxides (**c**), and particle number (**d**) for each phase of the test and in the entire RDE test for individual plug-in hybrid vehicles; the values were determined using an algorithm allowing for averaging in the measurement windows.

Much greater differences in individual phases of the test were recorded for the emission of carbon monoxide (Figure 7b). Similar to the previous case, the urban phase was characterized with zero emission of this component and in the rural phase, the differences were significant and amounted to 33% (for vehicle B, the emission of this component was 99 mg/km and for vehicle C, 67 mg/km). Vehicle A reached an intermediate value of 74 mg/km. An even greater divergence was recorded in the motorway phase of the test, where, similar to carbon dioxide, the highest values (259 mg/km) were recorded for vehicle A (downsizing) and the smallest (93 mg/km) for vehicle C (the greatest displacement). This translated into the final values that, for vehicle A (110 mg/km), were twice as high compared to vehicle C (53 mg/km). It is noteworthy that these values are 10–20 times lower than the admissible ones prescribed in the Euro 6d-Temp standard, where the limit is 1000 mg/km.

The road emission of nitrogen oxides was dependent on the engine displacement: the greater the engine displacement, the higher the emission of this component in individual phases of the RDE test (Figure 7c). The road emission of nitrogen oxides in the rural phase for all the tested vehicles was the result of the engine cold start in this phase. The recorded values of the road emission in the rural phase were 2.9 g/km, 4.9 mg/km, and 5.8 mg/km for vehicles A, B, and C, respectively. Vehicle C generated twice the mass of nitrogen oxides compared to vehicle A (assuming the same distance in the rural phase of the test). The values of this parameter in the motorway phase were even more varied: vehicle C

generated 2.5 times the mass of nitrogen oxides compared to vehicle A, which could have been caused by the mileage of this vehicle in the first place (MY 2019) and the reduced efficiency of the catalytic converter. The consequence is the final result of the emission of nitrogen oxides for vehicle A of 3.3 mg/km and vehicle C of twice as much (8 mg/km). These values are more than 10 times lower that the ones prescribed in the RDE standard: 60 mg/km (bNOx Euro 6d-Temp) × 1.43 (CFNOx) = 86 mg/km. The road emission of the particle number was quite the opposite compared to nitrogen oxides. The highest values in each phase of the test and in the entire RDE test were recorded for vehicle B (Figure 7d). These values were approximately 10–20% higher compared to vehicle A, whose values were 0, 4.4·10<sup>10</sup> 1/km, 1.7·1011 1/km, and 6.9·1010 1/km in the urban, rural, motorway phase, and the entire test, respectively. The lowest emission of the particle number was vehicle C, for which the value in the entire RDE test was 3.1·10<sup>10</sup> 1/km. All the obtained values were several times lower than the admissible limit of this component prescribed in the RDE test requirements, amounting to 1.5 times the limit of the Euro 6d-Temp standard (6·10<sup>10</sup> 1/km × 1.5 = 9·10<sup>11</sup> 1/km).

The above-presented procedure of determination of exhaust emissions under road conditions served to assess the environmental performance of three different plug-in hybrid vehicles. On this basis, the authors identified the exhaust emissions falling in the admissible limits, i.e., quantitative information was obtained. The qualitative information, however, was not assured, i.e., the result was not obtained against the environmental capabilities of a given plug-in hybrid vehicle model. The obtained results do not contain information on the scope of variability in the position of the results against the minimum and maximum obtainable values for each vehicle.

#### **5. Discussion**

The obtained on-road emission values from each vehicle were used as input parameters to determine the limits (where the values are possible). Specifying a maximum value is not questionable because it should be a limit:

$$\mathbf{b}\_{\text{j.max}} = \mathbf{b}\_{\text{j.Euro6d-Temp}} \times \mathbf{CF}\_{\text{j}} \tag{5}$$

where: bj,Euro6d-Temp denotes the admissible value of the road emission for the j-th exhaust component (bCO,Euro 6d-Temp = 1000 mg/km, bNOx,Euro 6d-Temp = 60 mg/km, bPN,Euro 6d-Temp = 6·10<sup>11</sup> 1/km and CFj—conformity factor for the j-th exhaust component (CFCO = 2.1, CFNOx = 1.43, CFPN = 1.5).

The minimum values that vehicles can theoretically obtain were determined with constant and variable emission rates. Constant emissivity should be assumed when changes in emissivity do not appear in individual phases of the RDE test. This means that a significantly small standard deviation occurs from the average value described with the coefficient CoV < 10%. The exact values of CoV for the emission intensity of all the exhaust components from all the investigated vehicles are presented in Table 6.

**Table 6.** CoV coefficient for all the tested exhaust components.


It can be seen from Table 6 that all CoV values are greater than 10%, so the emission intensity (or road emission value) must depend on a mean value that accurately describes the nature of changes in the said emission intensity (or road emission). In this paper, the minimum value of the exhaust emissions with both methods was determined irrespective of the above.

#### *5.1. Determining the Minimum Road Emissions*

#### 5.1.1. Constant Road Emission Intensity

For the determination of the theoretical values of the minimum exhaust emissions, the authors used a method described in Section 3.4 of this paper. As the input values, the authors used the emission intensities of a given exhaust component Ej,k, determined from the road emissions bj,k; j = CO2, CO, NOx, PN, for the rural and motorway phases). In the urban phase, the emission intensity was 0. The variable values of the algorithm were: urban, rural, and motorway test duration (tk) and the distance in the urban, rural, and motorway phases (Sk). The limitations were:


The initial values were: tU = tR = tM = 30 min and SU = SR = SM = 16 km.

The values of the emission intensity of individual exhaust components (Table 7) were obtained based on the average road emission, the time, and the distance covered in each test phase. The objective function had a form:

$$\mathbf{b}\_{\text{j,RDE}} = 0.34 \,\mathbf{b}\_{\text{j,U}} + 0.33 \,\mathbf{b}\_{\text{j,R}} + 0.33 \,\mathbf{b}\_{\text{j,M}} \tag{6}$$

and upon including the constant emission intensity Ej,k in each phase of the test:

$$\mathbf{b}\_{\mathrm{j},\mathrm{RDE}} = 0.34 \,\mathrm{E}\_{\mathrm{j},\mathrm{U}} \,\mathrm{t\_{U}/S\_{\mathrm{U}}} + 0.33 \,\mathrm{E}\_{\mathrm{j},\mathrm{R}} \,\mathrm{t\_{R}/S\_{\mathrm{R}}} + 0.33 \,\mathrm{E}\_{\mathrm{j},\mathrm{M}} \,\mathrm{t\_{M}/S\_{\mathrm{M}}}.\tag{7}$$

**Table 7.** The value of the constant emission intensity for the phases of the RDE test (rural, motorway) as the algorithm input data.


Using the Solver tool (Excel MS OfficeTM), for each of the vehicles, the duration of each of the test phases (tk), the distance covered in each of the test phases (Sk), and the share of each test phase in the entire RDE test (uk) were determined. Detailed data are presented in Tables A2–A4 (Appendix A), for vehicles A–C, respectively. From the comparison of the data in the tables, the theoretical minimum value of the exhaust emissions is greater than zero, which means that the plug-in hybrid vehicle in the RDE test will always use a combustion engine. For vehicle A, the minimum value of the road emission of carbon dioxide was 49 g/km; for carbon monoxide, 83 mg/km; for nitrogen oxides, 2.5 mg/km; and for the particle number, 5.2·10<sup>10</sup> 1/km. It should be noted that the obtained values of the parameters of time (tk), test phase duration (Sk), and test phase share (uk) were different for each exhaust component and the scatter of results in individual analyzed categories, as measured with the CoV coefficient, fell in the range from 1.1% to 5.7% (Table A2). For vehicle B, the following road emission values were obtained using the same pattern: bCO2 = 48 g/km, bCO = 65 mg/km, bNOx = 4.7 mg/km, a bPN = 6.6·1010 1/km, at the coefficient of variation CoV changing from 0.9% to 20% (Table A3). For vehicle C, the values of individual parameters were as follows: bCO2 = 49 g/km, bCO = 41 mg/km, bNOx = 6.2

mg/km, a bPN = 2.4·10<sup>10</sup> 1/km, at the coefficient of variation CoV changing from 0.0% do 12.4% (Table A4).

The obtained average values for individual vehicles and each exhaust component are shown in Figure 8. From this figure, the road emission of carbon dioxide for each vehicle falls in the range 60–80 g/km (Figure 7a) at the minimum value of approximately 49 g/km (Figure 8a). The value was adopted obligatorily on the level of 95 g/km (this is a target value for a manufacturer's vehicle fleet; however, this value may decrease in future years). The road emission of carbon monoxide for all the investigated vehicles was in the range 50–110 mg/km (Figure 7b) at the maximum value of 2100 mg/km and the minimum one of 40–80 mg/km (Figure 8a). The road emission of nitrogen oxides fell in the range 3.3 mg/km–8.0 mg/km (Figure 7c) at the maximum value of 86 mg/km, determined according to Equation (5).

**Figure 8.** Values of the minimum road emission intensity of carbon dioxide (**a**), carbon monoxide (**b**), nitrogen oxides (**c**), and particle number (**d**) in the RDE test obtained according to the algorithm for a constant emission intensity; data provided in Tables A2–A4.

The theoretical minimum value of the road emission of nitrogen oxides was in the range from 2.5 mg/km to 4.7 mg/km (Figure 8c). The last investigated exhaust component, the road emission of particle number, was in the range from 3.1·10<sup>10</sup> 1/km to 8.3·10<sup>10</sup> 1/km (Figure 7d). For this parameter, the maximum value determined in Equation (5) is 9·10<sup>11</sup> 1/km and the minimum determined value falls in the range from 2.4·10<sup>10</sup> 1/km to 6.6·10<sup>10</sup> 1/km (Figure 8d).

#### 5.1.2. Variable Road Emission Intensity

The determination of the minimum road emission using constant emission intensity of a given exhaust component does have its flaws: the engine warm up and increased catalyst efficiency after light-off are not allowed for. A more generalized approach may also be the use of the mathematical description of the curves presented in Figure 5, in relation to which the general form of Equation (2) assumes a form where individual values of the road emission for each exhaust component and each test phase will be dependent on the distance covered by the vehicle:

$$\mathbf{b}\_{\text{j},\text{RDE}} = 0.34 \text{ b}\_{\text{j},\text{U}}(\text{S}) + 0.33 \text{ b}\_{\text{j},\text{R}}(\text{S}) + 0.33 \text{ b}\_{\text{j},\text{M}}(\text{S}).\tag{8}$$

In this case, we need to apply the non-continuous function for each exhaust component that will allow for the operation of the electric motor (in the range Sk ≤ SEV) during the urban and (partially) rural phases. For the outstanding distance, each course of the road emissions was described with a square Equation (9):

$$\mathbf{b}\_{\mathbf{j},\mathbf{k}}(\mathbf{S}\_{\mathbf{k}}) = \begin{cases} 0 & \text{for } \mathbf{S}\_{\mathbf{k}} \le \mathbf{S}\_{\text{EV}}\\ \mathbf{x}\_{\mathbf{j},\mathbf{k}}(\mathbf{S}\_{\mathbf{k}})^2 + \mathbf{y}\_{\mathbf{j},\mathbf{k}}\mathbf{S}\_{\mathbf{k}} + \mathbf{z}\_{\mathbf{j},\mathbf{k}} & \text{for } \mathbf{S}\_{\mathbf{k}} > \mathbf{S}\_{\text{EV}} \end{cases} \tag{9}$$

where: j = CO2, CO, NOx, PN; S—distance [km]; SEV—distance covered by the vehicle using an electric motor [km]; k = Vehicle A, Vehicle B, Vehicle C; and xj,k, yj,k, zj,k—multinomial coefficients (Table 8).

**Table 8.** Values of the equation indexes of the z, y, z multinomial and the coefficient of determination (R2) for the road emission in the RDE test for each plug-in hybrid vehicle.


The values of the multinomial coefficients (xj,k, yj,k, zj,k) are presented in Table 8 and their analysis indicates an increase in the road emissions upon exceeding the distance (Sk > SEV), which is indicated by the negative value of each coefficient xj,k.

The positive value of coefficient yj,k indicates a shift of the course of the emissions to the right for the increasing distance from the start of the test and the negative value of coefficient zj,k confirms the assumptions of zero emission of each exhaust component during the operation of the electric motor. In Table 8, the authors also provide the coefficient of determination (R2), which indicates a very good fit of the adopted equation to the curves, showing the road emission of each exhaust component in the RDE test.

Utilizing Equation (9) and the data contained in Table 8, the minimum road emission of each of the exhaust components was determined in individual phases of the RDE test and in the entire test. The obtained detailed results regarding the duration of each phase of the test, the distance in each phase of the tests, and the share of each phase in the test are shown in Tables A5–A7 (Appendix A). The graphical presentation of the final relations is shown in Figure 9. From the figure, it is found that the minimum value of carbon dioxide for vehicles A and B is 0 g/km, which leads to the conclusion that the vehicle can cover the entire RDE test distance using an electric motor exclusively (which is compliant with the test detailed requirements). This is possible because the range of vehicles A and B on an electric motor was approximately 52 km and the minimum distance of the RDE test is 48 km. Due to the fact that the combustion engine was off, the emission of the outstanding exhaust components from vehicles A and B was also zero. Only in the case of vehicle C with the small battery capacity was it impossible to carry out the entire RDE test exclusively

using the electric motor. For this particular vehicle, the theoretical minimum values of the road emission are greater than zero for each of the exhaust components. However, when comparing the results obtained for constant and variable emission intensity, one can observe lower values for the latter method. It is probable that the theoretical minimum values of the road emission obtained using the variable emission intensity method during the road tests are closer to reality, which is why the authors recommend them for application in further research.

**Figure 9.** Value of the minimum road emission of carbon dioxide (**a**), carbon monoxide (**b**), nitrogen oxides (**c**), and particle number (**d**) obtained according to the algorithm for variable exhaust emission intensity; data provided in Tables A5–A7.

#### *5.2. Plug-in Vehicles Emission Category*

The environmental assessment of the vehicles was initiated upon the analysis of the obtained results of actual road emissions of individual exhaust components (p. 4) and upon adopting the maximum values (values adopted based on the emission standards—p. 5.1.1) and the minimum ones (values adopted based on the theoretical determination of the minimum for the measured emission intensity—p. 5.1.2). The environmental categorization (EC—ecological category) for each exhaust component was performed based on the determination of the percentage value of the obtained road emission depending on the minimum and maximum value according to the formula:

$$\rm{EC\_j} \, [\%] = 100\% \times (b\_{\hat{\jmath}} - b\_{\hat{\jmath}, \min}) / (b\_{\hat{\jmath}, \max} - b\_{\hat{\jmath}, \min}) . \tag{10}$$

Equation (10) describes the process of scaling that adapts the value of any exhaust component to the new limits determined with the minimum (0%) and the maximum value (100%). Such an approach enables each road emission to be presented as a value ranging from 0–100%, as shown in Table 9. According to the results presented below, one can confirm that the highest values (EC = 64–68%) are gained for the road emission of carbon dioxide, which indicates great potential for improvement in this matter. In the case of the emission category pertaining to the road emission of carbon monoxide, the determined values are in the range from 4% to 11%. This confirms a substantial reserve of approximately 90%, which is tantamount with the fact that the analyzed vehicles generate much less of this exhaust component compared to the emission standard prescribed for the RDE test. We have a similar situation for the road emission of nitrogen oxides and particle number: the obtained values from 4–7% also confirm the rule.


**Table 9.** Real values of the road exhaust emissions (bj), theoretically determined maximum value (bj, max) minimum value (bj, min), and the environmental assessment (ECj) for individual plug-in hybrid vehicles.

The values of emissions of individual exhaust components for individual vehicles were used for the overall environmental assessment of the analyzed vehicles. An arithmetic average relation was applied, yet the authors are aware that a valuation can be introduced in terms of the significance of each of the exhaust components. Such an action would require considering the hazard of individual exhaust components to human health and the criteria of its assessment would require further analyses. Therefore, the application of an arithmetic average allows all the discussed exhaust components to be treated equally. The obtained values, shown in Figure 10, show that all the vehicles under analysis obtain ecological values lower than 50%, which classifies them to the ecological category A.

**Figure 10.** Value of the minimum road emission in the RDE test (-—vehicle A, -—vehicle B, - vehicle C) obtained according to the algorithm for variable exhaust emission intensity; data provided in Tables A4–A6.

An explanation is required for the case in which the road emission of an exhaust component is greater than the product of the CF coefficient and the value provided in the emission standard. In such a case, the value of CF will be greater than 100%, which, for conventional vehicles, will result in the necessity to extend the range of qualification and emission categories.

#### **6. Conclusions**

In the paper, the authors presented the process of creation of a new tool that can be used in the environmental assessment of motor vehicles under actual traffic conditions. Currently, the certificate of type approval provides the final results as regards the compliance with the vehicle ecological category, yet without the emission-related valuation of the vehicle. The methodology presented in the paper and applied when investigating plug-in hybrid vehicles unveils its practical application, which is confirmation that the analyzed vehicles adapt well environmentally when it comes to the trends in electromobility, given their emission category (A). The division into emission categories of plug-in hybrid vehicles can be introduced in a different subjective way, e.g., by creating more categories from A to E.

The major conclusions of the paper can be summarized in the following points:


**Author Contributions:** Conceptualization, K.S. and J.P.; methodology, K.S. and J.P.; software, K.S.; validation, K.S. and J.P.; formal analysis, K.S; investigation, K.S. and J.P.; resources, K.S. and J.P.; data curation, K.S. and J.P.; writing—original draft preparation, K.S.; writing—review and editing, K.S. and J.P.; visualization, K.S. and J.P.; supervision, K.S. and J.P.; project administration, J.P.; funding acquisition, K.S. and J.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Poznan University of Technology, grant number 0415/SBAD/0319.

**Institutional Review Board Statement:** Not Applicable.

**Informed Consent Statement:** Not Applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**


#### **Appendix A**

**Table A1.** Characteristic data of the performed tests and their comparison with the admissible values.



**Table A1.** *Cont*.

**Table A2.** Road emissions for the RDE test (assuming constant exhaust emission intensity)—output data obtained from the algorithm and the proposed values for the generalized test (Vehicle A).


**Table A3.** Road emissions for the RDE test (assuming constant exhaust emission intensity)—output data obtained from the algorithm and the proposed values for the generalized test (Vehicle B).


**Table A4.** Road emissions for the RDE test (assuming constant exhaust emission intensity)—output data obtained from the algorithm and the proposed values for the generalized test (Vehicle C).



**Table A5.** Road emissions for the RDE test (assuming variable exhaust emission intensity)—output data obtained from the algorithm and the proposed values for the generalized test (Vehicle A).

**Table A6.** Road emissions for the RDE test (assuming variable exhaust emission intensity)—output data obtained from the algorithm and the proposed values for the generalized test (Vehicle B).


**Table A7.** Road emissions for the RDE test (assuming variable exhaust emission intensity)—output data obtained from the algorithm and the proposed values for the generalized test (Vehicle C).


#### **References**

