Stochastic MPC-Based Air-Fuel Ratio Regulation of Compressed Natural Gas Engines

In this paper, the air-fuel ratio regulation problem of compressed natural gas (CNG) engines is considered by employing stochastic model predictive control (MPC) technology. A stochastic model predictive regulator based on a discrete-time dynamic model of CNG engines is proposed, taking into account the residual gas, and the closed-loop system is deduced to be stochastically stable. A numerical simulation is performed to demonstrate the effectiveness of the proposed control scheme under two working conditions. The simulation results show that the performance of the proposed stochastic model predictive regulator is better than that of the open-loop controller.


I. INTRODUCTION
Natural gas is a widely acknowledged apposite alternative fuel that can help improve the environment and address energy issues, owing to its widespread distribution, clean-burning properties and higher proportion of hydrogen to carbon [1]. Consequently, natural gas engines have received considerable attention to realize fuel saving and reduce emissions.
The air-fuel ratio in the pre-combustion chamber of CNG engines, corresponding to various mixtures and fuel flow rates can be estimated using a model established in [2]. The emission and combustion performance of a natural gas engine with excess hydrogen, which is affected by the compression ratio under diverse air-fuel ratios has been described in [3]. The influence of the components of natural gas on the combustion and emission performance of natural gas engines has been researched, and the well-known results in this research domain has been reported in [4]. In [5], researchers investigated the particle emission characteristics of natural gas engines operating a traditional oil fueled engine with natural gas. The cyclic variation of the combustion in a pre-mixed natural gas engine with the mixture characteristics was investigated in [6]. In [7], the researchers performed experiments to improve the combustion efficiency of natural gas engines The associate editor coordinating the review of this manuscript and approving it for publication was Juntao Fei . enhancing the compression ratio to achieve a larger expansion ratio. In [8], to clarify the potential of CNG for transport applications, an experimental investigation of the laser ignition of a mixture of lean CNG and air under various compression ratios and excess air ratios has been expressed. The results of recent research pertaining to the performance of CNG engines in terms of the power, efficiency, and other factors have been provided in [9]. In [10], the combustion and emission performance of a CNG engine using a stratified air-fuel mixture were investigated, and overall engine efficiency was improved compared to that of the premixed type engine. To clarify the content of methane, the technology of fuel adaptive injection of CNG engines was investigated in [11]. A reduced order dynamic model of twin spool gas turbine was established for the tracking problem in [12], and a sliding mode controller was designed based on the theory of finite-time stabilization, e. g. [13]. To reduce the root and carbon dioxide emissions of diesel engines, performances of CNG engines with different replacement rates were compared in [14], and the hole size of the fuel injector was optimized. However, the fuel economy and emissions of CNG engines are considerably affected by the control accuracy of the air-fuel ratio. In practice, the performance of the air-fuel ratio control is affected by many factors, such as the so-called residual gas trapped in the cylinder at the end of the exhaust stroke, which reflects stochastic characteristics. Moreover, the inaccuracy of the dynamic models of CNG engines also considerably influences the control accuracy of the air-fuel ratio. An intuitive approach to solve the above problems is to employ the stochastic MPC technology, using which, the limitations of the stochasticity of the residual gas and inaccuracy of the dynamic model can be overcome.
Consequently, the stochastic MPC technology has been widely researched and applied to several practical systems. In [15], the researchers developed a control algorithm to enable the operation of an energy vehicle by combining stochastic MPC with learning, and the proposed control algorithm was validated by considering the energy distribution of a hybrid electrical vehicle. An energy distribution algorithm based on the stochastic MPC theory was developed in [16], by considering the vehicle allocation, driving direction, and transportation information of hybrid electrical vehicles operating in hilly areas with less traffic. A stratified stochastic control algorithm for the energy management of charging of plug-in electrical vehicles and wind energy in a micro-grid was presented in [17], to enable the coordination of plug-in electric vehicles and wind energy. The use of restriction fastening to anti conservatively ensure the stability and recursive practicability in a stochastic MPC was reported in [18]. In [19], the framework of a stochastic MPC is described in context of a certain real-time permission regarding the regulation of the energy and frequency markets for an immobilization battery. Considering the energy distribution of micro-grids involving double glazed units, a control algorithm combined with the MPC working under two different time axes was proposed in [20].
In this paper, the regulation problem of the air-fuel ratio of CNG engines is addressed by using a stochastic MPC algorithm based on the air path and fuel path dynamics. The stochasticity of residual gas is modelled as a Markov chain [21], and the stabilization of the whole system is considered. The control accuracy of the air-fuel ratio is improved, since the statistical information of the residual gas transitions is fully utilized, and the influence of the inaccuracy of the dynamic model is eliminated by using a stochastic model predictive regulator. A numerical simulation is performed to demonstrate the effectiveness of the employed air-fuel ratio stochastic model predictive regulator.

II. STOCHASTIC MPC ALGORITHM DESIGN
The stochastic air-fuel ratio regulator is designed as described in this section. Because the dynamics of CNG engines is the same as that of gasoline engines, a model of gasoline engines, which involves the air path and fuel path dynamics, as established in [21], is used: where M a (k) denotes the mass of the total air in the cylinder, M f (k) denotes the mass of the total fuel in the cylinder, λ d denotes the ideal air-fuel ratio, µ∈ (0, 1) denotes the efficiency of combustion, M an (k) denotes the mass of fresh air, and M fn (k) denotes the mass of fresh fuel, r (k) denotes the residual gas fraction, and is considered as a finite-state irreducible aperiodic Markov chain whose state space S and one-step transition probability matrix P can be expressed as follows: and where s i denotes the state, and p ij denotes the one-step transition probability. The error of regulation of the air-fuel ratio y (k) is defined as Rearranging system (1) and (4) yields, where Next, the detailed design process of the stochastic MPC algorithm will be given: Step 1. Prediction of y (k) and u (k): For system (5), the error of regulation of the air-fuel ratio y (k) and the residual gas fraction r (k) are available at sampling point k by estimation [21]. Based on y (k) and r (k), the future error of regulation of the air-fuel ratio and the future fresh fuel mass trajectories can be expressed as where N p and N c denote the corresponding horizons, and Based on (5) and the future fresh fuel mass trajectories (8), the future error of regulation of the air-fuel ratio can be expressed as follows: . . . where Step 2. Optimization within the horizon N p : Let J (y (k) , r (k) , u [k,k+Np−1]/k ) denotes the cost function as follows: [Q(k +n/k, r(k + n/k))y 2 (k +n/k, r(k + n/k)) + R(k + n/k, r(k + n/k))u 2 (k + n/k, r(k + n/k))] + (k + N p /k, r(k + N p /k))y 2 (k + N p /k, r(k where Q(k + n/k, r(k + n/k)) > 0, R(k + n/k, r(k + n/k)) > 0, ∞ > (k + n/k, r(k + N p /k)) > 0, and N p is a finite positive integer. Define [Q(k + n/k, r(k + n/k))y 2 (k + n/k, r(k + n/k)) + R(k + n/k, r(k + n/k))u 2 (k + n/k, r(k + n/k))] + (k + N p /k, r(k + N p /k))y 2 (k + N p /k, where u * [k,k+Np−1]/k is the optimal control sequence. Using the optimality principle, we have where α (k, s i ) > 0 satisfies Substituting (5) and (15) into (14), we can obtain Based on the first derivative of (17) with respect to u(k), the optimal control can be obtained as follows: Considering (17) and (18), we obtain α (k, s i ) Based on (6), the optimal fresh fuel mass M * fn (k) can be obtained as follows: Step 3. At sampling point k + 1, M * fn (k + 1) can be obtained by repeating step 1 and step 2.
Remark 1: The traditional control approaches, which usually employ the robust control method to address the mode transitions of a discrete-time system with jump parameters, do not take into account the statistical information of the mode transitions. To solve this problem, the mode transitions are modeled as a Markov chain, and the statistical information of the mode transitions is fully considered in the corresponding stochastic control algorithm. However, the control performance of the stochastic control algorithm is considerably influenced by the accuracy of the system model. In general, the proposed stochastic MPC exhibits a better control performance than that of the above mentioned control approach, as the statistical information of the mode transitions is fully utilized, and the accuracy of the system model does not have a strict requirement due to its receding horizon implementation mechanism. It is noted that the better control performance is usually achieved at the cost of increased online computational burden. However, with the improvement in the computational capabilities, the computational issues of the MPC are being gradually alleviated.
Remark 2: The optimal controller (18) at sampling point k is obtained by deriving (17) with respect to u(k) based on the optimality principle, and makes the cost function (12) reach the minimum value within the horizon N p .

IV. NUMERICAL SIMULATION
The effectiveness of the designed stochastic MPC regulator is demonstrated in this section. The numerical simulation which is performed on the Matlab/Simulink platform, involves the dynamic equations of the air and fuel paths (1) and models in mean value form [23]- [25], as follows: Jω e = T e − T l , VOLUME 8, 2020 where T m is 298.15K, V m is 5.897E-03m 3 , V d is 5.76E-04m 3 , s 0 is 3.5E-03m 2 , T a is 298.15K, and The numerical simulation control structure diagram is shown in Fig. 1. The simulation parameters corresponding to two working conditions W 1 and W 2 are as follows. In W 1 , the engine revolution is 1200rpm, external burden is 60Nm, and ideal air-fuel ratio is 17.4. In W 2 , the engine revolution is 1600rpm, external burden is 90Nm, and ideal air-fuel ratio is 17.4. For the two working conditions, N p and N c are 10 and 5, respectively. The state spaces and one-step transition probability matrices of W 1 and W 2 are as follows: By setting Q(k + n/k, r(k + n/k)) and R(k + n/k, r(k + n/k)) as 1 for all k ∈ [0, ∞) and n ∈ 0, n p , the values of α (k + n/k, s i ) for working conditions W 1 and W 2 can be obtained as follows:  Fig. 2 and Fig. 4 indicate that the air-fuel ratio can be regulated into a neighborhood of its ideal value by using both U SMPC and U OPEN , and the fluctuation range of the air-fuel ratio of U SMPC is smaller than that of U OPEN under  W 1 and W 2 . In contrast, the fluctuation range of the fresh fuel mass of U SMPC is larger than that of U OPEN under W 1 and W 2 , which can be observed from Fig. 3 and Fig. 5. To enable the quantitative examination of the control performances of U SMPC and U OPEN , the cost function J of W 1 and W 2 are listed in Table 1, where The values of the controllers U SMPC and U OPEN listed in Table 1 indicated that the cost function of U SMPC is smaller than that of U OPEN , and the values of ε for W 1 and W 2 are 0.503% and 0.644%, respectively. Therefore, the proposed stochastic MPC algorithm U SMPC exhibits a better performance under working conditions W 1 and W 2 . Remark 3: The mean value models reported in references [23], [24] and [25] contain the dynamic equations of the air mass flow leaving the manifold into the cylinders, torque generated, pressure in the manifold and air mass flow passing through the throttle, these equations, along with (1), constitute the dynamic model of the CNG engines.

V. CONCLUSION
This paper proposes a stochastic MPC regulator based on a dynamic model of CNG engines in the discrete-time form, which contains the air path and fuel path dynamics. The whole system is deduced to be stochastically stable. The effectiveness of the designed stochastic MPC regulator is demonstrated by performing a numerical simulation. The control performance of the proposed stochastic MPC depends on the computer performance since the online computational burden is augmented in this technique. The limitations of the increase in the computational burden are being progressively overcome via improvements of the computer performance. An on-line technique to estimate the masses of the total air and the total fuel in-cylinder should be designed, as these values can't be directly estimated. Furthermore, the control accuracy of the air-fuel ratio of CNG engines is affected by estimation errors. As an effective control algorithm, the stochastic MPC has been applied to the energy distribution of hybrid electrical vehicles, coordinated dispatch of power grid and other applications.