Figure 1.
HEV elements (FC Stack—Storage Element Stack).
Citation: Ruixia Yuan, Shujing Gao, Jicai Huang, Xinan Zhang. Mathematical modeling the dynamics of Clonorchiasis in Guangzhou City of China[J]. Mathematical Biosciences and Engineering, 2019, 16(2): 881-897. doi: 10.3934/mbe.2019041
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The number of registered vehicles is increasing, and although the new internal combustion engines are less polluting, pollution problems are becoming increasingly critical, particularly for greenhouse gas emissions that are responsible for global warming of the planet [1]. One way to fight against these emissions is to implement other kind of energy sources (renewable energy: wind, solar, and new environmental friendly sources: fuel cell...) [2]. All-electric vehicles have been presented as a promising solution, thanks to their independence from oil. But their lack of autonomy does not allow them today to permanently replace conventional vehicles; hence the development of another concept: hybrid vehicles [3,4,5].
A hybrid vehicle uses at least two different energy sources to take advantage of the benefits of each organ and thereby improve the overall efficiency of the transmission. The storage element can be recharged on board the vehicle, which avoids costly installation of new infrastructure: the autonomy of the vehicle depends then only on the capacity of its tank conventional vehicle [6]. Hybridization is one of the promising alternatives in the short and medium term. Having two types of source makes it possible to recover the energy and to choose the operating point with the best efficiency for the primary source [7,8].
In order to evaluate the contribution of hybridization and its potential in terms of fuel economy, an overall assessment must be considered by analyzing the cost of the energy flow to achieve the traction function. This cost depends on the operating modes and power distribution policies between different sources. The primary source is intended to provide power on long trips requiring high autonomy and to charge the storage element when needed [9,10]. The braking energy is recovered by the secondary source which also responds to power peaks during acceleration. This combination is supposed to meet the requirements of reducing fuel consumption and pollutant emissions.
The fuel cell produces electricity from hydrogen and oxygen without release of pollutants. This is why it is used in the automotive sector [11]. The Fuel Cell Stack is usually associated with almost one element of storage of electrical energy to power an electric motor in a full hybrid electrical structure. This secondary source of energy is composed of either batteries or super capacitors. Among the advantages of hybrid vehicle is that kinetic energy can be recovered and stored in order to be used later.
Energy management has been a hot topic since the 2000 and has been the subject of much publicity. Several approaches have been developed to minimize fuel consumption and reduce emissions of greenhouse gas by satisfying the power demand at every moment with full respect for the constraints of energy and power elements [12]. In this context, in [13], authors deal with the problems of energy management and control of energy sources. Also, Garcia-Arregui et al. and Zhou et al. propose a simple mission filtering approach [14,15]. Besides, Neffati et al. and Sabri et al. have developed management algorithms based on fuzzy rules [16,17].
There are two main strategies in order to define the distribution of power [18,19]:
1. Offline strategy: If the road section is known, it is possible to use offline strategies for global optimization; this approach is only possible in offline simulation [20,21].
2. Online strategy: In real time the road section is not known. The control strategies used are generally suboptimal but based on offline learned optimized references [22,23,24,25,26,27].
The proposed real time controller is based on fuzzy rules optimized offline on a given profile. Membership function positions are impacting the consumption [28]. It is clear that the phase of generating optimized fuzzy rule by genetic algorithm requires prior knowledge of the path traveled offline [15]. It would be wise to generate different rules in order to find a set of parameters that can be easily adjusted according to the nature of the mission profile in which the vehicle will be submitted. This paper focuses on the management of electric power of such hybrid electrical vehicle. It demonstrates if an optimization is made globally on a profile including two different phases it is quite the same to switch from two rules than applying the global one and can be made online.
Firstly, the different elements used in HEV are presented in part 2 with regards to their efficiency characteristics. Besides, three optimized fuzzy controllers are presented: Analyzing Urban, Road mixed and Highway power profiles. In part 3, segmentation and switching decision method is presented conducing to the discussion and comparison of results in part 4.
For this study, we will consider a generator set formed by a fuel cell and supercapacitors. The different elements of this system have been dimensioned a priori in order to satisfy the power demands, using either a source or two sources simultaneously (Figure 1). Energy management consists in splitting optimally the power demand between the available sources. Depending on the efficiency provided by each source, the minimum hydrogen consumption is reached when, on given profiles, each source delivers its power with a maximum global efficiency.
The Fuel cell Stack FC and the Storage Element Stack FS are characterized by their efficiency and energy sizing [18]. At each time the sum of power sources must provide the power profile demand. The optimization problem is resumed in Figure 1 where it is also noticed the size of each element. Each element size is defined and energy management will be made with decision controller dealing with these min and max power and energy limits (Figure 2).
The energetic model of the generator is characterized by different energetic flows. The storage of chemical energy is represented by the hydrogen reservoir while the storage of electrical energy is represented by the super-capacitors as shown in Figure 3.
The various modes of operation of the vehicle can be classified into three broad categories; depending on the environment of use. We have a set of typical circulation missions, established by the French Institute of Science and Technology for Transport, Development and Networks IFSTTAR (Institut Français des Sciences et Technologies des Transports, de l'Aménagement et des Réseaux) from a statistical study on a set of real missions [29,30]. We adopt the same classification used in the work done by Chapoulie in [31], these missions can be classified by type of path:
- An Urban mission: characterized by low power requirements and a succession of repeated accelerations and decelerations. Operating speeds are between 0 and 20 km/h, for power lines between −5 and 42 kW (Figure 4).
- A Road mission (between urban and highway cycle): it concatenates a series of transient and quasi-permanent patterns with speeds ranging between 40 and 80 km/h for power limit of −39 to 51 kW (Figure 5).
- A Highway mission: characterized by high power applications with a high concentration of operating point on a speed range of 70 to 120 km/h, with power limits −53 kW and 54 kW (Figure 6).
A Lot of studies can be found to recognize some typical information in such different profiles. Statistic approach can select the average power and the power variance to classify them. More complex pattern recognition (clustering, spectral analysis, power density computation…) can be found [32,33]. However in all cases there is a lot need of information to have the correct identification and such computation must be made offline and can't be used in real time to help the controller to switch from one profile to another. Moreover, a priori, splitting the power demand with regard to the dynamic of sources (low power variation manage by fuel cell, high transient manage by supercaps), in order to find a rational method for the characterization of power profiles, it is interesting to analyze the characteristic powers of each mission. The Table 1 illustrates the ratios between the maximum positive and negative powers, the average powers demanded on the entirety of each profile and the average of the positive powers.
| Mission type | Urban | Road | Highway |
| Max Power (kW) | 42 | 51 | 54 |
| Min Power (kW) | −17 | −39 | −53 |
| Average P (kW) | 1.3 | 5.1 | 11.3 |
| Normalized average P/Pmax (%) | 3.1 | 10 | 21 |
| Positive Power average (kW) | 5.6 | 13.8 | 17.7 |
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The problem of energy optimization for the hybrid vehicle is to find the best way to distribute, store and consume energy, while optimizing the consumption of hydrogen over the entire mission [21].
In general, a dynamic optimization problem is governed by the set of following equations [29]:
| ˙x(t)=f(x(t),u(t),t) | (1) |
| ∫tftiγ(x(t),u(t),t)dt | (2) |
| ψ(x(t),u(t))=0 | (3) |
| φ(x(t),u(t))≤0 | (4) |
where x(t) represents the state variables, u(t) the control variables and γ(t) the cost function. The constraints imposed on the system are represented by the functions ψ(t) and φ(t). The beginning of the bounded course is marked by ti and its end by tf.
In this work, we consider a power supply consisting of a fuel cell and a super-capacitor. We note PES(t) the power supplied/absorbed by the Storage Element. This variable is limited and its role is to make energy exchanges, resulting from the links between Pdem and PFC. The state variable x, represents the state of the energy of the storage element E(t). The equation governing the dynamics of the system is written in this case:
| ˙E(t)=−PS(t) | (5) |
PS represents the power supplied/absorbed by the super-capacitor pack. Its expression as a function of the useful power PES and the efficiency of the storage element ηES, is given by the following formula:
| PS(t)={PSE(t)ηSE(Psk(t))DischargingPES(t)ηES(PBS(t))Charging | (6) |
The instantaneous hydrogen consumption γ(t) depends on the power provided by the PFC fuel cell system and the total output of the ηFC generator set. The cost criterion is therefore expressed as follow:
| JH2=tf∫tiγ(x(t),u(t),t)dt | (7) |
| γ(x(t),u(t),t)=PFC(t)ηFC(PFC(t)) | (8) |
The sizing of the system imposes constraints on the powers and the energy levels.
| PSE(t)+PFC(t)−Pdem(t)=0 | (9) |
| PSE−min≤PSE(t)≤PSE−max | (10) |
| PFC−min≤PFC(t)≤PFC−max | (11) |
| ΔSOC=SOCf−SOCi=0 | (12) |
SOC: state of charge.
The hybrid generator group formed by the secondary source of energy and the fuel cell must ensure the instantaneous power demand of the powertrain (Pdem). Energy management strategy on line is based on a fuzzy system. The controller serves to identify the instantaneous power to be supplied by the fuel cell PFC for a given state of charge E and a power demand Pdem.
The fuzzy inference system is characterized by rules that are already optimized offline by a genetic algorithm GA [18] and generate the specific rules for each profile as shown in Figure 7. The GA makes it possible to optimize the choice of the parameters of the fuzzy controller and to generate its specific rules for each profile that link the input variables to the output variable in order to obtain the lowest hydrogen consumption [23]. Once the optimal parameters identified, the optimized fuzzy controller will be used in energy management online without prior knowledge of the power mission.
The main objective is to permanently ensure the requested power along a mission profile, minimizing the hydrogen consumption (H2) as much as possible and respecting the imposed constraints, especially those on the state of charge of the storage element. On the first, we propose a criterion based on two parameters of the specifications, in particular the consumption of hydrogen from the FC and the average energy error MSE, it is proposed to multiply the MSE by 100 so that it has influence on the criterion 1:
Criterion _1 = tot_Consumption + MSE*100
The constraint of maintaining the state of charge is not taken into account in this first form of the criterion. One way to improve the results obtained, in terms of accuracy of total cost calculation is to introduce the constraint on the state of charge and use a penalty function. The latter introduces an additional "artificial" hydrogen consumption when the state of charge is below the initial one, which results in a negative SOC variation 'when'.
Criterion _2 = tot_Consumption + MSE*100 + recharge artificielle (if Δ soc < 0)
To be able to identify the effectiveness of the energy management method applied to each power profile, we will refer to two consumption references:
a. Only use the fuel cell to provide traction (a required positive power): Negative powers will not be recovered (braking energy will be dissipated as heat in the brakes), and we may have energy errors if there are power demands that exceed the maximum power that can be provided by the FC (Table 2).
| Profiles | Profile 1 | Profile 2 | Profile 3 |
| Reference consumption without storage kJ | 5815 | 15451 | 23084 |
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b. Dynamic Programming DP: This strategy considers the FC and the supercapacitors as two sources of energy for the application of the dynamic algorithm. The goal of this method is to find the optimal consumption while respecting all the constraints imposed in particular the constraint on ΔSOC for maintaining the initial and final state of charge equal (Table 3).
| Profiles | Profile 1 | Profile 2 | Profile 3 |
| Total consumption with DP KJ | 5986 | 12667 | 20099 |
| Δ soc (kJ) | 0 | 0 | 0 |
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The segmentation method consists of decomposing the profile into several segments with three consecutive instant. It is necessary to note that interested is made on positive power, since the optimization consists in satisfying a requested power and not distributing a recovered power. The following Figure 8 illustrates the principle of segmentation; the red line denotes the average power of each segment (duration 3s).
The segmentation method allows adding an average power demand during three consecutive instant (window size of three instant). This sliding windows allows to avoid a lot of excessive switching at consecutive instant (chattering) and also can be seen as a predictive window that force the future to be the same as the present. If the two previous power demands indicate variation of profile let the controller switching and changing the rule to use.
In addition, a power profile can be found several characteristics and various power classes. From the previous analysis, and in order to establish a method for real-time optimization, a subsequent switching method of manipulating the rules according to the required power through the segmentation method is proposed. This approach is not really a filtering and not really pattern recognition but is the based-knowledge to conduce intelligent switching at the controller level.
The segmentation method is to decompose the space of power and each segment is characterized by its average power to indicate the membership of each segment to a specified mission.
For example: The beginning of any mission always belongs to the urban mission «Rule 1», since at startup power demand is generally low [0, X1]. Mission Road contains areas that are characterized by urban mission, and other areas that are characterized by the mission highway [X2, Pmax], because the mission Road is located between the two [X1, X2] (Figure 9).
After the optimal rules were identified (Rule 1, Rule 2, Rule 3), the optimized fuzzy logic controller will be used in the energy management, switching optimally from this three possibilities. In order to establish a method for real-time optimization, a method of switching consisting in an accurate segmentation method is proposed. The segmentation method splits the space power into segments and to characterize each segment by its average power to indicate the membership of each segment to a specified mission (Urban, Road and Highway). The switching of rules is done from a program that uses the power indicator. Then it treats the average power of segments to give everyone a clue, 'knowing that the values of the segments are the average power of a sliding window'. This will allow the segmentation algorithm to choose the optimal rule to be used among the existing rules (Figure 10).
The fuzzy switching algorithm FSA operates as follows:
If power segment is low, «Index» = 1 then use the «Rule 1».
If power segment is medium, «Index» = 2 then use the «Rule 2».
If power segment is strong, «Index » = 3 then use the «Rule 3».
The calibration algorithm which is reflected in the choice of position points 'index 1, 2 and 3' to the switching algorithm to switch between different rules is a tricky work. In order to validate the choice of these points, the genetic algorithm can be a good solution for this problem.
After generating optimization rules with genetic algorithm in [28], the application of these three different rules for each profile can also be optimized through the optimal computing of levels X1 and X2 using also Genetic Algorithm (GA) [21]. The genetic algorithm is a search heuristic that mimics the process of natural evolution. This heuristic is routinely used to generate useful solutions to optimization and search problems. A classical genetic algorithm (implemented in Matlab®) was used in order to generate a suitable set of parameters by choosing the best hydrogen consumption (fitness/cost function) for the different profiles to converge towards the two optimal X1, X2 shown in Table 4.
| X1 | X2 | Consumption kW.s | |
| Profile 1 | 13.9 | 15.1 | 3390 |
| Profile 2 | 6.1 | 32.3 | 11031 |
| Profile 3 | 14.8 | 15.2 | 19710 |
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Comments: optimizing X1 and X2 with genetic algorithm can decrease the hand-made choice difficulty. In some profile analysis, it is evident for human being but for some profile it is tricky and the Genetic Algorithm is the way used here to be sure to have an optimal choice on a given profile. eg: X1 = 13.89, X2 = 15.1 applied on Profile 1, provide 3390 kW consumption, which is the lowest consumption obtained with the optimized set X1, X2 other combination provide higher consumption.
The results obtained by applying the method of segmentation and switching rules on the different profiles are shown in the following Figures 11, 12 and 13.
The following Table 5 lists the consumption values obtained by applying:
| (The last two lines is the optimal result obtained with Dynamic Programming (DP) approach and Reference consumption without storage kW.s) | ||||||
| Profile 1—Urban | Profile 2—Road | Profile 3—Highway | ||||
| Fuzzy | 1 rule | FSA | 1 rule | FSA | 1 rule | FSA |
| FC Conso (kW.s) | 3390 | 3390 | 11020 | 11031 | 19660 | 19710 |
| Soc (kW.s) | −420 | −419 | −188 | −188 | −53 | −46 |
| Artificial Cost (kW.s) | 1132 | 1130 | 494 | 494 | 131 | 111 |
| Total Cost (kW.s) | 4522 | 4520 | 11514 | 11526 | 19791 | 19821 |
| D.P. (kW.s) | 5986 | 12667 | 20099 | |||
| Consumption without storage (kW.s) | 5815 | 15451 | 23084 | |||
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● A single rule for each profile.
● A fuzzy switching Algorithm FSA.
● A dynamic programming DP.
● The consumption without storage.
The results presented in Table 5, highlight that applying a rule optimized on a given profile provides the lowest consumption. If a different rule is applied, the consumption increases. But these results demonstrate also that switching accurately between three rules did not really change the consumption. That mean the controller is able to dynamically switch the rules and reach the optimal consumption. The goal to decide online what controller should be used is now open.
To be noticed: the fuzzy controller results are also compared to DP optimal solution. The D.P consumption is higher because it takes into account one more constraint made on the final State Of Charge (SOC) which must be the same as the initial (charge sustaining from start to end facilitating cycling missions).
In order to have a basis of comparison between the different management methods proposed, it is mandatory to have the same constraints and to bring the final state of charge to its initial state. One way to improve the calculation of the total cost is to introduce the constraint on the state of charge and to use a function of overconsumption. The latter introduces an artificial additional hydrogen consumption when the state of charge is below the initial one, this translates into a variation of negative SOC (Δsoc < 0), for the case ΔSOC > 0, the discharge does not cause extra cost of H2 consumption by discharging the elements in the garage for example either in resistors or in the distribution network.
The constraint of maintaining the state of charge is not taken into account in this first form of the criterion. One way to improve the results obtained, in terms of accuracy of total cost calculation is to introduce the constraint on the state of charge and use a penalty function. The latter introduces an additional 'artificial' hydrogen consumption when the state of charge is below the initial one, which results in a negative SOC variation 'when'.
The FSA end the profile with a difference (shown as ΔSOC), so an extra consumption is added to artificially recharge the storage element. To do this, the power is chosen artificially to be given by the Fuel Cell Stack at its maximum power (that provides a not so bad FCS efficiency). This condition adds an extra cost which is less than the cost used during the profile during D.P. computing. That explains the total consumption with fuzzy is a little bit less consuming than D.P. results.
To be also noticed in Figure 14, the difference is impacting the energy stored and delivered by the storage element. The switch at time (80 seconds) changes the rest of the profile due to switching that increases the consumption. But if the switching is not frequent, with a short duration, or if the switch did not affect high power (case in highway power demand) the consumption is sub-optimal but very closed to the optimal.
This work has been devoted to online management based on fuzzy rules. This method has been improved by the segmentation and switching method FSA. The switching between the rules is associated with an optimal distribution of the requested power online learned and optimized offline.
This work results from a succession of local optimizations, thus a decrease of the global consumption has been obtained. The results obtained show that the energy management based on fuzzy rules can be likened to a management of the state of charge of the storage element under stress of minimizing the energy delivered by the non-reversible source (FC). This energy management strategy makes it possible to take into account the evolution of the state of charge of the storage element at each instant and in addition forces the fuel cell to operate at its best points of efficiency.
A 'predictive' management strategy has been proposed; this method is inspired by the fuzzy rules method and has given satisfactory results in terms of consumption. A fuzzy decision layer makes it possible to switch the rules according to a prediction of the mission profile and to apply the right set of rules, especially on an unknown path or on path zones identified as closest to the known profiles.
This work shows that the fuzzy rules-based power management strategy has improved consumption compared to applying a single optimized rule to another profile.
The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program.
This work was supported by the Deanship of Scientific Research at King Khalid University for funding this work through research program [grant number G.R.P-235-40].
The authors declare that there is no conflict of interest in publishing this paper.
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Figures(4) / Tables(2)
Ruixia Yuan, Shujing Gao, Jicai Huang, Xinan Zhang. Mathematical modeling the dynamics of Clonorchiasis in Guangzhou City of China[J]. Mathematical Biosciences and Engineering, 2019, 16(2): 881-897. doi: 10.3934/mbe.2019041
| Mission type | Urban | Road | Highway |
| Max Power (kW) | 42 | 51 | 54 |
| Min Power (kW) | −17 | −39 | −53 |
| Average P (kW) | 1.3 | 5.1 | 11.3 |
| Normalized average P/Pmax (%) | 3.1 | 10 | 21 |
| Positive Power average (kW) | 5.6 | 13.8 | 17.7 |
DownLoad:
CSV
| Profiles | Profile 1 | Profile 2 | Profile 3 |
| Reference consumption without storage kJ | 5815 | 15451 | 23084 |
DownLoad:
CSV
| Profiles | Profile 1 | Profile 2 | Profile 3 |
| Total consumption with DP KJ | 5986 | 12667 | 20099 |
| Δ soc (kJ) | 0 | 0 | 0 |
DownLoad:
CSV
| X1 | X2 | Consumption kW.s | |
| Profile 1 | 13.9 | 15.1 | 3390 |
| Profile 2 | 6.1 | 32.3 | 11031 |
| Profile 3 | 14.8 | 15.2 | 19710 |
DownLoad:
CSV
| (The last two lines is the optimal result obtained with Dynamic Programming (DP) approach and Reference consumption without storage kW.s) | ||||||
| Profile 1—Urban | Profile 2—Road | Profile 3—Highway | ||||
| Fuzzy | 1 rule | FSA | 1 rule | FSA | 1 rule | FSA |
| FC Conso (kW.s) | 3390 | 3390 | 11020 | 11031 | 19660 | 19710 |
| Soc (kW.s) | −420 | −419 | −188 | −188 | −53 | −46 |
| Artificial Cost (kW.s) | 1132 | 1130 | 494 | 494 | 131 | 111 |
| Total Cost (kW.s) | 4522 | 4520 | 11514 | 11526 | 19791 | 19821 |
| D.P. (kW.s) | 5986 | 12667 | 20099 | |||
| Consumption without storage (kW.s) | 5815 | 15451 | 23084 | |||
DownLoad:
CSV
| Mission type | Urban | Road | Highway |
| Max Power (kW) | 42 | 51 | 54 |
| Min Power (kW) | −17 | −39 | −53 |
| Average P (kW) | 1.3 | 5.1 | 11.3 |
| Normalized average P/Pmax (%) | 3.1 | 10 | 21 |
| Positive Power average (kW) | 5.6 | 13.8 | 17.7 |
| Profiles | Profile 1 | Profile 2 | Profile 3 |
| Reference consumption without storage kJ | 5815 | 15451 | 23084 |
| Profiles | Profile 1 | Profile 2 | Profile 3 |
| Total consumption with DP KJ | 5986 | 12667 | 20099 |
| Δ soc (kJ) | 0 | 0 | 0 |
| X1 | X2 | Consumption kW.s | |
| Profile 1 | 13.9 | 15.1 | 3390 |
| Profile 2 | 6.1 | 32.3 | 11031 |
| Profile 3 | 14.8 | 15.2 | 19710 |
| (The last two lines is the optimal result obtained with Dynamic Programming (DP) approach and Reference consumption without storage kW.s) | ||||||
| Profile 1—Urban | Profile 2—Road | Profile 3—Highway | ||||
| Fuzzy | 1 rule | FSA | 1 rule | FSA | 1 rule | FSA |
| FC Conso (kW.s) | 3390 | 3390 | 11020 | 11031 | 19660 | 19710 |
| Soc (kW.s) | −420 | −419 | −188 | −188 | −53 | −46 |
| Artificial Cost (kW.s) | 1132 | 1130 | 494 | 494 | 131 | 111 |
| Total Cost (kW.s) | 4522 | 4520 | 11514 | 11526 | 19791 | 19821 |
| D.P. (kW.s) | 5986 | 12667 | 20099 | |||
| Consumption without storage (kW.s) | 5815 | 15451 | 23084 | |||
