Storage device sizing for a hybrid railway traction system by means of bicausal bond graphs

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1 Storage device sizing for a hybrid railway traction system by means of bicausal bond graphs G Gandanegara, X Roboam*, and B Sareni LEEI ENSEEIHT/INP Toulouse, Toulouse, France Abstract: In this paper, the application of bicausal bond graphs for system design in electrical engineering is emphasized. In particular, it is shown how this approach is very useful for model inversion and parameter dimensioning. To illustrate these issues, a hybrid railway traction device is considered as a case study. The synthesis of a storage device (a supercapacitor) included in this system is then discussed. Keywords: bond graph, bicausality, model inversion, dimensioning, electrical engineering, hybrid systems 1 INTRODUCTION complex energetic systems and is complementary to the analysis process. Two main issues can be Electrical engineering systems are more and more addressed from this approach: model inversion and complex and heterogeneous, consisting of elements of parameter sizing. On bond graph formalism, the different natures, strongly coupled between different properties of bicausality can help to solve these physical fields. Within this framework, the system issues [13 19]. analysis becomes complicated so that a unified This paper aims at showing how bond graph formalism such as the bond graph (BG) [1 4] is formalism associated with the bicausal approach can particularly useful. This graphical method illustrates be useful in constructing inverse models and solving the energetic transfers in the system. By analysing the system design problems. A hybrid railway traction causal bond graph, it is possible to know whether device is considered as a case study. Coupling energy the association is physically (and energetically) and power sources is firstly investigated when convenient and then numerically consistent. Homo- the consumption cycles vary significantly. Indeed, the geneous modelling and system analysis methods size of the energy source can be adjusted with directly applicable for bond graphs are the major the average power consumption, the power peaks interest of bond graph formalism [2 12]. being provided by the storage device. Furthermore, Having chosen the system architecture and para- if breaking modes appear during the driving mission, meter sizing, the system analysis process consists of it can be possible to the breaking energy. These verifying whether the device fulfils the requirements: issues are well known for hybrid vehicles, but it has this is usually done from a system s model and also recently appeared in railway traction. its simulation. On the other hand, the synthesis In the paper, the example of a supercapacitor process consists of choosing the system structure dimensioning by means of the bicausal approach is and its sizing, directly starting from the requirements. considered. To present this application, bicausality This inverse process is essential for the design of and model inversion are briefly discussed in the next section. Then, the model of a hybrid traction device * Corresponding author: LEEI ENSEEIHT/INP Toulouse, 2 Rue is presented and the storage element dimensioning is Camichel, BP 7122, Toulouse Cedex 7, 3171, France. carried out to demonstrate the interest of bicausality Xavier.Roboam@leei.enseeiht.fr synthesis. Finally, conclusions are given in section 4.

2 2 BICAUSALITY AND MODEL INVERSION test this property on the bond graph before applying the bicausality inversion process. For this purpose, In a causal bond graph, each bond has one causal several definitions are employed [17]. stroke. In fact, it can be divided into two causal halfstrokes: one for the flow variable and one for the Definition 1 effort variable. Thus, the assignment on the model Two single-input/single-output (SISO) causal paths can be examined by applying this type of causality, are disjoint if they do not pass through any common which is called bicausality. For graphical convention, variable. the flow information is on the bond side with a half-arrow (in the present examples, it is below Definition 2 the bond). The concept of bicausality was invented and A set S is disjoint if it consists of m disjoint SISO first published by Gawthrop [13]. This proposition causal paths. has opened a new research field in bond graph applications [14 19]. Definition 3 The order v (u,y) of an SISO causal path p between p i i 1. System inversion. If the bond graph structure, the an input u and an output y is defined as i i parameters, and the initial states are chosen and v (u, y )=n ( p) n ( p) (1) if the outputs are given from system requirements, p i i I D the necessary inputs can be directly defined from where n ( p) [respectively n ( p)] is the number of I D the bicausal solver. dynamic elements in integral (respectively differential) 2. State estimation. If parameters, inputs, and outputs causality crossed by this causal path. are given, the dynamic elements of initial states can be deduced. In this paper, the state Definition 4 estimation is not studied. Therefore the initial The order v(s) ofasetsof m disjoint SISO causal states of dynamic elements are considered to be paths p, i=1 tom, is known. i 3. Parameter estimation. If inputs, outputs, and v(s)= m v (2) pi initial states are given and if a set of parameters i=1 are fixed, the other parameters values can be By using these definitions on the direct bond synthesized. The number of parameters that can graph, the invertibility condition for an MIMO be determined depends on the degrees of freedom (multi-input/multi-output) model with m inputs in the system (number of provided inputs/outputs). and m outputs is: 1. If there is no choice for the set of m disjoint SISO In a bicausal bond graph, the causality of each causal paths, the model is structurally not variable is separately examined. With causal half- invertible. strokes, there are four possibilities. The conventional 2. If there is only one choice for the set of m disjoint or physical causality is a particular case of the SISO causal paths, the model is structurally bicausality where both causal half-stokes are placed invertible. on the same side of the bond. Note that, if the con- 3. If there are several choices, the modified sequential ventional causality gives considerable information causality procedure for inversion (MSCAPI) should about the physical meaning of associations, the be applied [16, 17]: bicausality is purely conceptual and is consequently (a) Determine a set S whose order is the smallest only useful for the design process. among them. In a bicausal bond graph, it is preferred to replace (b) Replace all sources and detectors associated the notation Sf, Se, De, and Df by SS (source sensor with the control variables or inputs and outelements) [2], even if their causality is not changed puts by SS elements. (both causal half strokes are on the same side of (c) Assign the effort source flow source causality the bond). In this case, these SS elements are called to the SS output elements and propagate the the effort source flow sensor (for Se and Df) or the causal information to the SS input elements. flow source effort sensor (for Sf and De). This propagation has to arrive and impose An inverse model can only be obtained if the direct the effort detector flow detector causality model is invertible. Therefore, it is necessary to firstly on the input SS elements. Other elements take

3 the causality due to the bicausality propagation Generally, some derivative causalities appear when the and junction conventions. These conventions inversion process is applied. An implicit differential are: algebraic equation (DAE) model is obtained and the $ For 1 junctions: inverse formulation with integral (int) and derivative effort side: only one bond without a (der) causality is as follows half-stroke close to the junction; d flow side: only one bond without a half- dt x int = f inv (x int, y spec,, yn spec ) stroke close to the junction. $ For junctions: u =g (x, y,, yn ) effort side: only one bond with a halfwhere x is the state variable corresponding to the int inv inv int spec spec stroke near the junction; flow side: only one bond with a halfbond graph. The specified inputs ( y ) of the spec element that keeps an integral causality in the inverse stroke near the junction. (d) If there is at least one causality conflict, the inverse model have to be sufficiently differentiable. model is not invertible. In the opposite case, In particular, if the inversion is processed along an it is invertible. SISO causal path with a length of n, the correspond- ing input of the inverse model must be n times When the model is invertible, the following pro- differentiable if no symbolic manipulations are made. cedure can be applied to construct the inverse model Some propositions have been made in reference [21] (or the synthesis model) using bicausality [16]. to facilitate the numerical derivation. This issue can 1. Replace all source and detector elements by SS also be solved by filtering the specified inputs of the elements. inverse model. From a specified data file, several 2. Apply the bicausality effort source flow source on methods of filtering can be used with different cutthe output elements and the effort sensor flow off frequencies. In particular, causal (lowpass filter) sensor on the input elements or on the elements or acausal (centred weighted moving average) filter- whose parameters have to be synthesized, in ing methods can be applied. In the first case (causal) relation to the degrees of freedom. the signal phase is shifted following the cut-off 3. Propagate the bicausality from outputs to inputs. frequency. In order to avoid this delay an acausal Other elements take the causality due to the filtering can be used. bicausality propagation with respect to junction conventions. 3 APPLICATION ON A RAILWAY TRACTION The obtained bond graph is then called a bicausal DEVICE bond graph. In order to apply these methods, a railway traction Model derivation system is considered. Basically, the model of this A direct formulation with integral causality can be device is a simplified vision of the traction part of expressed as the Alstom BB36 locomotive [22] (see Fig. 1). The considered structure is composed of a direct current d (d.c.) voltage source, an RLC input filter feeding dt x= f (x, u); y=g(x, u) an induction machine that drives the mechanical Fig. 1 The BB36 simplified block diagram

4 transmission line [22]. It has been modelled in the Second choice: v (a, f )= bond graph and several analyses have been carried L =a e e L DC f f e e I out [1 12] as a model simplification or stability analysis. f f f f C acc e e e In this paper, the induction motor with a d.c. e I 18 5 f f f f K e motor is replaced in order to simplify the bicausal 21 approach of the model inversion. This equivalent e e e I f f f f motor is sized to present the same power balance K 5 e e e e I f f (identical efficiency) as the one actually obtained 3 31 with the induction motor. The previous voltage f f K jac e e e e I source inverter is then also replaced by a d.c. d.c. chopper. The direct causal bond graph corresponding to this system is presented in Fig. 2. To test the f f f f f f K ess e e e I eq device behaviour with real conditions, the Central Business District (CBD) cycle has been retained as Third choice: v (a, f )= the system driving mission [23]. The CBD cycle is L =a e e L DC f f e e I considered as a reference for the design of traction systems. Each cycle includes a velocity acceleration f f f f K acc e e e phase, a constant velocity phase at 2 mile/h, a e I 18 5 f f f f K e velocity deceleration (or braking) phase, and a phase 21 with zero velocity. The curves of velocity and e e e I f f f f power source applied to the present case study are K 5 e e e e I f f illustrated in Fig. 3. It can be seen that negative 3 31 powers (i.e. the regenerative phase) are obtained. f f C jac e e e e I f f f K ess e e e I eq 3.1 The inverse model f f f In the direct causal bond graph, the d.c. voltage Fourth choice: v (a, f )= source (U ), the duty cycle a of the d.c./d.c. con- cont verter and the resistive force (Fres ) are considered as L =a e e L DC f f e e I the inputs and the obtained source current and f f f f C acc e e e velocity as the outputs. In particular, consider the SISO causal path L from the control variable a to e I 18 5 f f f f K e 21 the flow variable (train velocity) at the bond 42 f 42 e e e I f f f f (see Fig. 2): there are four choices of SISO causal paths, where X indicates that the element X is K 5 e e e e I f f 3 31 crossed by the causal path: f f C jac e e e e I First choice: v (a,f )= L1 =a e e L 7 9 DC f f e e I f f f f K acc e e e e I 18 5 f f f f K e 21 e e e I f f f f K 5 e e e e I f f 3 31 f f K jac e e e e I f f f K ess e e e I eq f f f f 36 f 37 f 38 K ess e 38 e 39 e 4 I eq f f f Given that several choices of the SISO causal path exist, it cannot be directly deduced whether the model is invertible or not. In this case, the MSCAPI procedure has to be applied. The last choice is associated with the SISO causal path with the smallest order. In this way, this path is examined. After having replaced the input (in this case, the right bond of the MTF )byaneffort sensor flow sensor and a the output (the detector velocity) by an effort source- flow source SS element with e=, the bicausality propagation does not imply any causality conflicts (see Fig. 4). Therefore, the model is invertible. The model obtained by the MSCAPI procedure is also the

5 Fig. 2 Direct causal bond graph of a railway traction device with an equivalent d.c. motor

6 Fig. 3 CBD cycles: (a) velocity and (b) power source curves inverse model. In this inverse model, the resistive causality). Therefore, the input must be differentiable force and the velocity information related to the CBD at least 1 times. The 2 Sim build-in backward cycles are injected. differentiation formula (BDF) [18] is used as an Note that by considering the driving mission integration method. (CBD cycles) as requirements, the electrical con- As a validation of the inversion process, Fig. 5 shows straints can directly be synthesized by means of the duty cycle obtained from the inverse model this bicausal approach from the model inversion. superimposed with the one obtained from the direct This example emphasizes the design capacity of this model. In this particular case, the validation process methodology in electrical engineering in the context consists in using the output of the direct model of a top-down systemic approach. (i.e. the train velocity) as the input of the inverse Simulations are carried out by means of the 2 Sim model. Therefore the velocity is differentiable at simulation software [24, 25] by using a modified least 1 times and is obtained in the direct model library developed in the authors laboratory, in order through 1 dynamic elements with integral causality. to take bicausality into account. In the case study, In the general case, the specified input signal must be having an SISO causal path length of 1, a set of DAE sufficiently filtered to inverse and differentiate the is obtained for which the output (i.e. the duty cycle a) model. Some symbolic manipulations of the derived is expressed versus the input (i.e the train velocity). model can also be made, as given in reference [21]. Without any symbolic manipulation [21] of the Note that this includes the control strategy, contrary equations, this input is differentiated 1 times in the to the inverse model. This result proves that the inverse model (1 I and C elements are in derivative inverse model is validated.

7 Fig. 4 Inverse model to obtain the duty cycle a

8 Fig. 5 Comparison between the direct model and inverse model responses: duty cycle a 3.2 Bicausal synthesis of the storage element in a resistance (representing storage losses) in series with hybrid system an important capacitance. Even if this model is Hybrid systems rather simplified, it is energetically significant and sufficiently accurate to demonstrate the synthesis In numerous devices, such as the one considered process. This storage element is connected to the here as a case study, the loading power is very time- primary bus with a d.c./d.c. boost converter including variable (see here the CBD cycle of Fig. 3) so that the boost inductance. The storage current (I )is the peak power is far from the average consumed then controllable (see Fig. 6). power. For such systems, the main source must be The proposed energy management strategy is the oversized to take into account the maximum peak following. power demand, which presents a great drawback from 1. For a given and nearly constant d.c. bus voltage an economic point of view. The general idea in hybrid (V ), the loading power variations are equivalent systems is well known for electrical vehicles but it bus to the loading current variations (I ). also becomes applicable in other fields. For railway load 2. A low pass filtering (LPF) of the loading current is traction systems, hybridization can offer advantages considered, the idea being to control the system such as a reduction of energy consumption and of in such a way that the primary energy source pollution (carbon emissions) when diesel electric only provides the lowpass filtered loading currents devices are used. (ILPF ). As a result, the primary source only pro- In hybrid systems, it is necessary to associate the load vides the sum of the average loading power, the main energy source with a storage device, such as average value of the system losses, and the lowsupercapacitors, batteries of accumulators, or inertia frequency power harmonics. On the contrary, the wheels. With such components, the primary energy storage device must provide the high-frequency source will only have to furnish the average loading loading currents (IHF ). The bandwidth of the LPF power and the average system losses. The variations load is determined according to the primary source of the consumed power can be provided by the specifications. storage element(s). In this type of system, the con- 3. The system is controlled to ensure that the storage cept of energy sources is used for the sources supplyelement provides the high-frequency (HF) part of ing the average loading power and power sources for the loading current. Note in this case that the those furnishing the loading power variations in average power produced by the storage element short periods according to the driving mission time. is null on a time interval superior to the inverse Energy management and storage control of the filtering cut-off frequency. In this way, the loading state of the storage device remains almost A supercapacitor will be considered here as the constant: the loading management of the storage storage element. This device is modelled by a low element is therefore really simplified.

9 Fig. 6 Synoptic of a hybrid system with a storage element In Fig. 3(b) the maximum loading power is 1 MW. values are determined as An energy source needs to be applied that is sized up to 4 kw (4 per cent of the maximum loading power). This reduction value can be obtained by C( p)=k 1 A1+ T i pb (4) tuning the filter bandwidth. To accomplish the objective, a second-order filter is applied with a with bandwidth (cut-off frequency) of.1 Hz. This frequency is also convenient for filtering the power variations of the considered system with the CBD T = L i R and K= L T CL cycles. where T is the desired closed-loop time constant. CL The control of the storage element is defined in Fig. 7. The storage element should provide the HF Storage synthesis part of the consumed current (I ). Respecting the load power transfer orientation gives In the particular case of a supercapacitor, an analytical approach is possible to synthesize the C value. I =I +I +I (3) This analytical process is used as a comparison with source cf in load the bicausal approach which is the main aim of where this paper. I #ILPF source load Analytical approach. The energy that can be d I # or supplied by the storage element is defined as cf The HF current I is indirectly set by controlling the E (t)= P t [V (t)i (t)]dt+e () (5) in bus in storage current I filtered by the boost inductance from the duty cycle of the d.c./d.c. converter. A PI The d energy of the supercapacitor E is controller is used to control I in order to comply C with the desired specifications for I. The parameter E (t)=1c[v (t)]2=1c[b(t)v (t)]2 (6) in C 2 2 bus

10 Fig. 7 Schema of the storage controller by HF filtering In fact, E and E are almost equivalent if the C chopper and the resistance R losses are neglected (the average d energy in the inductance L is also zero) E (t)#e (t) (7) C First assumption If the bus voltage is constant, then E(t)=V busp t I (t)dt+e () (8) in Second assumption If the discharge state of the supercapacitor is 5 per cent of the nominal voltage, there are two extreme cases. 1. The case where V is maximal (1 per cent): max(v )=V when b =1 bus max The associated energy is E =1C(V )2 (9) max 2 bus 2. The case where V is minimal (5 per cent): min(v )=.5V when b =.5 bus min The associated energy is The current profile can be directly obtained from the relation I =ILPF in load I load =IHF (12) load Finally, the minimal value of the supercapacitor C can be calculated as C = 8 min 3V bus A max AP t I (t)dt in AP t B min I (t)dt BB in (13) Bicausal approach. Bicausality is another way to determine this supercapacitor value. Both voltage (V ) and current (I ) should be provided by the bus in storage element and can easily be deduced from the bicausal bond graph of Fig. 8. In this BG, the loading current (SS element) has been synthesized from the inverse BG model developed in section 3.1. For this purpose, the train velocity has been filtered with a causal filtering so that the inverse system can be derived with the BDF algorithm of 2 Sim. From the inverse model presented in Fig. 4, the current f and 6 the associated voltage e can be obtained at the 6 output of the LRC filter (see Fig. 9). It can be seen that the voltage of the input filter is almost constant. Therefore, the power can be tuned if the current is itself controlled. E = 1 min 2 C A V bus 2 B 2 = 1 8 C(V )2 (1) bus To generalize, if the discharge depth is x%, then the minimal energy will be E min =1 2 C(x{%}V bus )2 (11) From these assumptions, E max E min V bus = 3 8 CV bus #max CP t I (t)dt D in min CP t I (t)dt D Fig. 8 Inverse model with a submodel to calculate the in minimal value of C

11 Fig. 9 Loading current and voltage for CBD cycles The bicausal BG of Fig. 4 allows the direct calcu- appropriate and it is interesting to integrate a storage lation of the d energy to be made at any given element. time. It is sufficient to take the maximal and minimal values of the d energy during the CBD cycles. Numerical applications. Applying both methods For a 5 per cent discharge depth, the supercapacitor previously presented with a discharge depth of 5 value is obtained using the following formula per cent gives: With the analytical approach C = 8 min 3(V )2 (E max E min ) (14) bus max APt I (t)dt B min in AP t I (t)dt The general expression for a discharge depth of x%is in 2 Then C = min (1 x{%})2(v )2 (E max E min ) (15) bus C = =2.18F Note that the obtained minimal value C is only min min valid for a certain driving mission (here the CBD With the bicausality approach cycle). If the power profile is changed, this value should be recalculated. C =2.22F min With the CBD cycles (cf. Fig. 3), the loading power Note that these methods give quite similar results. is extremely time variable (from 4 kw to 1 MW). For this kind of application, a hybrid structure is The difference is related to the accuracy of both integration methods.

12 3.3 Storage synthesis validation Fig. 9(a). The storage element is represented by a supercapacitor of C = 3 F with internal resistive In order to validate the bicausal approach based losses R = 5mV. The boost inductance is fixed at on storage synthesis, the direct model of Fig. 6 is L = 1 mh in order to smooth the current ripple simulated with the loading current illustrated in correctly. The responses of this direct BG model are Fig. 1 Analysis of the entire hybrid system modelled by the direct causal bond graph: (a) loading and primary source powers; (b) loading, LPF, and source currents; (c) bus and C voltages

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