Analytical approach for the optimal design of combined energy storage devices in ship power system E. Fasano*, D. Lauria*, F. Mottola**, R. Rizzo* * ** University of Naples Federico II, via Claudio 21, Naples, (Italy) University of Naples Parthenope, Centro Direzionale di Napoli Isola C4, Naples, (Italy) modern paradigm of integrated power system allows itself the naval ship survivability due to the intrinsic ability in adapting the electrical distribution system, exploiting the redundancy of the structure: the zonal survivability, regarded as the ability to supply loads in zones not affected by faults, can be realized in a feasible way [3], [4]. Furthermore it permits a reduction of the prime movers and a noticeable improvement of their efficiency and then the same efficiency of the propulsion systems can be significantly enhanced. At this purpose, it has to be noted that the integrated power system could allow fixed pitched propellers in place of the traditionally employed controlled pitched propellers. Medium-voltage direct current (MVDC) power systems, conceived as a promising candidate to implement the integrated power system on ships, have been deeply investigated in the last decade since it allows to realize many remarkable advantages over alternating current (AC) distribution systems [4]. MVDC systems show a high level of flexibility due to the massive penetration of the innovative components as power electronic devices capable of performing better and better performances in terms of reliability and power quality. Silicon carbide power electronics in particular is a promising evolving technology for reducing dramatically the size and the weight of the capacitive and magnetic devices, due to high switching speeds. The intrinsic ability in withstanding high voltages allows lowering the number of series modules requested for MVDC technology. Another potential advantage of MVDC system relies in the inherent ability in counteracting the fault currents, even if this topic requires also deeper investigations. One of the interesting advantages of MVDC is the inherent ability in integrating energy storage systems for improving fuel efficiency while maintaining the desired level of reliability. Energy storage systems have been extensively investigated, since they play a crucial role in all the sectors of the electrical systems, thanks to their versatility and to optimal performances obtainable through the employment of electronically-coupled devices [5]. The optimal sizing of these devices is not a trivial task and many proposals are available in the relevant literature aimed to determine the optimal size. Focusing on the DC applications, a new analytical methodology for sizing energy storage systems has been Abstract— In the last years a great interest has been paid in the relevant literature to the benefits deriving from the use of direct current for ship power systems. This is in compliance with the most recent concept about all-electricship and smart electrical distribution for achieving better performances by exploiting as far as possible the potentiality of integrated power systems. In this context the storage devices play a crucial role in improving the reliability, the efficiency and the dynamic performances of the whole electric system. Moreover, compared to conventional ship propulsion plants, this approach has a significant impact on ship design (particularly when combined to azimuthing thruster or podded thrust units) as regards comfortable (noise and vibrations), flexibility (for general plan), consumption and environment protection. The variety of pulsed and intermittent loads also requires a proper choice of the storage devices and of the techniques for the optimal system design. Focusing on DC ship power systems, in this paper two different methodologies for the design of combined storage systems constituted by batteries and supercapacitors are proposed. Based on some typical loads, some relevant case studies are analyzed and discussed. Index Terms-- Energy storage, design optimization, allelectric-ship. I. INTRODUCTION The worldwide challenge to environmental friendly systems, with consequent more stringent requirements for emissions, has pulled the researchers to investigate more efficient solutions for the large variety of electrical applications. The marine sector is universally recognized as one of the most significant energy consumer needed for ensuring the electrical propulsion demand and the other appliances. This is confirmed also by the recent advances of IEEE Standards [1]. The revolutionary concept of all-electric allows improving the design and the management of the ship power systems through the widespread integration of all of the electrical and electronic subsystems. The rationale behind this solution is a power station able to supply all loads [2]. Two primary aspects have to be highlighted in the design philosophy. The first refers to the fact that the overall electrical load consequent to the high-power combat system and the gradual electrification of auxiliary equipment is critically increased. The second one is related to the more stringent requirements in terms of survivability and continuity of the electrical service. The 978-1-5090-4682-9/17/$31.00 ©2017 IEEE 761 recently presented in the literature [6]. In the present paper an effort is paid at the aim of extending the design procedure proposed in [6] to ship power system, where the network exhibits various independent generation buses. The original contribution of this paper is to discuss the proper application of the design procedure to the case of onboard ship applications and, particularly, to the variety of loads typically supplied by the ship power systems. At this regard, two different applications of the sizing procedure are discussed: - the storage system is used to make more efficient the use of the vessel diesel generators; - the storage system is used to provide the onboard ship power system to face with severe pulse loads. Based on these applications, the use of batteries and supercapacitors are discussed and an original analytical procedure for sizing the supercapacitors is also proposed. The rest of the paper is organized as follows. In Section II, possible architectures of the power distribution of all-electric-ships are proposed and discussed. By focusing on different kind of load profiles, two different analytical optimization are solved for the optimal sizing of storage devices in Section III; based on the derived optimal storage capacity, the features of battery and supercapacitors are derived in Section IV. In Section V a discussion on the use of the analytical approach is proposes with reference to some relevant case studies. Finally, our conclusions are drawn in Section VI. levels of security and reliability of the onboard ship power system while guaranteeing cheaper operation, basically relies in the optimal integration of storage devices with the other system components and their portioning in the frame of the zonal loads system. At this aim, both control and sizing of storage devices must be properly tailored with respect to different typologies of loads and generators. Two goals are expected: to prevent undesired operative condition (e.g., undesired peak power request) and to allow cheaper operation (e.g., reduction of fuel consumption). Generator Generator storage storage Zonal loads Zonal loads Zonal loads Zonal loads M M Propulsion Propulsion Pulsed loads Fig. 1. DC Ring-bus microgrid for ship power system. II. MVDC ONBOARD SHIP POWER DISTRIBUTION In this framework this paper focuses on the optimal design of the storage system by proposing an analytical optimal sizing of storage system aimed at guarantee an efficient operation of the grid. More in depth the optimal storage sizing is exploited considering the objective to level the power production of the on board generators against pulsed loads’ profiles; the second approach is devoted to the classical network loss minimization. In the power systems of all electric ships the use of DC allows better integrating power sources and loads able to make more efficient and cheaper energy management on the ships [7]. The integration of loads and resources can be further improved by including storage devices in the system. This is even more apparent in the presence of critical loads which require high level of secure power supply. At this regard, the ring-bus based microgrid architecture has been evidenced as DC architecture for ship application (e.g., [3], [8]) which better fits the needs of more reliability and safety of the on board electrical distribution networks. The ring bus architecture is characterized by different zones that can be independently monitored and controlled. As shown in Fig. 1, different loads and resources can be connected at the ring-bus, such as generators, energy storage devices, propulsion, navigation apparatus and instruments, pulsed loads (e.g., weapons systems) and zonal load centers (lighting, refrigeration, etc.). The ring-bus architecture implies the possibility to guarantee higher levels of survival in case of single points failures by avoiding fatal interruptions in the network operation. This is reached by partitioning the ship distribution network in properly identified zones that can be supplied according to different paths selected through a proper combination of switches operating modes. The ability of ring-bus architecture to improve the III. ENERGY STORAGE OPTIMAL CURRENT PROFILE In this section the current profiles requested to the storage devices (i) to lowering the power output of the on board generators and (ii) to minimize the network power losses are analytically derived. The current profiles are derived by solving two optimization practices formulated in terms of minimization problems combined with isoperimetric constraints. The objective functions and constraints are properly tailored to face with the peculiarities of ship power systems. The derived current profiles will be then used for the optimal design of the storage systems, as discussed next section, with reference to the case of supercapacitors. The first formulation is based upon the assumption of negligible resistances, since the design objective requires the lowering as far as possible of the current supplied by the generators, while taking into account the cost of the storage components. 762 The second formulation is based on the same approach proposed in [6]. Based on the peculiarities of ship power systems, the optimal current profiles of the storage devices in the onboard MVDC network is investigated. The purpose is the identification of the analytical expression of the current profile able to minimize network losses over a specified time period while accounting both the costs of losses and storage devices. with k 2' = Esto k 2 . By simple manipulations the following solution is obtained: A. Lowering current drawn from the power generators k' J sto (t ) + 2 sign J sto (t * ) = 2k1 ng J sto (t ) = j =1 ng ³ ¦ ³E sto J sto (t ) dt ng J al = ³E sto J sto (t ) dt (1) ³¦J j (ξ ) dξ 0 j =1 T ng ³¦J j (ξ ) dξ 0 j =1 Jal(A) 2000 0 -2000 0 0 8 t(h) 16 24 200 Jsto(A) (2) 0 -2000 -200 -100 0 (A) 100 Jsto 200 -100 0 (A) 100 Jsto 200 200 0 -200 0 8 t(h) 16 24 0 -200 -200 Fig. 2. Graphical interpretation of the solution of (5). In Section IV, details on a possible methodology for the optimal sizing of supercapacitors based on the desired current profile (5) is discussed. B. Minimizing losses in the Onboard Ship Power Network The MVDC onboard ship network’s line resistances are considered and the aim of the storage sizing problem is devoted to the network losses minimization. In order to formulate the considered problem in mathematical terms, let us consider the following classification for the MVDC busses: nG generation busses; operations of the generators, k 2 is related to the per unit cost of the energy charged and discharged during T by the storage devices. The isoperimetric constraint (2) allows obtaining a null net energy exchanged by the storage device during T . The assumption of zero resistances implies that the voltage at the storage device’s terminal and generators’ terminal is equal Esto = E gen . According to the theory of n L load busses; n S storage busses. ) Thus, nL + nG + nS = n , with n the number of internal busses of the network. The busses are considered in the following order: {1, …, nG , nG +1, … nG +nL , nL +nG +1, …, n} . The loads are described in terms of their power profiles, assumed to be repetitive signals over a certain time period T and independent of the respective bus voltages. The network can then be described in a compact way as (for the sake of simplicity, the time dependence is not evidenced): the calculus of variations, the following system to be solved is derived from the minimization problem (1)(ޤ2): 2 º · k 2' ¸ I j (t ) + J sto (t ) + λ J sto (t ) »» = 0 ¸ 2 j =1 ¹ »¼ (3) ng ¦ T sto (t ) dt 1 T 2000 is the current of the jth generator (j=1,…ng) at time t , E sto is the voltage at the storage’s terminal, k1 and k 2 are the weights of the two terms of the objective function (1): k1 is related to the additional cost due to variable ³J ¦ T ng The solution of this equation can be graphically interpreted as shown in Fig. 2. where T is the considered time period, J sto (t ) is the current of the storage device at time t (for the sake of clearness, only one storage device is considered), J j (t ) ª d « §¨ k1 J sto (t ) − dI c « ¨ «¬ © ¦ J j (t * ) − j =1 0 ( 1 J j (t ) − T j =1 * T = E sto J sto (t ) dt = 0 ³ ng The solution of this equation can be graphically interpreted as shown in Fig. 2, where it has been assumed T 0 0 j =1 k 2' signJ sto (t ) (4) 2k1 (5) s.t. T ³¦ J j (ξ )dξ − * Jal(A) 2 · k J j (t ) ¸ dt + 2 ¸ 2 j =1 ¹ T ng Jsto(A) § min k1 ¨ J sto (t ) − ¨ 0© 1 T By rewriting (4) for an assigned t = t * as: Two hypotheses are made in this subsection: (i) the cost of storage system are expressed in terms of unit storage capacity of the single charge/discharge cycle; and (ii) network power losses are neglected. By keeping this last on mind, the whole system can be regarded as a common bus to which generators, loads and storage devices are connected. The objective function is given in terms of a weighted sum of the cost to be sustained for purchasing the storage device and the squared deviation of the currents of storage devices and generators. In mathematical terms, the problem can then be formulated as follows: T ¦ J j (t ) − =0 0 763 T ªJ G º ªEº « J » = G «V » , ¬ ¼ ¬ ¼ min J T RJ dt (6) ³ where E is the vector of voltages at the generators busses, J G is the vector of the injected currents at the generator buses, G is the network’s conductance matrix, V and J are the vectors of voltages and injected currents, respectively, at the remaining busses of the network. By partitioning G , the equivalent representation of (6) can be derived: ªJ G º ªG 00 G 0E º ª E º »« » « J » = «G ¬ ¼ ¬ E0 G EE ¼ ¬V¼ s.t. T ³ diag(E P = diag(J)V (7) (8) d T T ° dJ (J RJ + Ȝ J sto ) = 0 ° sto ® T ° diag(Esto )J sto dt = 0 ° ¯ 0 ³ (9) Ploss = E J G + V J d T T ° dJ (J RJ + Ȝ J sto ) = 0 ° sto T ® ° J sto dt = 0 ° 0 ¯ (10) By taking into account (7), the right hand side of (10) can ³ be rewritten as −1 −1 ETG 00E + ETG 0EG EE J − ETG 0EG EE G E0E + −1 −1 J TG EE J − ETG TE0G EE J (11) Since G 0E = G TE0 , (11) reads: E G 00 E − E T T −1 −1 G 0EG EE G E0 E + J TG EE J (12) generators busses are the same: (13) (19) (20) Since J load , that is the vector of injected currents at the load busses, does not dependent on J sto , the previous expression becomes: (21) where I is the identity matrix. By transposing (21), a suitable equivalent formulation is derived: (14) ª J* º 2[0 I ]R J + Ȝ = 2[0 I ]R « »+Ȝ= «¬ J sto »¼ ª J* º 2[R 1 R 2 ]« »+Ȝ=0 ¬« J sto ¼» thus implying that −1 Ploss = J T G EE J dJ + ȜT = 0 dJ sto ª0º 2J T R « » + Ȝ T = 0 ¬I ¼ with 1 is the vector of all ones, it can be easily deduced that: −1 E TG 00 E − E TG 0EG EE G E0 E = 0 (18) From the matrix differentiation theory, it follows that the solution of the system (19) can be derived by solving the equation: 2J TR It is worth to note that, in case the voltages at all E = E1 , (17) Under the hypothesis of constant voltage at the storages’ busses, the equation system can be rewritten as: By observing that network power losses can be obtained as: T ) J sto dt = 0 , −1 with R = G EE , E sto and J sto the n S -vector of voltages and injected currents, respectively, at the storage busses. In the case hypothesis (13) is not satisfied, further power losses have to be considered as consequence of circulating currents due to the different voltages at the generators’ terminals. The isoperimetric constraint (17) allows obtaining a null net energy exchanged by the storage device during the considered time T (e.g., the day). The solution of this isoperimetric problem is provided by the solution of the system: with: 0 0º ª J nG +1 « 0 J nG +2 0 »» diag(J ) = « « 0» » « 0 0 Jn ¼ ¬ 0 sto 0 where the matrix G EE is clearly not singular. The vector of the injected powers at all the busses P can be derived from J and V as: T (16) 0 (15) Under this hypothesis, the optimization problem that is aimed to minimize power losses, can then be formalized as follows: (22) where J * = J load . The elements of R 1 and R 2 can be 764 easily derived as: R 1i , j = R(nL +i , j ) i = 1,..., nS , j = 1,..., nL (23) R 2i , j = R(nL +i ,nL + j ) i = 1,..., nS , j = 1,..., nS (24) DC ship network Jsto Thus (22) becomes: 2R1 J* + 2R 2 J sto + Ȝ = 0 Vsto (25) DC Once the Lagrange multipliers are evaluated, the optimal storage current profiles which realizes the minimization of the network power losses can finally be derived as: DC a) T · §1 J sto = R −21R1 ¨ J*dt − J* ¸ ¸ ¨T ¹ © 0 ³ Vsc Once having determined J sto , the actual voltages at the load and storage busses can be easily derived. Furthermore, the estimation of injected currents at the generator busses are given by: ( ) −1 J G = G 00 − G 0EG EE G E0 E + −1 G 0EG EE ª § T ·º «J * , R 2−1 R 1 ¨ 1 J *dt − J * ¸» ¨T ¸» « © 0 ¹¼ ¬ DC ship network T ³ (27) Vsto Jsto DC Based on the required current profiles at the storage busses, the size of the storage capacity can be easily derived based on the considered technology. Details on economic implications can be found in [6], where the problem of finding the capacity of the storage device is discussed. AC b) AC DC supercap IV. SUPERCAPACITOR OPTIMAL DESIGN dVsc dt ESS Vsc In this section, the optimal sizing of supercapacitors in DC networks is discussed. The procedure is able to select the size of a suparcapacitor able to guarantee the current profile evaluated by (26). In Fig. 3, two possible configurations of a supercapacitor is showed: to connect the storage device to the network an interfacing DC/DC converter (Fig. 3.a) or AC/DC converters coupled with a transformer (Fig. 3.b) can be used. In both case of Fig. 3, it is assumed that the supercapacitor operates at a specific voltage (Vsc) which is different from the voltage value of the bus where the ESS is connected (Vsto). The optimal sizing of the supercapacitor consists in evaluating the value of the capacitance which is able to provide the desired current profile: J sup = C (Vsc ) ESS supercap (26) Fig. 3 Possible configurations of the connection of supercapacitors to the DC network. A widely accepted model of the capacitance is based on a simplified first order equivalent circuit which includes a constant term (C0 ) and a term linearly variable with the voltage (C1 ) [9], [10], which reads: J sup = (C 0 + 2C1Vsc ) dVsc dt (29) Under the hypothesis of ideal conversion, the following relationship applies: (28) (C0 + 2C1Vsc ) dVsc where J sup is the supercapacitor’s current, Vsc is the dt supercapacitor’s voltage and C (Vsc ) is the capacitance value; the latter depending on the value assumed by the supercapactor’s voltage. which can be rewritten as: = Vsto J sto Vsc 1 2 C0 dVsc 2 + C1dVsc 3 = Vsc J sto dt 2 3 (30) (31) By integrating (31), the values of the terms of 765 capacitance and voltage can be expressed in terms of the energy stored in the supercapacitor at the generic time t. In particular, two given times can be selected during the time period T: the first refers to the maximum energy stored in the supercapacitor; the latter refers to the minimum energy stored in the supercapacitor: ( ) ( ) C0 2 2 V0 − Vsc2 ,min + C1 V03 − Vsc3 , min = 2 3 ³ ( ) ( ) ³ 3 0 where V0 is the initial value of the voltage, Vsc, min is the ) ) ( ) (34) ( ) ( ) (35) C0 2 1 C0 V0 − Vsc2 , max + V03 − Vsc3 , max = K min 2 6 Vsc , rated t K max = max Vsto J stodt t 0 t K min = min Vsto J sto dt t ³ (V 2 0 − Vsc2 , max ) 2 K min 1 1 + V03 − Vsc3 , max 3 Vsc , max ( ) (39) 0 V. DISCUSSION Eqs. (34) and (35) consist in a third order system where the values of C 0 and V0 are unknown. In practical cases, it is assumed that the maximum value of the voltage is the rated voltage of the supercapacitor and the minimum voltage is that corresponding to a specified percentage of the rated voltage, being its value corresponding to the specified maximum value of the depth of discharge. As is well known, this type of equations has a real solution. In this case, the solutions can be derived from a third order algebraic equation obtained by the ratio of (34) and (35): The solution of the design problem has been obtained by considering the cost coefficients as variable parameters at the aim to make it usable by the designers, who may choice the most proper value of size for the storage . This allows also to perform a sensitivity analysis which could be also employed to derive off-line control trajectories which can be successively employed for realtime control, for which suitable stability margins and robustness are required. The sensitivity analysis is particularly simple due to the linearity of the obtained solution: it is expected a behavior close to the absolute optimum referring to the rated parameters. The randomness of the various parameters related to the load uncertainties can be characterized by describing them by means of probabilistic density functions. At this regard, it has to be noted that the assumption of multivariate Gaussian distribution is widely performed in the technical literature. Then, it can be observed that, as a consequence of the linearity of the solution, a Gaussian multivariate distribution results for the storage current profiles. Therefore, a risk-based design can be afforded in a correct way, by ensuring the required robustness against 1 § 3 27 3 § 2 9 2 · 1 · Vsc , rated ¸ ¨V0 − Vsc ,rated ¸ + ¨V0 − 25 125 © ¹ 3 Vsc ,rated © ¹ = ξ (36) 1 1 2 2 3 3 V0 − Vsc, rated + V0 − Vsc, rated 3 Vsc, rated ( (38) 3 Finally, the value of the rated value of the capacitance is derived as C = 3 / 2C0 . Based on the capacitance value, the number of supercapacitors’ elements can be found, as well as the size of the interfacing converter and/or transformer depends on the voltage levels of both supercapacitor and network bus. Moreover, a correct sizing of an ESS should imply the consideration of the uncertainties in managing both loads and generators, the economic aspects related to the cost sustained for the installation and operation of ESSs and the benefits obtained in terms of losses reduction. However, this is out of the scope of this paper. where ³ 2 Vsc3 ,rated § 162 · ¸. ¨ 4ξ − (1 − ξ ) © 125 ¹ C0 = to the discharging stage. In most of the practical cases, the suitable approximation C1 = C0 / 4Vsc, max [11] is ( 2 3 Once V0 is evaluated, C 0 can be derived as: voltage corresponding to the charging stage of the supercapacitor and Vsc, max is the voltage corresponding considered. Thus, (32) and (33) read: C0 2 1 C0 V03 − Vsc3 , min = K max V0 − Vsc2 , min + 2 6 Vsc , rated 3 where α = 3V sc , rated β= ( (37) β· β · §α · α § −α § −α − ¸− ¨ − ¸ −¨ ¸ − ¨ 27 2 27 2 9 3 © ¹ © ¹ © ¹ 3 (33) t min Vsto J sto dt t ∈ [0, T ] t 2 § −α3 β · §α2 · § −α3 β · − ¸ −¨ ¸ + V0 = 3 ¨ − ¸+ ¨ 2¹ 2¹ © 9 ¹ © 27 © 27 0 C0 2 2 V0 − Vsc2 , max + C1 V03 − Vsc3 , max = 2 3 1 162 · § Vsc3 ,rated ¨ 4ξ − ¸=0 125 ¹ (1 − ξ ) © whose solution is: (32) t max Vsto J sto dt t ∈ [0, T ] t V03 + 3Vsc,rated V02 + ) ( ) where Vsc,max = Vsc,rated , Vsc, min = 3 / 5 Vsc,rated (i.e., it is assumed that the maximum voltage is assumed to be the rated voltage and its minimum value is assumed 60% of the rated value) and ξ = K max / K min . Eq. (36) depends only on the unknown variable V0 and can be rewritten in the following suitable expression: 766 [3] Z. Jin, G. Sulligoi, R. Cuzner, L. Meng, J. C. Vasquez and J. M. 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Funaki, "Evaluating energy storage efficiency by modeling the voltage and temperature dependency in EDLC electrical characteristics," IEEE Trans. on Power Electronics, vol. 25, issue 5, 2011, pp. 1231–1239 the loads uncertainties. This is particularly important in case of ship applications where complexities related to the survivability in case of system failure have to be handled through effective and accurate approaches. VI. CONCLUSION The complexity of the on board power networks of modern ship requires the use of new paradigms of the planning and control of loads and generators. At this aim, in this paper efforts have been made in order to discuss on the usefulness of analytical approaches for the optimal sizing of storage devices used in the context of the all electric ship. Based on some requirements particularly debated in literature, which are aimed at reducing power losses and lowering the power supplied by the on board generator, two different optimization problems are formulated and analytically solved. Some details and hypotheses are discussed and the implications analyzed with reference to the use of different storage technologies. With reference to the supercapacitors, which can be used in the case of severe pulsed loads, an analytical sizing approach for the capacity determination is also detailed. Future research efforts will be devoted at applying these analytical methods to real case study where the use of methods based on numerical solution is discouraged by the complexities of the system. REFERENCES [1] N. Doerry, "Naval Power Systems: Integrated power systems for the continuity of the electrical power supply," IEEE Electrification Magazine, vol. 3, no. 2, June 2015, pp. 12-21. [2] R. E. Herbner, K. Davey, J. Herbst, D. Hall, J. Hahne, D. 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