On Optimizing Sensor Placement for Spatio-Temporal Temperature Estimation in Large Battery Packs

On Optimizing Sensor Placement for Spatio-Temporal Temperature Estimation in Large Battery Packs Philipp Wolf 1, Scott Moura 2 an Miroslav Krstic 2 Abstract This paper optimizes the number of sensors an their locations for estimating the 2-D spatio-temporal temperature ynamics in large battery packs. Monitoring temperature in battery packs is crucial for safety, efficiency, an longterm enurance. The temperature ynamics in large battery packs evolve in space an time, whereas sensors only provie pointwise ata. Moreover, the number of sensors shoul be minimize to reuce costs. The temperature ynamics are moele by a system of linear two-imensional heat partial ifferential equations (PDEs). In this paper we perform eigenecomposition of the PDEs to prouce a finite-imensional moel. Moal observability is efine from the magnitue of these eigenmoes. The process of optimizing the number an location of sensors involves two steps: First, a binary optimization minimizes the number of sensors. Secon, a constraine nonlinear programming problem is solve to optimize the previously foun sets with respect to a min-max-type objective function. The optimization proceure is inepenent of the estimator esign. I. INTRODUCTION This paper presents a systematic metho to optimally place a minimally-size sensor set for estimation of battery pack thermal ynamics. Monitoring an controlling battery pack temperature ynamics is a critically important problem. Large-scale evices, such as electric vehicles an gri-scale energy storage, require high energy an power. Consequently, battery packs are often constructe with thousans of cells. One important challenge with such battery packs is that an iniviual cell may become thermally unstable. The heat generate from this unstable cell can transfer to neighboring cells, thereby causing a chain reaction commonly calle thermal runaway. Even when cells o not enter this unstable region, high temperatures can accelerate egraation mechanisms an low temperatures limit available power/energy [1]. Consequently, accurately monitoring an regulating battery pack temperature is a crucial component of battery management systems [2], [3], [4]. CFD moeling, moel reuction, an analysis of battery pack thermal ynamics is currently an active research area [5], [6], [7], [8]. Intelligent sensor placement is the next step towars avance thermal management. Nonetheless, sensor placement in istribute parameter systems bears several challenges. First, placing multiple sensors is NP- This work was supporte by the German Acaemic Exchange Service (DAAD) an the National Science Founation 1 P. Wolf is with the Institute for System Dynamics, Faculty 7, University of Stuttgart, 70550 Stuttgart, Germany 2 S. Moura an M. Krstic are with the Department of Mechanical an Aerospace Engineering, University of California, San Diego, CA 92093, USA (email: smoura@ucs.eu; krstic@ucs.eu) har [9]. The current work aresses this issue. Secon, the optimization problem is non-convex, in general. The engineering literature on optimal sensor placement examines various optimization criteria. These can be classifie into open-loop an close-loop consierations. Open-loop methos often focus on observability metrics, e.g. the spatial H 2 norm [10], [11]. Close-loop methos simultaneously esign the observer gains an sensor locations [12], [13], an consier metrics such as estimation error regulation, isturbance rejection, an robustness [14]. The survey paper [15] provies an excellent analysis of various aspects of actuator/sensor placement. The current work focuses on placement of a minimally size sensor set, with respect to a particular observability metric, inspire by eigenanalysis. We focus on a 2D PDE moel for battery pack thermal ynamics recently propose in [16]. Our main iea is to erive a finite-imensional moal moel from the aforementione PDE moel via eigenanalysis. The moal setting provies a natural efinition for an observability measure we use for sensor placement. In aition, we focus on computing the minimal sensor set to achieve a given observability criterion. This problem is solve by capitalizing on a submoularity property [17], which substantially reuces computational effort. Finally, the sensor placement problem is mathematically cast as a nonlinear optimization problem. In Section II, the moal moel is erive from the PDE moel. Next, a systematic metho for optimal sensor placement is presente in Section III, followe by some exemplary results in Section IV. The paper s main ieas are summarize in Section V. II. MODELLING We use the PDE moel evelope in [16]. The two spatial imensions are enote x an y, with pack bounaries at x = ±L x an y = ±L y, so that the battery pack resembles a rectangular shape within a rectangular omain Ω = [ L x, L x ] [ L y, L y ]. (1) Time is assume to run from t = 0 to t =. A. PDE moel The moel for temperature istribution in the battery pack has three states, Θ, T h, T c, which respectively represent temperature in the pack material, cells, an cooling channels.

Fig. 1. Illustration of the PDE approximation for battery packs with a large number of cells. Cells are illustrate in re, cooling channels in blue, an pack material in white. Note that states Θ, T c, T h exist over the entire omain Ω in the final PDE moel. It is mathematically given by t Θ(x, y, t) = a Θ(x, y, t) + b [T h(x, y, t) Θ(x, y, t)] + c [T c (x, y, t) Θ(x, y, t)], (2) t T h(x, y, t) = [Θ(x, y, t) T h (x, y, t)] + eφ h (x, y, t), t T c(x, y, t) = f [Θ(x, y, t) T c (x, y, t)] gφ c (x, y, t), where = 2 x + 2 2 y 2 are constant coefficients, is the Laplacian operator, (3) (4) a, b, c,, e, f, g R, (5) Θ(x, y, 0), T h (x, y, 0), an T c (x, y, 0), (6) enote the initial conitions, an Θ(x, y, t) x = λ Θ(x, y, t) x=±lx k p (7) x=±lx Θ(x, y, t) y = λ Θ(x, y, t) y=±ly k p (8) y=±ly enote the Robin-type bounary conitions. The variables Φ h an Φ c enote the heat generate by the cells an the heat taken by the cooling system, respectively. The PDE moel was erive in [16] by iviing the battery pack omain into subomains, each of which containe either a cell, cooling channel, or the pack material. Then the imensions of these subomains were taken to zero. This represents a large number of cells (see Figure 1). Please note that we have moifie the moel by subtracting the ambient temperature from each state, thus maintaining the structure of the PDE system. This positions us to perform an eigenanalysis an erive the moal moel, escribe next. B. Moal moel We achieve the moal version of the PDEs by calculating an orthonormal basis of eigenmoes over the omain Ω an projecting the PDEs onto this basis. This process prouces an infinite set of ODEs, which we summarize next (for more etails please refer to [18]). Assuming Θ(x, y, t) = µ Θ (t)z(x, y), the homogeneous part of (2) becomes t µ Θ + b + c = 2k = Z aµ Θ a Z with the separation constant 2k. We separate this into two inepenent ODEs governing the temporal an spatial parts of the solution as t µ Θ(t) = (2ka + b + c)µ Θ (t), (10) Z(x, y) = 2kZ(x, y). (11) The solution of (10) is given by (9) µ Θ (t) = µ Θ (t)e (2ka+b+c)t. (12) For the spatial part, we assume Z(x, y) = X(x)Y (y) an (11) can be arrange as 2 2 x X 2 X + k = h = y Y 2 k. (13) Y which is separate into two Sturm-Liouville equations with the bounary conitions X + (k h)x = 0, (14) Y + (k + h)y = 0 (15) X (±L x ) = λ k p X(±L x ), (16) Y (±L y ) = λ k p Y (±L y ), (17) erive from (7) an (8). The solution of (14), (16) for X is an infinite set (i N) of eigenfunctions { cos(γxi x) for o i X i (x) = (18) sin(γ xi x) for even i where γ x = k h solves for the cosine eigenfunctions an tan(γ x L x ) = λ k p 1 γ x (19) tan(γ x L x ) = k p λ γ x (20) for the sine eigenfunctions. We get a structurally ientical solution for Y an in combination, these sets form the eigenmoes for (2), consisting of the moe shapes Z ij (x, y) = X i (x)y j (y), i, j = 1,..., with corresponing separation constants k ij = (γ 2 xi + γ2 yj )/2. Note from (12) that the eigenvalues, given by (2k ij a + b + c), are real, negative, an increase in an approximately quaratic fashion towars as the inices i, j increase. The eigenmoes are L 2 -orthogonal [18], [19] an are normalize by pre-multiplying by the constant K ij = 1 Ω Z2 ij Ω, (21)

forming an orthonormal basis K ij Z ij, i, j = 1,.., for (2). Equations (3) an (4) are ODEs in time, parameterize over space. Since they are couple with (2), their solutions inherit the same spatial structure. For this reason, an the fact that the eigenmoes provie an orthogonal basis, the whole system of PDEs is projecte onto the eigenmoes. Thus, we obtain an infinite set of systems of ODEs (n = 1,..., ), t µ Θn = (2ak n + b + c) µ Θn + bµ Th n + cµ Tcn (22) t µ T h n = µ Th n + µ Θn + ek n Φ h Z n Ω (23) Ω t µ T cn = fµ Tcn + fµ Θn + gk n Φ c Z n Ω (24) Ω where n inexes the Cartesian prouct of i = 1,..., an j = 1,...,. The moal coefficients µ Θn, µ Th n an µ Tcn characterize the temporal part of the solution. A finite-imensional ODE moel is obtaine by taking a finite subset of (22) - (24), containing only the first N eigenmoes. This truncation is justifie by the fact that the moel is stable an the eigenvalues increase approximately quaratically towars, as the inex number increases. III. SENSOR PALCEMENT Next we investigate optimal in-omain placement of temperature sensors for a moal moel-base estimator. Sensor placement, in general, has many ifferent aspects, which inclue, but are not limite to observability, robustness, an noise/isturbance rejection. We choose one of these aspects to explicitly account for in our optimization: the moal observability. In view of this requirement, the task is two-fol. First, etermine the minimal number of sensors. Secon, place the sensors over the omain in some optimal sense. This problem is, in general, an NP-har, constraine, mixeinteger, nonlinear program. For the approach presente here, the problem is split into two steps: First, we efine a fixe minimum threshol for the observability an a gri on the omain. On the gri points, we perform a binary integer search, minimizing the number of sensors an fining sets that satisfy the minimum observability requirement. In the secon step, we take all sensor sets from the first step an optimize their location, now with respect to maximizing the observability in a constraine nonlinear optimization. A. Moal observability measure Moal observability is a well known concept for spatially istribute systems. It can be efine accoring to open-loop specifications (e.g. the spatial H 2 norm [10], [11]), or closeloop specifications (e.g. LQG metrics [14]). We investigate the following efinition of moal observability for the n th eigenmoe at location (x p, y p ), M(n, x p, y p ) := K nz n (x p, y p ) max (x,y) Ω K n Z n (x, y), (25) where M : N [ L x, L x ] [ L y, L y ] R. Equation (25) can be simplifie to M(n, x p, y p ) = Z n (x p, y p ). (26) Fig. 2. Sample visualization of S x(x s) for two sensors. The global optima are enote by black circles. using (18) an (21). Note that M(n, x p, y p ) [0, 1], where M(n, x p, y p ) = 0 signifies no observability an M(n, x p, y p ) = 1 signifies best case observability. Conceptually, the moel observability for an eigenmoe at a particular spatial location is given by the magnitue of the corresponing eigenfunction. This efinition is novel, an is motivate by the iea that one shoul place sensors where the eigenfunction magnitue is large. B. Optimization scheme 1) Separation an Symmetry: The structure of the moal moel contains the separation property, which consequently reuces the complexity of the optimization. We ecompose the moe shapes of the N eigenmoes into their constituting eigenfunctions Z n (x, y) = X i (x)y j (y) an from this point on, we work on the x imension only. The proceure for the y imension involves ientical steps. At the en, the results are recombine. The moal observability measure (26) also separates in two parts M x (i, x p ) := X i (x p ) an M y (j, y p ) := Y j (y p ), (27) since M(n, x p, y p ) = M x (i, x p )M y (j, y p ). Aitionally, we reuce the optimization to the positive half of the x subomain (the interval (0, L x )) to avoi multiple isomorphic solutions since the moe shapes are symmetric with respect to the origin, in terms of moal observability. 2) Objective Function Analysis: The nonlinear program, escribe in Section III-B.4, optimizes the set, x s, of sensor locations on the continuous interval (0, L x ) with respect to maximizing the lowest moal observability over all eigenfunctions. Mathematically, the objective function is efine as S x (x s ) = min i {1,...,N} { max x p x s M x (i, x p ) }. (28)

Conceptually, each moe is assigne the sensor which is locate at the greatest eigenfunction magnitue. Then the objective function value, S x (x s ), gives us the lowest moel observability over the set of moes. Our goal is to maximize the worst case moal observability. Figure 2 provies a sample visualization of S x for two sensors in the x subomain. This problem is rile with multiple local optima. Multiple global optima exist (enote by circles), an we wish to ientify these sensor locations - for two or more sensors. This motivates the following metho for selecting optimization starting points. First, we seek to etermine the minimal number of sensors require to achieve a given level of observability. Secon, we optimize the location of this minimal set of sensors. 3) Minimal Number of Sensors: The first part is formulate as a combinatorial problem. Thus, we establish a binary integer approximation by imposing a gri of Q points on (0, L x ). For every eigenfunction X i, we efine a binary observability criterion for a hypothetical sensor at the gripoint x q, q {1,..., Q} with respect to a moal observability threshol τ int B x (i, q) := x : { 1 for Mx (i, x q ) τ int x 0 else. (29) The value for τx int is chosen a priori. Since this value epens on the structure of the involve eigenfunctions, it must be chosen appropriately. The gri structure requires careful consieration to trae off accuracy with complexity. Specifically, the eigenfunction with the greatest spatial frequency provies the appropriate inicator for the gri imensions. It has feasible regions where its observability is above the threshol τx int. We use an equal-istance gri with a istance small enough to cover each of its feasible regions by at least two gri points. This is a rather conservative choice, but in our tests turne out to be a goo balance between coverage an problem size. One reason is that the optimal sensor locations ten to be at the bounaries of the feasible regions. With this setting, there is at least one gri point less than half of the region away from every bounary for all eigenfunctions. The binary observability information can be organize in matrix form B = [B x (i, q)] i,q i = 1,..., N q = 1,..., Q, (30) an this matrix is the input to the search algorithm. Our immeiate goal is to select a set of sensor locations with minimal carinality, such that for each moe i, there exists at least one sensor location q in this set where B x (i, q) = 1. The brute-force metho to solve this problem is full enumeration. This metho woul fin a complete set (i.e. a set of sensors covering all eigenfunctions), establishing the minimal number of sensors. Since there are likely multiple locations that satisfy the minimum observability criterion, all combinations must be evaluate. However, this metho is computationally expensive an oes not take avantage of the problem s submoularity, a critical property for our metho. TABLE I COMPARISON OF ITERATIONS PERFORMED BY FULL ENUMERATION AND MODIFIED GREEDY ALGORITHM τx int Full Enumeration Moifie Greey Reuction Factor 0.30 820 245 3.3 0.65 32567 1796 18.1 0.82 4912381 15843 310 A problem is submoular, if it has a iminishing return property. In our case we choose one gri point after another to buil a set of sensor locations. Each gri point x q, q 1,..., Q satisfies (29) for a fixe number of eigenfunctions that it can contribute to the set of covere eigenfuntions. The later we a a gri point to a sensor location set, the less aitional eigenfunctions it will contribute. This is because more an more eigenfunctions are alreay covere by previously chosen sensor locations. Base on submoularity, we have esigne a moifie greey algorithm (for etails please refer to [18]) that spans a much smaller tree than the complete combinatorial tree, while still covering all branches of interest. This is crucial for achieving a practical algorithm for placing multiple sensors. Full enumeration an the moifie greey algorithm both return the minimal number of sensors P, an the corresponing sets of gri points that satisfy the binary observability criterion. 4) Sensor Location Optimization: In the secon step, the sets of sensors from the first part are use as starting points x s0 for the constraine nonlinear programming problem (NLP) we escribe next. The NLP optimizes the sensor locations x s on the continuous interval (0, L x ) with respect to maximizing the lowest moal observability over all eigenfunctions, S x (x s ) efine in (28). The optimization problem is thus formulate as max S(x s ), (31) x s (0,L x) P s. to x s,1 x s,2 x s,p, (32) for every set of starting points x s0. Constraint (32) prevents permutations in the sensor orer. Note that multiple starting points may converge to ientical optima. This optimization problem can be solve with graient-base algorithms, such as sequential quaratic programming. IV. EXAMPLE RESULTS To illustrate some results, we assume a battery pack with imensions efine by L x = L y = 10. We set the ratio between the thermal conuctivity insie the pack material an the heat transfer rate from pack material to surrouning air to λ k p = 20, (33) corresponing to moerate heat conuctivity insie the pack in combination with high heat transfer over the bounaries.

Fig. 3. Example: The first 31 eigenfunctions X i, for the x imension. Green an blue correspon to sine an cosine eigenfunctions, respectively. Fig. 4. Optimal locations for (a) two, (b) three, an (c.1) four sensors. For each number of sensors, multiple optima with ientical objectives S x(x s) exist. The sensor locations in (c.2) are suboptimal, yet exhibit more equally istance spacing after reflecting two sensors onto the left-half of the subomain. The finite subset of eigenmoes is efine by selecting the banwith of interest. Since the eigenvalues ecrease quaratically towars, we truncate the high frequency moes an retain the ominant low frequency moes. In this example, 185 eigenmoes are retaine. This example set of eigenmoes is ecompose into the eigenfunctions. Figure 3 shows the first 31 eigenfunctions for the x imension. Table I shows a comparison of the require iterations for the moifie greey algorithm an full enumeration. In this example, capitalizing on the submoularity property provies a reuction of up to two orers of magnitue. Figure 4(a)-(c) shows the resulting optimal sensor sets. In each case, multiple sensor sets are shown because multiple equivalent optima were ientifie. The sets of two sensors in Figure 4(a) where obtaine by setting τx int = 0.3 for the integer optimization. The maximum observability following the nonlinear optimization is S x (x s) = 0.5. One may see that the NLP successfully ientifies the global optimain shown in Fig. 2,. Setting τx int = 0.65 requires one aitional sensor. The optimal locations for several sets of three sensors are provie in Figure 4(b). These sensor sets achieve S x (x s) = 0.77. Further increasing the initial τx int = 0.82 provokes a fourth sensor an increases the observability objective to S x (x s) = 0.88, as shown in Figure 4(c.1). Note that the total observability for the eigenmoes is S(x s, y s) = S x (x s)s y (y s). Therefore, S(x s, y s) = 0.25 if S x (x s) = S y (y s) = 0.5, an S(x s, y s) = 0.76 for S x (x s) = S y (y s) = 0.87. A. Implementation Issues Choosing one of the optimal sensor sets from Fig. 4 is base on the requirements of the application. For estimating temperatures in large battery packs, one critical aspect is the worst case time elay between a local failure an a sensor, ue to thermal conuctivity limits. To minimize this effect, we woul prefer sets with the most equally space istribution. Another aspect is the way the sensor set is moifie to cover the right half of the x subomain. One choice is to uplicate a sensor set by mirroring at the origin, proviing two reunant sensors for each eigenmoe while accepting a lower minimal moal observability. For example, a total of four sensors woul be achieve by mirroring the first location set from Figure 4(a). The other option is to take a sensor set from the results with more sensors an a higher minimal moal observability an completely move half of the sensors to corresponing locations on the other half of the subomain. Figure 4(c.2) illustrates such an example. This set has a slightly lower minimal moal observability (0.87 vs. S x (x s) = 0.88), but is the most equally istance set for the case of moving two sensors to the left half of the x subomain. The key ifference is the former esign choice provies sensor reunancy whereas the latter provies higher minimal moal observability. V. CONCLUSION This paper examines sensor placement for estimation of battery pack thermal ynamics. Specifically, we seek the

location of a minimal set of sensors that optimize a particular observability criterion. The critical ieas can be summarize into four points. First, we erive a finite-imensional moal moel via eigenanalysis. Secon, we quantify the observability of an eigenmoe at a given spatial location by the eigenmoe s value. Thir, we etermine the minimal set of sensors which all satisfy a particular observability criterion. This proceure capitalizes on the submoularity property of the optimization problem. Finally, we optimize the locations of this minimal sensor set via a nonlinear program. This paper provies a systematic methoology for placing temperature sensors in large battery packs for estimating thermal ynamics. ACKNOWLEDGMENT P. Wolf thanks Prof. O. Sawony for proviing support uring his tenure as a visiting scholar at UC San Diego. REFERENCES [1] K. Smith an C.-Y. Wang, Power an thermal characterization of a lithium-ion battery pack for hybri-electric vehicles, Journal of Power Sources, vol. 160, no. 1, pp. 662 673, 2006. [2] S. J. Moura, N. Chaturvei, an M. Krstic, PDE Estimation Techniques for Avance Battery Management Systems - Part I: SOC Estimation, in Proceeings of the 2012 American Control Conference, Montreal, Canaa, June 2012. [3], PDE Estimation Techniques for Avance Battery Management Systems - Part II: SOH Ientification, in Proceeings of the 2012 American Control Conference, Montreal, Canaa, June 2012. [4] S. J. Moura, J. L. Stein, an H. K. Fathy, Battery Health-Conscious Power Management for Plug-in Hybri Electric Vehicles via Stochastic Control, vol. 1, Cambrige, MA, Unite states, 2010, pp. 615 624. [5] C. Park an A. Jaura, Dynamic thermal moel of li-ion battery for preictive behavior in hybri an fuel cell vehicles, SAE transactions, vol. 112, no. 3, pp. 1835 1842, 2003. [6] M. Guo an R. E. White, Thermal moel for lithium ion battery pack with mixe parallel an series configuration, Journal of the Electrochemical Society, vol. 158, no. 10, pp. A1166 A1176, 2011. [7] X. Hu, S. Lin, S. Stanton, an W. Lian, A state space thermal moel for hev/ev battery moeling, SAE 2011 Worl Congress an Exhibition, 2011. [8] H. Sun, X. Wang, B. Tossan, an R. Dixon, Three-imensional thermal moeling of a lithium-ion battery pack, Journal of Power Sources, 2012. [9] D. Uciński, Optimal measurement methos for istribute parameter system ientification. CRC, 2005. [10] D. Halim an S. Reza Moheimani, An optimization approach to optimal placement of collocate piezoelectric actuators an sensors on a thin plate, Mechatronics, vol. 13, no. 1, pp. 27 47, 2003. [11] A. Armaou an M. A. Demetriou, Optimal actuator/sensor placement for linear parabolic PDEs using spatial H2 norm, Chemical Engineering Science, vol. 61, no. 22, pp. 7351 7367, 2006. [12] Y. Lou an P. D. Christofies, Optimal actuator/sensor placement for nonlinear control of the kuramoto-sivashinsky equation, IEEE Transactions on Control Systems Technology, vol. 11, no. 5, pp. 737 745, 2003. [13] K. K. Chen an C. W. Rowley, H2 optimal actuator an sensor placement in the linearise complex Ginzburg-Lanau system, Journal of Flui Mechanics, vol. 681, pp. 241 260, 2011. [14] J. Borggaar, J. Burns, an L. Zietsman, On using lqg performance metrics for sensor placement, in Proc. American Control Conf. (ACC), 2011, pp. 2381 2386. [15] C. Kubrusly an H. Malebranche, Sensors an controllers location in istribute systems a survey, Automatica, vol. 21, no. 2, pp. 117 128, 1985. [16] A. Smyshlyaev, M. Krstic, N. Chaturvei, J. Ahme, an A. Kojic, PDE Moel for Thermal Dynamics of a Large Li-ion Battery Pack, in Proc. American Control Conf. (ACC), 2011, pp. 959 964. [17] A. Krause, J. Leskovec, C. Guestrin, J. VanBriesen, an C. Faloutsos, Efficient sensor placement optimization for securing large water istribution networks, Journal of Water Resources Planning an Management, vol. 134, no. 6, pp. 516 526, 2008. [18] P. Wolf, Spatio-temporal temperature estimation for large battery packs, Diploma Thesis, Institute for System Dynamics, University of Stuttgart, 2012. [19] G. Cain an G. Meyer, Separation of variables for partial ifferential equations: an eigenfunction approach. CRC Press, 2006.