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Contract No. DE-AC36-08GO28308

A Multi-Node Thermal System 

Model for Lithium-Ion Battery 

Packs

Preprint

Ying Shi, Kandler Smith, Eric Wood,

and Ahmad Pesaran

Presented at the 2015 American Control Conference

Chicago, Illinois

July 1–3, 2015

Conference Paper

NREL/CP-5400-63898

September 2015
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Abstract— Temperature is one of the main factors that 

control the degradation in lithium ion batteries. Accurate 

knowledge and control of cell temperatures in a pack helps the 

battery management system to maximize cell utilization and 

ensure pack safety and service life. In a pack with arrays of 

cells, a cell's temperature is not only affected by its own 

thermal characteristics but also by its neighbors, the cooling 

system and pack configuration, which increase the noise level 

and the complexity of cell temperature prediction. This work 

proposes to model lithium-ion packs' thermal behavior using a 

multi-node thermal network model, which predicts the cell 

temperatures by zones. The model was parametrized and 

validated using commercial lithium-ion battery packs. 

Index Terms— battery pack thermal management, lithiumion battery, system modeling

I. INTRODUCTION

Hybrid electric vehicles (HEV) bring opportunities to 

reduce gasoline consumption and emission. Widespread 

electric vehicle adoption calls for more robust hybrid systems 

to work in different terrains and climates. This has brought 

more challenges to battery pack management and vehicle 

powertrain control. Among all the environmental variables, 

lithium-ion batteries' performance and service life are heavily 

affected by operating temperatures. Therefore, most lithiumion battery performance models and battery life prognostic 

models are temperature-dependent. A battery pack's thermal 

performance is a result of the cell thermal properties, the 

cooling system configuration, and the pack packaging design. 

For vehicle system engineers, the detailed internal 

temperature map of a production pack is often not available 

due to non-disclosure agreements or the intellectual property 

of the pack manufacturers. Thus, vehicle system control 

engineers are often left to generate the thermal model of 

battery packs for macro-scale hybrid system controller 

design, implementation, simulation and testing based on only 

minimum, maximum, and average cell temperatures in the 

pack.

Computational fluid dynamics (CFD) models are popular 

tools for cell thermal behavior study and pack thermal 

management system design and analysis [1, 2, 3]. They have 

also been applied to safety simulation under abuse conditions

[4]. Those models contain many details of the batteries and 

cooling paths to capture the internal temperature map of a 

lithium-ion cell or pack [5]. 

*This work is supported by DOE Vehicle Technologies and ARPA-e 

AMPED Programs.

The authors are with the National Renewable Energy Laboratory, 

Golden, CO, USA 80401 (e-mail: ying.shi@nrel.gov; 

kandler.smith@nrel.gov; eric.wood@nrel.gov; ahmad.pesaran@nrel.gov).

However, the nonlinearities, high dimensionality, and 

complex battery physics involved in CFD models all 

contribute to the high computation cost of such models. In 

addition, CFD models are closed systems which cannot be 

easily integrated with online controllers that requests realtime access of model internal states. Hence, large-scale 

hybrid powertrain controller design and simulation creates 

the need for high fidelity, low-order, computationally cheap 

and control-oriented system thermal models. A lumped 

thermal model is one suitable candidate [6, 7]. Such models 

have been applied to single cell systems [8] and battery 

modules [9]. Thermal network models were proposed to 

develop a low-order battery pack thermal model that can 

capture the heat transfer of a pack sitting in an HEV [10, 11, 

12].

This work adopted the thermal network model idea to 

model large HEV and plug-in hybrid electric vehicle (PHEV) 

lithium-ion battery pack by partitioning battery packs into 

multiple zones and lumping each zone into a thermal node. 

Three-node and Four-node thermal models were proposed for 

commercial HEV and PHEV packs. The models were 

parameterized and validated with experimental pack thermal 

testing data.

Figure 1. A schematic of the thermal network and the lumped 

thermal model

II. MODEL CONSTRUCTION

The temperature distribution in a battery pack is affected 

by its environment, radiation, cabin temperature, and ambient 

A Multi-Node Thermal System Model for Lithium-Ion Battery 

Packs

Ying Shi, Kandler Smith, Eric Wood, Ahmad Pesaran
2 

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

temperature, as shown in Figure 1. The correlation can be 

modeled as a thermal network, with the battery pack 

represented as a lumped thermal mass and one temperature,

= ( ) ( ) + (1)

= ( ) ( ) (2)

or in the state-space form,

=

+

+ +

0

(3)

However, in a battery pack with a long string of cells, 

instead of the overall average temperature, maximum and 

minimum cell temperatures are more of interests because 

they probably will grow into the worst and the least degraded 

cells, which define the boundaries of pack life estimates. To 

predict maximum and minimum cell temperatures, the battery 

pack can be partitioned into multiple zones. The cells in each 

zone should have little temperature variation; cells in 

different zone should have large temperature variation. Each 

zone can be presented with a lumped thermal mass (called a 

"node" in this work) and heat transfer coefficients. Zone 

division needs to be based on the pack design and thermal 

configuration. A straightforward division can have three 

zones, one containing the cells that have the most efficient 

cooling (most outside or upstream cells), one containing the 

cells that have the least efficient cooling (most inside or 

downstream cells), and one containing the rest of the cells in 

between. This three-node pack thermal model can be 

formulated as:

= ( ) ( ) + (4)

= ( ) ( )

( ) +

(5)

= ( ) ( ) + (6)

where and represent the lumped thermal mass and heat 

transfer coefficients, respectively, of each zone, indicate 

the zone temperature, and is the heat generation in that 

zone. The state-space formation of the model is:

=

+

0

+ +

0

+

+

1

0 0

0

1

0

0 0

1

(7)

= max( , )

min( , ) (8)

In this model, the only nonlinearity comes from the output 

equations.

If a pack is sitting in an isothermal environment, e.g. 

soaked in an environment chamber, the three-node pack 

model (described by Eq.7) will be sufficient. If in a vehicle 

and the pack is configured to use cabin air as the coolant, 

combining the pack model with the external thermal network 

(described by Eq.3) will be necessary. So the overall thermal 

model will be fourth-order with states { , , , }.

III. EXPERIMENT VALIDATION

The three-node pack thermal model was applied to two 

types of lithium-ion battery packs: HEV and PHEV packs. 

Thermal characterization tests were performed to regress the 

model.

A. Experiment Setup

Two HEV packs, from the same make but aged 

differently, were soaked solely in the chamber. One of the 

packs was wrapped with 2-inch thick insulation and fitted 

with the three-node pack thermal model. The PHEV pack 

was in a chamber with another pack and its convection and 

inlet air temperature were affected. Thus, it was fitted with a 

fourth-order pack thermal model with cabin temperature 

considered.

1) Thermal characterization test protocol

The thermal characterization test protocol consists of two 

parts: (1) step change of chamber (ambient) temperature, and 

(2) charge-sustaining (CS) cycles. The step test started by 

soaking the pack at 0o

C, then setting the chamber temperature 

to 30o

C for a day, and then returning the chamber set point 

back to 0o

C. In the CS cycling, the pack was cycled under 10-

second CS pulses at multiple C-rates, 0.5C, 1C, 1.5C, 2C, 

2.5C, and3C. During the CS cycling, the environment 

chamber was set to a fixed temperature of 30o

C, 20oC, and 

0o

C.

From the test, the thermal capacitance and the heat

transfer coefficients were identified for both packs. For CS 

cycling, heat generation rate was estimated from the data as

= + + = ( )

where is estimated from a look-up table of =

( ).

2) Pack thermal system setting

All the packs tested were cooled by forced air from the 

environmental chamber. The HEV packs had the fan on/off 

control as an inaccessible internal state. The fan turned on or 

off in the middle of CS cycling. Thus, all the CS cycling test 

results on HEV packs had both periods of fan on/off. The 

PHEV pack allowed the user to enable/disable cooling, but 

the fan was not turned on until control threshold was met. 

Separate CS cycling tests were conducted to run the pack 

with and without cooling. 

B. Results and Discussion

1) HEV packs with three-node pack thermal model

The HEV packs were of the same make but had different 

aging histories. Nevertheless, the overall thermal mass of the 
3 

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packs was expected to be the same and thus their thermal 

behaviors under step change tests should be very similar. 

Therefore, first, a single-node pack model was adopted to 

find the overall thermal mass of the packs using step change 

test data. The fixed pack total thermal mass was 42,100 / . 

Second, in the three-node model, each node was assigned a 

portion of the total thermal mass. Based on the pack 

configuration, 10% was assigned to the two nodes 

representing the maximum and minimum and 80% was 

assigned to the node representing the rest of the cells. The 

models for each pack were regressed with both the stepchange test and the CS cycling test to identify the thermal 

coefficients. Note that the pack thermal characteristics were 

much altered by the fan, so the model was independently 

fitted for the cases with and without cooling.

Figure 2. Ambient step change test of HEV pack 2: experiment 

data (dash) and model prediction (solid); max. cell 

temperature (red) and min. cell temperature (blue).

Figure 2 plots the experiment results of HEV Pack 1 and 

the pack thermal model prediction for the cell temperatures 

subject to step changes of the chamber temperature.

Figure 3. CS cycling test of HEV pack 1 at 0oC: experiment 

data (dashed lines) and model prediction (solid 

lines); maximum cell temperature (red) and 

minimum cell temperature (blue).

Figures 3 and 4 plot the experiment results and the model 

prediction of HEV Pack 1 subject to CS cycling tests at 

ambient temperatures of 20o

C and 0oC. The top two subplots 

of each figure show the cell temperature measurements and 

prediction. The bottom two subplots are the corresponding 

C-rate and fan status. With the chamber set to 0o

C, Pack 1 

started at 1C and went through 1.5C and 2C CS cycles. The

fan was switched on shortly after the 1.5C CS cycling began 

and stayed on for the rest of the test. In the 20o

C test, the 

pack underwent 0.5C, 1C, 1.5C, 2C, and 1.5C CS cycles. 

The C-rate was reduced because Pack 1 reached the 

maximum temperature. The cooling fan was switched on 

during 1C cycling and stayed on.

Figure 4. CS cycling test of HEV pack 1 at 20oC: experiment 

data (dashed lines) and model prediction (solid 

lines); maximum cell temperature (red) and 

minimum cell temperature (blue).

From the results, the three-node pack model of Pack 1 was 

able to capture the maximum and minimum cell 

temperatures. It followed the transient during step change 

and the temperature swings during CS cycling for most of 

the time.

Table 1 

Fitted Thermal Coefficients for HEV Pack Thermal Models

Pack Fan

( ) (w/K) (w/K) (w/K) (w/K) (w/K)

RMS 

err

(

o

C)

OFF 20 0.074 3e-11 4.2e-5 0.683 0.125 0.988

A 0 0.081 0.124 0.089 0.94 0.274 1.176

ON 20 5.1e-7 1.444 2.823 11.978 1.8e-6 1.884

0 0.658 0.818 4.262 14.358 9.4e-8 1.208

OFF 20 9e-11 0.062 0.289 2.245 1e-14 1.884

B 0 0.087 0.368 0.148 1.539 0.385 1.208

ON 20 0.926 4.24 1.9e-6 13.896 1.789 1.773

0 2.798 1.014 8e-5 31.249 1.975 1.709

The resolution of the pack temperature measurements was 

1o

C. The average root-mean-square (RMS) errors for the 

with-cooling and without-cooling cases are 1.1o

C and 1.6oC, 

respectively. The model was only off by 2o

C at 20oC after 

5.3 hours. This was caused by the missing controller area 

network (CAN) data between 5.3 and 5.5 hours which made 

the model think the pack was at rest and cooled down, but 

the pack was actually not.

Figures 5, 6 and 7 show the experiment results and model 

prediction of HEV Pack 2 under step change and CS cycling. 

Pack 2 went through the similar CS cycling tests as Pack 1, 

only differing at the end of the 20o

C test where the 

maximum cell temperature in Pack 2 was below the 

threshold and therefore allowed the test to go into the higher 

2.5C-rate cycles.
4 

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Figure 5. Ambient step change test of HEV pack 2: experiment 

data (dash) and model prediction (solid); max. cell 

temperature (red) and min. cell temperature (blue).

Figure 6. CS cycling test of HEV pack 2 at 0oC: experiment 

data (dashed lines) and model prediction (solid 

lines); maximum cell temperature (red) and 

minimum cell temperature (blue).

Figure 7. CS cycling test of HEV pack 2 at 20oC: experiment 

data (dashed lines) and model prediction (solid 

lines); maximum cell temperature (red) and 

minimum cell temperature (blue).

Pack 2's thermal model also captured the pack thermal 

performance well. With the fan on and off, the 

corresponding averaged RMS errors were 1.05o

C and 1.5oC, 

respectively. The largest error occurred when the cooling fan 

turned off shortly during 1.5C cycling at 0o

C. Table I lists all 

the fitted parameters and RMS error for the two HEV packs.

2) PHEV pack with fourth-order pack thermal model

A PHEV pack was characterized under the thermal test 

protocol described in Sec. III-A.1. A fourth-order pack 

thermal model with states { , , , } was fitted to the 

data. The model was formed by plugging the states of Eq.7 

into Eq.3. For this pack, a different fitting scheme was 

adopted. First, fix the assigned mass ratio of the three cell 

nodes, was set in this case, to be 12%, 76%, and 12%. 

Second, the thermal coefficient ratio of the three nodes was 

fixed, i.e., = , = , = , 

and = 1. Similarly, the same for , and were 

used. Based on this pack's configuration, the ratios were 

estimated to be 0.24, 0.7, and 0.06. Then the experimental 

data were used to identify , , and . Note that the 

average cell temperature plotted in the following figures was

calculated by averaging measured maximum and minimum 

cell temperatures.

Figure 8. Ambient step change test of PHEV pack: experiment 

data (solid lines) and model prediction (dashed 

lines); maximum cell temperature (blue), average 

temperature (black), minimum cell temperature 

(red), and inlet air temperature (green).

Similar to the step change tests for HEV packs, the PHEV 

pack was soaked at 0o

C, warmed up to 30oC for 24 hours, and 

then cooled down to 0o

C for 24 hours. Figure 8 shows the 

measured cell temperature change along with step changes in 

chamber temperature.

Figure 9. CS cycling test of PHEV pack at 0oC: experiment 

data (solid lines) and model prediction (dashed 

lines); maximum cell temperature (blue), average 

temperature (black), minimum cell temperature 

(red), and inlet air temperature (green).
5 

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Figure 10. CS cycling test of PHEV pack at 30oC: experiment 

data (solid lines) and model prediction (dashed 

lines); max. cell temperature (blue), average 

temperature (black), min. cell temperature (red), 

and inlet air temperature (green).

The stair-like temperature curves were due to the 0.5oC 

resolution for the cell temperature measurements.

In CS cycling tests, the pack was cycled under 1C, 2C, and 

3C currents at 0o

C and 30oC. Figures 9 and 10 show the cell 

temperatures measured and predicted by the model. The 

model tracked the cell temperatures tightly at 0o

C and 30oC.

Judging from the results, the fourth-order pack thermal 

model was able to capture the pack temperature change under 

step change and CS cycling. The fitted parameters for this 

pack are listed in Table II.

Table II

Fitted Parameters for PHEV Pack

IV. CONCLUSION

This work modeled lithium-ion battery pack thermal 

performance using a thermal network with lumped thermal 

nodes. The resulting multi-node thermal model is low-order 

and can be explicitly described by ordinary differential 

equations. This makes it a convenient tool for vehicle system 

engineers to use in battery management system design. A 

three-node thermal model was fitted and validated for two 

HEV packs, focusing on pack thermal performance with 

directly heat exchange between a pack and the ambient 

temperature. The thermal masses in the models were fixed 

and the thermal coefficients were regressed from the 

experimental data. If considering heat exchange between a 

pack and the cabin, a fourth node representing the cabin can 

be added. Such a fourth-order thermal model was applied to a 

PHEV pack. The mass ratio and thermal coefficient ratio 

were fixed for the model. The overall pack thermal mass and 

the thermal coefficients with the cabin and the ambient 

temperature were identified using testing results. Both the 

three-node and four-node models successfully capture the 

maximum and minimum cell temperatures of the packs under 

step change tests and charge sustaining cycling. These 

models can be integrated with lithium-ion battery life model 

or vehicle powertrain model for large-scale system 

simulation.

ACKNOWLEDGMENT

The initial thermal models for this work were developed 

by support from the Vehicle Technology Office, Office of 

Energy Efficiency and Renewable Energy, U.S. Department 

of Energy. Recent development for multimode development 

was supported by the DOE's Advanced Research Projects 

Agency-Energy, Advanced Management and Protection of 

Energy Storage Devices Program.

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(J/K) _ (w/K) (w/K)

84,375 0.4 16