. 2024 Dec 26;19(12):e0312856. doi: 10.1371/journal.pone.0312856

Table 2. Description of STL-LSTM algorithm.

Algorithm: LSTM model based on data characteristics and space-time attention machine
Input: The input variable $x (k) = [x_{1} (k), x_{2} (k), \dots, x_{T} (k)]$ represents the k-th cycle’s short-term data, where T represents the dimensionality of the short-term data. L(k) represents the real capacity value at the k-th cycle. Output: The output variable $\tilde{Q} (k)$ represents the SOH estimated value for the k-th cycle’s capacity.
1.Algorithm: Long Short-Term Memory Network Based on Temporal Attention Mechanism(STL-LSTM) procedure: 1.Standardize the data set 2.Initialize the network parameters, including the weight matrix W and the bias vectors U,V 3.Execute for each time step T = 1 to k: a.Short-time feature extraction: For each time step k = 1 to T i. Calculate the activation values of the input, forget, and output gates Input gate: $i_{k} = σ (W_{i} [s (k - 1), c (k)] + b_{i})$ Forget Gate: $f (k) = σ (W_{f} [s (k - 1), c (k)] + b_{f})$ Output gate: $h (k) = σ (W_{o} [s (k - 1), c (k)] + b_{o})$ Calculate cell state ${\tilde{Q}}_{k} = \tanh (W_{c} [s (k - 1), c (k)] + b_{c})$ Update the hidden status $d (k) = f_{k} d (k - 1) + i_{k} {\tilde{C}}_{k}$ b. Spatial attention mechanism: For each time step k = 1 to T: i.Calculate the time attention value at moment k: $x s (k) = \sum_{i = 1}^{T} α^{i} (k) x_{i} (k), 1 \leq i \leq T$ ii.The time weighted sum of all encoder hidden states is calculated to obtain the final feature input: $α^{i} (k) = \frac{\exp \| x_{i} (k) - Q (k - 1) \|}{\sum_{i = 1}^{T} \exp \| x_{i} (k) - Q (k - 1) \|}, 1 \leq i \leq T$ b. Temporal attention mechanism: i.Calculate the time attention value of the hidden state at time k: $β_{1}^{i} (k) = \exp \| x s_{i} (k) - s (k - 1) \|, 1 \leq i \leq M - 1$ ii.Final feature input c(k) $c (k) = \sum_{i = 1}^{M} β^{i} (k) x s_{i} (k), 1 \leq i \leq M$ c. Capacity estimation: i.The average capacity is trend data L(k) $L (k) = \frac{1}{M} \sum_{i = 1}^{M} Q (k - i)$ ii. c(k)、L(k)、 $\tilde{Q} (k - 1)$ 、Q(k−1) as input of LSTM model as input, Estimate subsequent capacity $\tilde{Q} (k)$ $\tilde{Q} (k) = f_{l} [c (k), L (k), Q (k - 1), \tilde{Q} (k - 1)] + b_{v}$

Algorithm: LSTM model based on data characteristics and space-time attention machine

Input: The input variable

x (k) = [x_{1} (k), x_{2} (k), \dots, x_{T} (k)]

represents the k-th cycle’s short-term data, where T represents the dimensionality of the short-term data. L(k) represents the real capacity value at the k-th cycle.
Output: The output variable

\tilde{Q} (k)

represents the SOH estimated value for the k-th cycle’s capacity.

1.Algorithm: Long Short-Term Memory Network Based on Temporal Attention Mechanism(STL-LSTM)
procedure:
1.Standardize the data set
2.Initialize the network parameters, including the weight matrix W and the bias vectors U,V
3.Execute for each time step T = 1 to k:
a.Short-time feature extraction:
For each time step k = 1 to T
i. Calculate the activation values of the input, forget, and output gates
Input gate:

i_{k} = σ (W_{i} [s (k - 1), c (k)] + b_{i})

Forget Gate:

f (k) = σ (W_{f} [s (k - 1), c (k)] + b_{f})

Output gate:

h (k) = σ (W_{o} [s (k - 1), c (k)] + b_{o})

Calculate cell state

{\tilde{Q}}_{k} = \tanh (W_{c} [s (k - 1), c (k)] + b_{c})

Update the hidden status

d (k) = f_{k} d (k - 1) + i_{k} {\tilde{C}}_{k}

b. Spatial attention mechanism:
For each time step k = 1 to T:
i.Calculate the time attention value at moment k:

x s (k) = \sum_{i = 1}^{T} α^{i} (k) x_{i} (k), 1 \leq i \leq T

ii.The time weighted sum of all encoder hidden states is calculated to obtain the final feature input:

α^{i} (k) = \frac{\exp | x_{i} (k) - Q (k - 1) |}{\sum_{i = 1}^{T} \exp | x_{i} (k) - Q (k - 1) |}, 1 \leq i \leq T

b. Temporal attention mechanism:
i.Calculate the time attention value of the hidden state at time k:

β_{1}^{i} (k) = \exp | x s_{i} (k) - s (k - 1) |, 1 \leq i \leq M - 1

ii.Final feature input c(k)

c (k) = \sum_{i = 1}^{M} β^{i} (k) x s_{i} (k), 1 \leq i \leq M

c. Capacity estimation:
i.The average capacity is trend data L(k)

L (k) = \frac{1}{M} \sum_{i = 1}^{M} Q (k - i)

ii. c(k)、L(k)、

\tilde{Q} (k - 1)

、Q(k−1) as input of LSTM model as input, Estimate subsequent capacity

\tilde{Q} (k)

\tilde{Q} (k) = f_{l} [c (k), L (k), Q (k - 1), \tilde{Q} (k - 1)] + b_{v}