An adaptive forgetting factor RLS algorithm is proposed based on fuzzy algorithm and RLS algorithm to identify the mechanism model online, and the structure of this algorithm is shown in Figure 2. We continuously detects the real-time input value α ₁, α ₂, …, α _n of model and the energy consumption Y in the process of online operation, and then we obtain the time series average W of the residual and its change rate ΔW between the deviation of the system dynamic characteristic value and the identified model value, taking them as the inputs of the fuzzy algorithm. In this way, we can acquire the adjusted forgetting factor’s value μ in real time, and bring it into Eq. 4 to get the parameters θ ̂ ( k ) online.

Through the section 2, the real-time output y _m (k) of the simplified mechanism model can be obtained. However, in the process of mechanism modeling, the mechanism model is deviated from the actual characteristics. In addition, in the case of production process or external disturbances, it will cause errors between the actual model and the simplified mechanism model.

Therefore, the kernel partial least squares algorithm is used to establish the deviation compensation model of the simplified mechanism model. it makes the calculated deviation y _d (k) be the output of the deviation compensation model.

If the variable X ∈ R ^n×p , Y ∈ R ^n×q , and p is the number of independent variable, q is the number of dependent variable, n is the number of observed samples, the nonlinear mapping from the original input space { x j } j = 1 p to the feature space H is recorded as Φ : x _j ∈ R ⁿ → Φ(x _j ) ∈ H (Jiang et al., 2020).

In the mapping space, the weight vector ω ₁ and c ₁ of X are determined according to the mapped sample matrices Φ and Y. Under the constraint of ω 1 T ω 1 = 1 , c 1 T c 1 = 1 , finding the maximum value of ω 1 T Φ T Y c 1 . J 1 = max ω 1 = c 1 = 1 C o v ( Φ ω 1 , Y c 1 ) = max ω 1 = c 1 = 1 1 n ( ω 1 T Φ T Y c 1 ) (9)

The lagrangian function is established for the above. L ( ω 1 , Φ , Y , c 1 ) = ω 1 T Φ T Y c 1 − η 1 ( ω 1 T ω 1 − 1 ) − η 2 ( c 1 T c 1 − 1 ) (10)

η ₁, η ₂ are the Lagrangian multiplier, and the partial derivatives of L(ω ₁, Φ, Y, c ₁) to ω ₁, c ₁, η ₁, η ₂ are respectively divided into 0. ∂ L ∂ ω 1 = Φ T Y c 1 − 2 η 1 ω 1 = 0 ∂ L ∂ c 1 = ω 1 T Φ T Y − 2 η 2 c 1 = 0 ∂ L ∂ η 1 = − ( ω 1 T ω 1 − 1 ) = 0 ∂ L ∂ η 2 = − ( c 1 T c 1 − 1 ) = 0 (11)

Therefore Φ 0 T F 0 F 0 T Φ 0 ω 1 = λ 1 ω 1 (12) where “0” is the original data, ω ₁ is the eigenvector λ ₁ corresponding to the largest eigenvalue of Φ 0 T F 0 F 0 T Φ 0 . And on the basis of introducing the kernel matrix K 0 Φ = Φ 0 T Φ 0 , multiply both sides of the Eq. 12 by Φ₀, and substituting t ₁ = Φ₀ ω ₁ to obtain K 0 Φ F 0 F 0 T t 1 = λ 1 t 1 (13)

Then the kernel matrix is K Φ = Φ Φ T = K 1,1 ⋯ K 1 , n ⋮ ⋱ ⋮ K n , 1 ⋯ K n , n (14) where K _α,β is the selected kernel function.

In this paper, the kernel matrix is used Gaussian kernel function to gotten. its expression is: K α , β = ∑ j = 1 p exp − x j ( α ) − x j ( β ) 2 2 σ 2 (15) where σ is the width of the Gaussian kernel function, x is the data sample, α, β is the the data sequence.

After obtaining the first kernel principal component t ₁, and the centralized kernel residual matrix K i Φ can be expressed as K i Φ = D i K i − 1 Φ D i (16) where D i = I n − t i t i T / ( t i T t i ) .

The residual matrix F _i of the centralized dependent variable data matrix is F i = ( I − t i t i T / t i T t i ) F i − 1 (17)

So the remaining kernel principal components are obtained: K i − 1 Φ F i − 1 F i − 1 T t i = λ i t i (18)

Similarly, the kernel principal component u _i can be obtained according to the independent variable data matrix E ₀. And the matrix t _i , u _i are composed of T, U.

Therefore, the output of the KPLS algorithm can be expressed as: Y ̂ = K Φ U ( T T K Φ U ) − 1 T T Y (19)

Through the above mentioned, the deviation compensation model of mechanism model is established by kernel partial least squares algorithm.

However, with the changes in working conditions and mechanism model parameters, the nonlinear relationship between the deviation of mechanism model and its inputs are constantly changing. So this paper uses the model updating strategy with sliding window to update the kernel matrix of the deviation compensation model in real time, and improve the prediction accuracy of the model. The sliding window updating strategy is to set a fixed-length sample as a training sample, and to make the window slide forward with the acquisition of the new sample, thereby implementing the update of the model. The purpose of the deviation compensation model updating is to update the elements in the kernel matrix to adapt the adaptability of the model when operating conditions change. It is only necessary to change the position of some elements and calculate a small number of new elements during the updating process without having to perform a lot of double counting.

The updating process of the deviation compensation model is shown in Figure 5. Consuming the window length to be g,then the kernel is K _g×g , and the sliding length is q. So when the sliding window slides forward, the new input and output data are got to form the updated samples. The updated input samples are recorded as X* and the updated output samples are recorded as Y*. Firstly, the kernel matrix is renewed by the updated samples X* and Y* through the kernel function. The kernel matrix updating process is as follows: to begin with, deleting the matrix elements corresponding to the discarded q samples, then changing the position of the reserved elements, and finally calculating the corresponding elements of the q new samples and adding them to the kernel matrix to complete the updating of the model.

1) Deleting old samples

For the variable x ∈ R ^g , the expression of the kernel matrix K g × g Φ is: K g × g = Φ x 1 , x 1 Φ x 1 , x 2 ⋯ Φ x 1 , x g Φ x 2 , x 1 Φ x 2 , x 2 ⋯ Φ x 2 , x g ⋮ ⋮ ⋱ ⋮ Φ x g , x 1 Φ x g , x 2 ⋯ Φ x g , x g (20)

For the current running state of the modeled object, the first q group samples are old sample points with no value, and the elements containing x ₁ ∼ x _q in the kernel matrix should be deleted. After deleting the corresponding elements, the kernel matrix is recorded as K ( g − q ) × ( g − q ) Φ . K ( g − q ) × ( g − q ) Φ = Φ x q + 1 , x q + 1 Φ x q + 1 , x q + 2 ⋯ Φ x q + 1 , x g Φ x q + 2 , x q + 1 Φ x q + 2 , x q + 2 ⋯ Φ x q + 2 , x g ⋮ ⋮ ⋱ ⋮ Φ x g , x q + 1 Φ x g , x q + 2 ⋯ Φ x g , x g (21)

2) Adding new samples

After deleting the old samples, in order to maintain the order of the kernel matrix, it is necessary to add the same number of new samples. Since the updating model adopts the timing update strategy, the newly added matrix elements need to consider the sample timing, and the kernel matrix after adding the new elements is recorded as K g × g Φ * . K g × g Φ * = Φ x q + 1 , x q + 1 Φ x q + 1 , x q + 2 ⋯ Φ x q + 1 , x q + g Φ x q + 2 , x q + 1 Φ x q + 2 , x q + 2 ⋯ Φ x q + 2 , x q + g ⋮ ⋮ ⋱ ⋮ Φ x q + g , x q + 1 Φ x q + g , x q + 2 ⋯ Φ x q + g , x q + g (22)

Secondly, the new score vector T* and U* can be gained by Eqs 9–18 according to the updated samples X* and Y*.

Finally, the output y _d (k) of the adaptive deviation compensation model through the KPLS algorithm and the model updating strategy with sliding window is: y d ( k ) = K g × g Φ * U * ( T * ) T K g × g Φ * U * − 1 ( T * ) T Y * (23) where y _d (k) is the output of the adaptive deviation compensation model.

Through the content introduced above, the recursive least squares algorithm with adaptive forgetting factor, kernel partial least squares algorithm and the model updating strategy with sliding window are used to established the adaptive hybrid model. And the modeling steps are as follows:

1) Obtaining the running real-time/ historical data, and excluding outliers.

Under working condition 2, there are 1087 sets of actual data, and 965 sets of global data are selected as training data, and 122 sets of global data are used as test data. Respectively, through mechanism model, data model (Kernel Partial Least Squares model) and adaptive hybrid model to compare the model accuracy. The training and test effects of mechanism model is shown as Figure 6, the effects of KPLS model is shown as Figure 7, and the effects of adaptive hybrid model is shown as Figure 8.calculation accuracy of different models under signal working condition (working condition 2) is shown in Table 3.

It can be concluded from Figures 6, 7, 8 that the adaptive hybrid model has higher fitting accuracy and better generalization ability, and can track the energy consumption change in time. From Table 3, we can see that the accuracy of the mechanism model is poor, because the mechanism model ignores some factors that affect the power consumption of water chillers, resulting in larger errors in the mechanism model. The data (KPLS) model has a higher Accuracy, but the prediction accuracy of the model is low, the model has over-fitting phenomenon and thus affects the prediction accuracy of the model. Comparing with the mechanism model and the data model, the root mean square error of the adaptive hybrid model is reduced by 87.6 and 88.2% respectively, and the average absolute percentage error is reduced by 89.9 and 90.2% respectively. The model has higher accuracy.

This paper further verifies the prediction accuracy, generalization and adaptability of the adaptive hybrid model under variable working conditions by using the actual data of centrifugal chiller under working conditions 1 and 3. There are 954 sets of actual data in working condition 1 and all of them are used as training data. There are 590 sets of actual data in working condition 3 and all of them are used as test data. The accuracy of mechanism model under variable working conditions is shown as Figure 9. The accuracy of data (KPLS) model under variable working conditions is shown as Figure 10. The faccuracy of adaptive hybrid model under variable working conditions is shown as Figure 11.the calculation accuracy of different models under multiple working conditions is shown in Table 4.

From the Figure 10, it can be concluded that the calculation accuracy of data (KPLS) model has a large deviation under variable working conditions, because the dynamic characteristics of chillers will change when the external temperature changes, while the training samples contain limited working data. When the data contain the number of new working conditions, data (KPLS) model will occur to lose effectiveness, which seriously affects the accuracy of the model. However, the adaptive hybrid model has better generalization and adaptability when the working condition changes from working condition 1 to working condition 3. The root mean square error is reduced by 61.8 and 77.2% respectively compared with the mechanism model and the data model, and the average absolute percentage error is reduced by 85.4 and 92.6% respectively compared with the mechanism model and the data model.

From the analysis of the above results, it can be seen that the adaptive hybrid model not only guarantees high fitting accuracy, but also has high generalization ability and adaptability. Compared with the mechanism model, the adaptive hybrid model accurately describes the dynamic characteristics of the centrifugal chiller, avoiding the negative impact of the lack of input variables on the accuracy of the model. Compared with the data model, the accuracy of the adaptive hybrid model is not affected by the coverage of training samples, and it also avoids the over-fitting phenomenon. In addition, the adaptive hybrid model also has high adaptability to variable working conditions.