The Markov prediction model is a probabilistic prediction method about the occurrence of an event. It predicts the states of the event at various times in the future based on the current states of the event. In each step of the Markov prediction model, the system can realize mutual migration between states according to the probability distribution of state transitions. This section provides an overview of the steps included in the traditional Markov prediction model. Suppose a certain system state sequence: X = x 1 , x 2 , ⋯ x n (5)

The Markov prediction model emphasizes that the predicted state is only related to the previous state, so the transition probability between any states can be obtained based on this principle, that is, the one-step state transition matrix can be obtained: P 1 = p 11 p 12 ⋯ p 1 n p 21 p 22 ⋯ p 2 n ⋮ ⋮ ⋱ ⋮ p n 1 p n 2 ⋯ p n n (6)

According to the current state x n of the state sequence and the one-step state transition matrix, the next state P 1 can be predicted: x n + 1 = x n P 1 (7)

Based on this, the prediction of the next stage of state can be obtained. According to the traditional Markov prediction model, the subsequent prediction is described by the k-step state transition matrix. x n + k = x n P k = x 1 P n + k (8)

The basic steps of the traditional Markov prediction model can be obtained from comprehensive formulas (5-8). Certain limitations are inherent in the traditional Markov model. When the curve change rate experiences sudden changes and the numerical value undergoes violent fluctuations, the complex changes that occur in the prediction process cannot be handled by a single state transition matrix. The traditional Markov model, which predicts based solely on the current state, is directly led to an inability to have aftereffects. The state value of the model will pass through the same state interval multiple times, causing the model to enter a stationary distribution and rendering it incapable of making effective predictions. In other words, the convergence of subsequent prediction data is excessively strong.

As noted in section 3.2.1, the traditional Markov model, on one hand, exhibits poor sensitivity to rate mutations and data fluctuations, and is incapable of handling complex changes that occur in the prediction process. On the other hand, its data training is based solely on historical data and lacks the reuse of prediction data, limiting the number of steps that can be accurately predicted to three to five steps. This directly results in the state prediction falling into a stationary distribution. In light of the shortcomings of the traditional Markov model, the proposal in this section is to perform first-order difference processing on the state sequence based on the accurate prediction in the previous stage of the traditional Markov model and update the prediction data to the original sequence for iteration. The first-order difference value is able to reflect not only of reflecting data fluctuations, but also of reflecting the changes in the sequence, thereby preventing the sequence prediction from developing to extremes. The scheme is depicted in Figure 2.

As can be seen from Figure 2, the improved Markov model based on first-order difference and sequence iterative update includes three steps.

(1) First, the numerical sequence of power distribution and consumption tasks resource demand is constructed as:

According to the resource imbalance state matrix and Equation 10, the one-step state transition matrix can be obtained by Table 1.

In the table: p 11 , p 12 respectively represent the number of transitions from state M c o n d = 1 to state M c o n d = 1 , M c o n d = 2 . p 21 , p 22 respectively represent the number of transitions from state M c o n d = 2 to state M c o n d = 1 , M c o n d = 2 . P 1 , P 2 respectively represent the number of each state in the resource imbalance state matrix M c o n d . In order to facilitate the calculation of subsequent state transition matrices, the initial value of the current unbalanced state Δ η b a x n is described by matrix Δ η b a x n , where Δ η b a x n is a 1 × 2 boolean matrix, as follows: Δ η b a x n = 1 , 0 , M c o n d Δ η b a x n = 1 Δ η b a x n = 0 , 1 , M c o n d Δ η b a x n = 2 (11)

In the formula: Δ η b a x n = 1 , 0 , Δ η b a x n = 0 , 1 respectively indicate that the current states is a decreasing resource imbalance segment or a increasing resource imbalance segment. According to the one-step state transition matrix P c o n d 1 and the current imbalance state Δ η b a x n obtained in Table 1, the next stage state prediction value Δ η b a x n can be calculated: Δ η b a x n + 1 = Δ η b a x n P c o n d 1 (12)

After the state transition prediction calculation, Δ η b a x n + 1 is a non-Boolean matrix. For subsequent calculations, Δ η b a x n + 1 needs to be converted into a 1 × 2 Boolean matrix: Δ η b a x n + 1 = 1 , 0 , ⁡ max Δ η b a x n + 1 = Δ η b a x n + 1 1 Δ η b a x n + 1 = 0 , 1 , ⁡ max Δ η b a x n + 1 = Δ η b a x n + 1 2 (13)

According to Eq. 13, the predicted state probability can be equivalent to the segment in which the current predicted state is in resource imbalance.

(3) Finally, the predicted state is added to the original state sequence to form a new state sequence. According to the one-step state transition matrix P c o n d 2 updated in Table 1 and equations (11-13), a continuous iterative prediction of the power distribution and consumption tasks resource demand state is formed:

In the formula: Δ η b a x n + k represents the predicted state at step k. Δ η b a x n + k − 1 represents the Boolean matrix at step k-1. P c o n d k represents the one-step state transition matrix after k updates.

Eventually, the number of steps accurately predicted by the traditional Markov front stage covers the iterative prediction sequence. In this case, accurate coverage is achieved in 3 steps. The improved Markov prediction, which is based on first-order difference and iterative sequence updating, combines the short-term accurate prediction characteristics of traditional models with the data-sensitive identification characteristics of existing models. Concurrently, the convergence of the traditional k-step state transition matrix in prediction is overcome by the addition of state iterative updates and the iteration based on a state transition matrix.

The unbalanced increasing segment of the resource curve is attributed to the overlap of cyclical tasks, resulting in a significant gap between the resource demand curve and the average resource demand in that time cycle. In order to ensure the optimal balance of resource demand, this section combines the characteristics of cyclical tasks with fixed-interval sampling, and performs iterative updates for cyclical tasks to minimize resource imbalance based on the principle of peaking shifting. Despite the different time scales of millisecond-level cyclical tasks and second-level cyclical tasks, the adjustment ideas and principles are unified. The adjustment of second-level cyclical tasks is taken as an example for analysis in this context. The process is depicted in Figure 3.

As can be seen from Figure 3, this section divides the adjustment of second-level cyclical tasks into six steps:

(1) The first step is to define the first time point with the lowest resource demand of all second-level cyclical tasks in the increasing resource imbalance segment as t l o w s _ 1 .

a. When t l o w s _ 1 is less than the cyclical response time of the task: P E c a l s _ 1 = ∑ j ∑ α 2 P E j c a l _ ⁡ max ε t − t l o w s _ 1 − α 2 T j − ε t − t l o w s _ 1 − α 2 T j − τ j (15)

In the formula: P E c a l s _ 1 represents the total resource demand of all second-level cyclical tasks after the first peaking shifting.

b. When t low s _ 1 is greater than the cyclical response time of the task, the latest response time in each cycle is used as the target point for peaking shifting: P E c a l s _ 1 = ∑ j ∑ α 2 P E j c a l _ ⁡ max ε t − t M = 2 − T j − τ j − α 2 T j − ε t − t M = 2 − T j − τ j − α 2 T j − τ j } (16)

In the formula: t M = 2 represents the starting point of the resource imbalance increasing segment.

(4) The fourth step involves redefining the first time point with the lowest resource demand, following the procedure outlined in step (1) and utilizing Eqs 15 and 16. Following this logic, the minimum resource demand time point sequence t l o w s _ m can be obtained one after another.

In the formula: P E c a l s _ m represents the total computing resource demand curve of second-level cyclical tasks after adjustment in the mth step.

(6) The sixth step is to define minimizing the imbalance degree as the objective optimization function and corresponding constraints:

In the formula: η b a s represents the sum of squares of the difference between the computing resource demand of all second-level cyclical tasks in the segment and the average computing resource demand. ω t s represents the time proportion of each resource value. P E c a l s _ a v e represents the average computing resource demand of all second-level cyclical tasks in the segment. Constraint C 1 means that the maximum total computing resource demand of the second-level cyclical tasks after the mth step peaking shifting adjustment must not exceed the resource supply amount C cal ; Constraint C 2 means that peaking shifting must ensure that cyclical tasks are sampled within the response time of each cycle.

Determine whether the iteration meets the termination condition. If so, terminate it. If not, return to step (4). In the same way, the peaking shifting processing and parameter expression of millisecond-level cyclical tasks are similar to the second-level cyclical tasks.

The previous dynamic balancing model of cyclical tasks based on historical data for state prediction has effectively reduced resource imbalance. However, due to the burst characteristics of non-cyclical tasks, the balance of the original leveling model will be broken when such tasks moves. To address this issue, this section initially distinguishes the response cycles of cyclical tasks during the non-cyclical tasks execution time. Subsequently, it performs differential leveling based on the results of this differentiation. The specific diagram is depicted in Figure 4:

As can be seen from Figure 4, the main principle of the non-cyclical tasks dynamic balancing model is to avoid cyclical tasks stagnation during its execution time: on one hand, for short-response cyclical tasks, partial avoidance can be achieved by leveraging the characteristics of their short cycle response time. on the other hand, for long-response cyclical tasks, all non-cyclical tasks can be avoided based on the characteristics of long cycle response time. According to this idea, this model uses the short response cyclical tasks back-moving time Δ T d b m and the long response cyclical tasks back-moving time Δ T l b m as decision variables to optimize the non-cyclical tasks segment to minimize the load imbalance. The specific model is as follows.

To implement this idea, the adjustable cyclical step waveform during the non-cyclical tasks time is shifted backward: Δ P E c a l d A C = P E p c a l ε t − α 3 T p − Δ T d b m − ε t − α 3 T p − Δ T d b m − τ p Δ P E c a l l A C = P E q c a l ε t − α 4 T q − Δ T l b m − ε t − α 4 T q − Δ T l b m − τ q (20)

In the formula: Δ P E c a l d A C , Δ P E c a l l A C respectively represent the expressions under the final step waveform adjustment of short response cyclical tasks and long response cyclical tasks. P E p c a l , P E q c a l respectively represent the computing resource demand of short response cyclical task p and long response cyclical task q . The meaning of parameters α 3 , T p , τ p and α 4 , T q , τ q is consistent with the previous ones, indicating the number of steps, execution cycle and execution time of short response cyclical tasks and long response cyclical tasks respectively. Reference formula (18) defines minimizing load imbalance as the objective optimization function and corresponding constraints: A C c a l = A C k c a l ε t − t a r − ε t − t a r − τ k P E c a l d _ l = Δ P E c a l d A C + Δ P E c a l l A C + A C c a l + ∑ P E c a l d ¬ A C + ∑ P E c a l l ¬ A C min ⁡ η b a d _ l = ∑ ω t d _ l P E c a l d _ l − P E c a l d _ l _ a v e 2 (21) s.t. C 1 : ⁡ max P E c a l d _ l ≤ C cal C 2 : Δ T d b m ≤ τ p Δ T l b m ≤ τ q C 3 : Δ T d b m + α 3 T p ≤ t M = 2 e n d − τ p Δ T l b m + α 4 T q ≤ t M = 2 e n d − τ q (22)

In the formula: A C c a l represents the computing resource demand curve of non-cyclical tasks. P E c a l d _ l represents the total computing resource demand curve in the segment after the adjustment of the non-cyclical tasks dynamic balancing model. P E c a l d ¬ A C and P E c a l l ¬ A C respectively represent the computing resource demand curves of the unadjusted parts of the short response cyclical tasks and long response cyclical tasks. ω t d _ l represents the time proportion of each resource value after adjustment of the non-cyclical tasks dynamic balancing model. P E c a l d _ l _ a v e represents the average computing resource demand after adjustment of the non-cyclical tasks dynamic balancing model. Constraint C 1 indicates that the adjusted maximum total computing resource demand must not exceed resource supply C cal . Constraint C 2 means that the backward movement time does not exceed the execution time of the cyclical tasks. Constraint C 3 means that cyclical tasks cannot cross the current segment when avoiding non-cyclical tasks.

In order to further verify the applicable scope and influencing factors of the dynamic balance model, the adaptive analysis of the application of the dynamic balancing model in the resource imbalance decreasing segment is carried out. In order to facilitate the adaptability analysis between different scenarios in the following, this section only considers the role of photovoltaic scenarios, and selects the resource imbalance decreasing segment 316–317s as the research object, as shown in Figure 9 and Table 7.

It can be seen from Figure 9 that the number of peaks in this decreasing segment has dropped significantly after being adjusted by the dynamic balancing model, with the highest peak difference reaching 16MB, and the adjusted computing resource demand data points are more concentrated near the average demand line. It can be seen from Table 7 that the resource imbalance degree dropped by nearly 6 times, which is 10 times smaller than the previous increase in resource imbalance. Based on the above results, it can be seen that the dynamic balancing model is universally applicable to both the increasing and decreasing segments of resource imbalance.

On the basis of the adjusted resource imbalance decreasing segment, what factors will cause differences in dynamic balancing effects and the pros and cons brought by these differences will continue to be analyzed. Here, a simulation analysis is conducted on the change amount and change rate of computing resource demand in the two segments, as shown in Figure 10.

It can be seen from Figure 10: In terms of the change in computing resource demand, the reduction in computing resource demand in the two segments is basically synchronized. However, in terms of the increase, whether it is the horizontal time width or the vertical numerical scale, the segment where resource imbalance decreases are significantly higher than the segment where resource imbalance increases. At the same time, in terms of the change rate of computing resource demand, the fluctuation in the decreasing resource imbalance segment is more obvious than that in the increasing resource imbalance segment. Similarly, the change rate of increasing computing resources is significantly greater than that in the increasing resource imbalance segment.

Based on the above results, the composition of power distribution and consumption tasks in the two segments is further analyzed. Since millisecond-level cyclical tasks have similar distribution in the two segments, no analysis is performed here. It can be seen from the enlarged display part of the figure that the types of second-level cyclical tasks in the resource imbalance increasing segment are significantly more than those in the resource imbalance decreasing segment. The reason is that the execution cycle of second-level cyclical tasks is large, and aggregation usually occurs during its common cycle, and the segment where aggregation occurs naturally forms a increasing segment. Therefore, in the decreasing segment where there is no large accumulation of second-level cyclical tasks, the baseline value of computing resources is relatively low. Correspondingly, the possibility and rate of change of computing resources under dynamic balancing will also be relatively high.

To sum up, the execution cycle of large time-scale cyclical tasks is the main factor affecting the dynamic balancing effect. Subsequent adjustments are mainly targeted at the segments where resource imbalance increases. At the same time, the analysis of the common tasks cycle must be strengthened to achieve optimal adjustment results.

In order to further verify the effectiveness of the dynamic balancing model in different scenarios, this section considers comparing the photovoltaic-only scenario and the wind power-only scenario, and selects the resource imbalance increasing segment 315–316s as the research object, as shown in Figure 11.

As can be seen from Figure 11, whether it is a separate photovoltaic scenario or a separate wind power scenario, the adaptive changes in computing resource demand reflect the universality of the dynamic balancing model in different scenarios, and also prove that the good dynamic balancing effect in the comprehensive energy scenarios in Sections 4.3 and 4.4 is not due to the role of a particular energy scenario only. Similarly, it can also be observed from Figure 11: the demand in the front stage of the wind power scenario is greatly reduced, while the demand in the back stage is increased; the overall photovoltaic scenario is relatively balanced. The reason is that the wind power scenario has a large number of proprietary tasks and a long execution cycle. Before adjustment, the demand in the front stage is large and the demand in the back stage is slightly smaller. After adjustment, there will be obvious changes in the two stages. In subsequent adjustments, consider strengthening the coordination of scenario-specific tasks features to achieve optimal adjustment results.