Proposed optimal control algorithms for photovoltaic arrays reconfiguration

. , 𝑛𝐴𝑆} ( 2-9 ) 7 2.3.2 Munkres' Assignment Algorithm (MAA) This is the first task division problem proposed by author James Munkres. It is stated as follows: Contents of the problem: Given nM workers (iM = 1, 2, ... , nM) and nM tasks (jM = 1, 2, ... , nM). In order to assign the worker iM to perform the task jM, cost of CiMjM ≥ 0 as required. The problem is to assign which task to which worker (each worker only performs one task; each task is performed by one worker only) to incur the lowest total cost? General C matrix in Figure 2-7: Worker Task 1 2 ... nM 1 C11 C12 𝐶1𝑛𝑀 2 C21 C22 𝐶2𝑛𝑀 ... nM 𝐶𝑛𝑀1 𝐶𝑛𝑀2 𝐶𝑛𝑀𝑛𝑀 Figure 2-7. Matrix general C The mathematical model of this problem is as follows: 𝑧𝑀 = ∑ ∑ 𝐶𝑖𝑀𝑗𝑀𝑥𝑖𝑀𝑗𝑀 𝑛𝑀 𝑗𝑀=1 𝑛𝑀 𝑖𝑀=1 → 𝑚𝑖𝑛 ( 2-14 ) Provided that: Each worker only performs one task: ∑ 𝑥𝑖𝑀𝑗𝑀 𝑛𝑀 𝑗𝑀=1 = 1 , 𝑖𝑀 = 1, , 𝑛𝑀 ( 215 ) Each task is performed by one worker only: ∑ 𝑥𝑖𝑀𝑗𝑀 𝑛𝑀 𝑖𝑀=1 = 1 , 𝑗𝑀 = 1, , 𝑛𝑀 ( 216 ) 𝑥𝑖𝑀𝑗𝑀 = 0 ℎ𝑎𝑦 1 , 𝑖𝑀 = 1 , , 𝑛𝑀 ; 𝑗𝑀 = 1 , , 𝑛𝑀 ( 217 ) due to the availability of conditions (2-15) (2-16), the conditions (2-17) can be replaced with 𝑥𝑖𝑀𝑗𝑀 integer ≥0, iM = 1 , 2 , ... , nM ; jM = 1 , 2 , ... , nM ( 218 ) 2.4 Conclusions of Chapter 2 Chapter 2 provides an overview of the optimal control problem, thereby proposing the optimal control method and formulating the optimal control problem used in the thesis. The first section gives an overview of the optimal control problem, provides definitions, limiting conditions, and classification of the optimal control problem. In the next section, the author formulates an optimal control problem used in the restructuring unit. For the last section, there are two optimization problems presented as the basis for proposing the optimal algorithms for the thesis: Subset sum problem and Munkres' Assignment Algorithm. The application of optimal control in the photovoltaic panels connection restructuring problem will increase the efficiency of the solar system under lighting conditions. The static optimization problem, with the open control system, is proposed to be applied by the author to build a structure that is fast acting and applicable to large solar systems. 8 Chapter 3: DEVELOPMENT OF A SYSTEM RESTRUCTURING STRATEGY USING OPTIMAL CONTROL THEORY 3.1 Irradiance equalization strategy with TCT connection circuit Irradiance equalization method for TCT connection circuit (Figure 1-13d) is the rearrangement of photovoltaic panel connection positions in order to balance the total solar irradiance at the parallel connections in TCT circuit (CT1,5). The Irradiance equalization method, improving the efficiency of the solar system can be generalized according to the diagram in Figure 2-6. This method is designed to make the solar system always operate at the highest efficiency with repetition in a certain period of time. 3.2 Measurement of current and voltage of photovoltaic panels During operation, the photovoltaic panels are connected to each other, and their current and voltage are interdependent as analyzed in section 1.1.4. It is big challenge to measure the current and voltage exactly generated by each photovoltaic cell as a basis for estimating the solar irradiance received by each photovoltaic panel. In this thesis, the measurement method shown in Figure 3-2 (for example, the measurement circuit for 4 photovoltaic panels) (CT9) is used. Figure 3-2. Current and voltage acquisition circuit 3.3 Solar irradiance estimation After measuring the current and voltage at each photovoltaic panel, the solar irradiance calculation formula (3-1) of each photovoltaic panel is applied. 𝐺𝑆 = 𝐺𝑆𝑇𝐶 𝐼𝐿𝑆𝑇𝐶 + 𝜇1𝑠𝑐(𝑇𝑐 − 𝑇𝐶𝑆𝑇𝐶) [𝐼 + 𝐼0(𝑒 𝑉+𝐼𝑅𝑆 𝑛𝑆𝐴𝑑𝑘 𝑇𝑐 𝑞 − 1)+ 𝑉 + 𝐼𝑅𝑆 𝑅𝑆ℎ ] (3-1) 3.4 Proposal of the mathematical model (s) and 02 algorithms for Seeking the irradiance equalization configuration The mathematical model, DP algorithm application method and SC algorithm prposal are announced in (CT8), (CT1) and (CT3), respectively. 3.4.1 Mathematical model establishment 𝐸𝐼 = max 𝑖=1,𝑚 (∑𝐺𝑖𝑗 𝑛𝑖 𝑗=1 ) − min 𝑖=1,𝑚 (∑𝐺𝑖𝑗 𝑛𝑖 𝑗=1 ) → 0 (3-4) Constraints: { 𝑛1 + 𝑛2 + 𝑛3 + . . . +𝑛𝑚 = 𝑛 𝐺𝑖1 + 𝐺𝑖2 + 𝐺𝑖3 + . . . + 𝐺𝑖𝑛𝑖 = 𝐺𝑖 𝑛𝑖 > 0 𝐺𝑖𝑗 ≥ 0 𝑖 = 1,𝑚 ; 𝑗 = 1, 𝑛𝑖 (3-5) 9 Where: EI : Equalization index, n : total number of photovoltaic panels, m : number of rows in TCT circuit, 𝑛𝑖 : number of photovoltaic panels in row i, 𝐺𝑖𝑗 : irradiance of PV panels in row i, column j, 𝐺𝑖 : total irradiance at row i. The objective function (3-4) is aimed at selecting a configuration with the smallest irradiance difference in the rows, that is, the difference between the row with the highest total solar irradiance and the row with the lowest total solar irradiance, whereby an EI of 0 is ideal. 3.4.2 Application of Dynamic programming algorithm (CT1) 3.4.2.1. Application method Considering the general TCT connection circuit in Figure 2-c. The system consists of m rows subject to serial connection. The row i consists of ni photovoltaic panels subject to parallel connection. Solar irradiance given by each photovoltaic cell is Gij with i, j being the row and column index of the photovoltaic panel placement, respectively. Total solar irradiance received at row i: 𝐺𝑖 = ∑ 𝐺𝑖𝑗 𝑛𝑖 𝑗=1 (3-2) Total number of photovoltaic panels: 𝑔 = ∑ 𝑛𝑖 𝑚 𝑖=1 (3-6) The number of rows of the solar system m, after seeking the optimal connection configuration, may differ from the number of rows k (of the initial structure), depending on the calculation of input voltage range for the structuring unit. The total solar irradiance per row in the ideal optimal connection configuration is equal to the average of the total solar irradiance at all photovoltaic panels divided by rows:: 𝑎𝑣𝑔 = ∑ 𝐺𝑖 𝑚 𝑖=1 𝑚 (3-7) The method applied is shown in the flowchart of Figure 3-4. Based on results of Dynamic Programming algorithm for the Subset sum problem, it can be noticed at each step that: 𝐺_𝑂𝑃𝑖 = ∑ 𝐺_𝑂𝑃𝑖𝑗 𝑛_𝑜𝑝𝑖 𝑗=1 → 𝑎𝑣𝑔 (3-8) Infer: 𝐸𝐼 = max 𝑖=1,𝑚 (𝐺_𝑂𝑃𝑖) − min 𝑖=1,𝑚 (𝐺_𝑂𝑃𝑖) → 0 (3-9) G_OP is the optimal connection and irradiance equalization configuration that satisfies the objective function (3-4) presented in section 3.4.1. Figure 3-4. Flowchart of the method of applying the Dynamic Programming in the problem of finding Irradiance Equalization 10 3.4.2.2. Illustrative examples Considering the solar system consisting of 16 photovoltaic panels, with TCT connection, under heterogeneous lighting conditions, each photovoltaic panel receives different solar irradiance, as shown in Figure 3-5, irradiance equalization configuration shown in Figure 3-13. Figure 3-5. PV system under non-homogeneous irradiance Figure 3-13. Configuration of irradiance equalization PV energy system corresponding to the G_OP matrix The equalization index is calculated according to the objective function (3-9) (m = 4): 𝐸𝐼 = max 𝑖=1,𝑚 (𝐺_𝑂𝑃𝑖) − min 𝑖=1,𝑚 (𝐺_𝑂𝑃𝑖) = 10 ( 3-1 ) 3.4.2.3. Comparison and assessment (CT1) Storey et al. 2013 proposed the BWSA algorithm - an algorithm for iterative arrangement, with very fast processing speed, low number of loops but poor results in the majority of cases. An example in Figure 3-14, shows the BWSA algorithm handles with very few iterations and high processing speed. However, the results obtained are not so good as those obtained from DP algorithm proposed above for the same input data. EIBWSA = 110 while EIDP = 10. 𝐸𝐼𝐵𝑊𝑆𝐴 = 1740 − 1630 = 110 ( 32 ) Figure3-14. Example of the BWSA algorithm 8 640 W/m2 1 170 W/m2 2 200 W/m2 3 250 W/m2 4 490 W/m2 5 520 W/m2 6 680 W/m2 7 480 W/m2 9 720 W/m2 10 410 W/m2 11 550 W/m2 13 150 W/m2 14 830 W/m2 15 140 W/m2 16 180 W/m2 + = 1110 W/m2+ + + + + = 2320 W/m2 + + = 1970 W/m2 + + + = 1300 W/m2 12 290 W/m2 + 9 720 W/m2 11 550 W/m2 15 140 W/m2 13 150 W/m2 14 830 W/m2 16 180 W/m2 4 490 W/m2 12 290 W/m2 7 480 W/m2 5 520 W/m2 6 680 W/m2 8 640 W/m2 3 250 W/m2 2 200 W/m2 1 170 W/m2 + = 1670 W/m2+ + + + + = 1680 W/m2 + + = 1680 W/m2 + + + = 1670 W/m 2 10 410 W/m2 + 11 Regarding the processing speed, applied to the system including g photovoltaic panels, m rows and SG as the total solar irradiance, the calculated complexity of algorithm is O(mgSG), taking at least 30.72 ms to deal with Intel Core i5 2.5 Ghz configuration CPU to reconfigure 16 photovoltaic panels with 4 rows. This is the appropriate processing speed for practical solar systems, for example in Palermo (Italy) with a maximum wind speed of 6.4m/s, meaning that for a solar system covering an area of 10-20m2, cloud shading for a few seconds is the worst case for that solar system, requiring a quick switching speed; accordingly, compared to DP's processing speed, the application of DP in practice is more satisfactory. DP algorithm has been analyzed and evaluated in Krishna's published work (SCI-Q1-2019) as one of the few "State of the art" methods that prove the topicality of the proposed algorithm. 3.4.3 Proposal of SmartChoice (SC) algorithm (CT3) SmartChoice is a smart selection algorithm published at (CT2,3) to complement the disadvantages of the DP algorithm. Better results of the two algorithms DP and SC are selected as the “optimal connection configuration”. 1.1.1.1. Algorithm description - Select the photovoltaic panels with the highest solar irradiance, and put them in the row with the lowest total solar irradiance. - In case there are many rows with the same lowest total solar irradiance, put that photovoltaic panel into the row with the lowest index. 3.4.3.1. Application method Figure 3-15. Flowchart of SC algorithm 3.4.3.2. Illustrative examples Considering the solar system consisting of 16 photovoltaic panels, with TCT connection, under heterogeneous lighting conditions, each photovoltaic panel receives different solar irradiance, as shown in Figure 3-16, irradiance equalization configuration shown in Figure 3-22. 12 Figure 3-16. PV system under non-homogeneous irradiance Hình 3-22. Configuration of irradiance equalization PV energy system corresponding to the G_OP matrix The equalization index is calculated according to the objective function (3-9) (m = 4): 𝐸𝐼 = max 𝑖=1,𝑚 (𝐺_𝑂𝑃𝑖) − min 𝑖=1,𝑚 (𝐺_𝑂𝑃𝑖) = 0 (3-13) ( 3-3 ) as the optimal arrangement method. 3.4.3.3. Comparison and assessment (CT3) SC algorithm is proposed by the PhD student for the purpose of overcoming special cases of DP algorithm. The DP algorithm gives very good results in most cases; however, in some special cases, it produces results that are not as good as those of SC algorithm, such as the cases in Figure 3-23 and Figure 3-24: EIDP = 1000 while EISC = 850: 150 150 1000 1000 1000 150 150 = 2300 1000 1000 1000 1000 1000 = 2000 1000 1000 1000 1000 1000 1000 = 3000 Matrix G Matrix G_OP by using DP Figure 3-23. Example of irradiance equalization by using DP 150 150 1000 1000 1000 150 150 = 2300 1000 1000 1000 1000 1000 = 2000 1000 1000 1000 1000 1000 1000 = 3000 Matrix G Matrix G_OP by using SC Figure 3-24. Example of irradiance equalization by using SC The SC algorithm has the advantage of a small number of iterations, with O(glogg) complexity. The combination of SC and DP algorithms into a hybrid algorithm with the selection of better results in the two algorithms as the final results will help us find optimal results for the irradiance equalization problem. 8 480 W/m2 1 620 W/m2 2 500 W/m2 3 600 W/m2 4 300 W/m2 5 540 W/m2 6 420 W/m2 7 320 W/m2 9 280 W/m2 10 460 W/m2 11 400 W/m2 13 360 W/m2 14 340 W/m2 15 300 W/m2 16 240 W/m2 + = 2020 W/m2+ + + + + = 1760 W/m2 + + = 1700 W/m2 + + + = 1240 W/m2 12 560 W/m2 + 15 300 W/m2 1 620 W/m2 6 420 W/m2 11 400 W/m2 16 240 W/m2 3 600 W/m2 10 460 W/m2 7 320 W/m2 12 560 W/m2 8 480 W/m2 13 360 W/m2 5 540 W/m2 2 500 W/m2 14 340 W/m2 15 300 W/m2 + = 1680 W/m2+ + + + + = 1680 W/m2 + + = 1680 W/m2 + + + = 1680 W/m2 9 280 W/m2 + 13 3.5 Proposal of mathematical model and 02 algorithms for Selecting the optimal switching method The mathematical model, MAA algorithm and improved MAA algorithm are announced in CT8, CT1 and CT3, respectively. 3.5.1 Introduction to the Dynamic Electrical Scheme (DES) switching matrix An example of the operation of DES switching matrix is shown in Figure 3-26. (a) (b) (c) (d) Figure 3-26. Dynamic Electrical Scheme (b-d) corresponds to the connection structure (a-c) Considering the general DES switching matrix for g photovoltaic panels, with a change in connection in m parallel circuits as shown in Figure 3-27. Figure 3-27. Dynamic electrical scheme Figure 3-28. Array Q and matrix S represents the number of opening and closing times of the switching matrix An overview of the number of opening and closing times applied for switching matrix key is shown in Figure 3-28. During the operation of the solar system and the restructuring unit, there is a change in the number of opening and closing of the switching matrix key after each session of restructuring. Convention on the number of opening and closing times - At the initial time: 𝑆𝑖𝑗 = 0 ∀ 𝑖 = 1, . . . , 𝑚; 𝑗 = 1, . . . , 𝑔 (3-14) - During the operation process, when the position of a photovoltaic panel p (p = 1..m) is changed from row i to row ik, the number of opening and closing times of the matrix S will be changed as follows: 𝑆𝑖𝑝 = 𝑆𝑖𝑝 + 1 𝑆𝑖𝑘𝑝 = 𝑆𝑖𝑘𝑝 + 1 (3-15) Given zP as the number of solar panels subject to change in position in each session of restructuring, the number of opening and closing times of the switching matrix will be 2 x zP. 14 - Given MI the number of opening and closing times in the session of restructuring stepk, we have: 𝑀𝐼𝑠𝑡𝑒𝑝 𝑘 = ∑(𝑆𝑖𝑗)𝑠𝑡𝑒𝑝 𝑘 𝑖=𝑚 𝑗=g 𝑖=1 𝑗=1 −∑(𝑆𝑖𝑗)𝑠𝑡𝑒𝑝 𝑘−1 𝑖=𝑚 𝑗=g 𝑖=1 𝑗=1 (3-16) 3.5.2 Proposal of mathematical model (CT8) Figure 3-29. An example of an irradiance equalization configuration with different switching times is given The problem is to select the optimal switching method with the purpose of controlling the switching matrix from the initial configuration G to the G_OP irradiance equalization configuration so as to achieve the minimum number of opening and closing times of S switching matrix key. Figure 3-29 shows an example of the optimal switching method. 3.5.2.1. The problem of finding a configuration with the minimum number of opening and closing times after each session of restructuring. MI is the number of opening and closing times in each session of restructuring, Sij is the number of opening and closing times of a key with row i and column j in the Switching matrix. The objective function sets the minimum number of opening and closing times after each session of restructuring. Objective function: (𝑀𝐼𝑚𝑖𝑛 )𝑠𝑡𝑒𝑝 𝑘 = ∑(𝑆𝑖𝑗)𝑠𝑡𝑒𝑝 𝑘 𝑖=𝑚 𝑗=g 𝑖=1 𝑗=1 −∑(𝑆𝑖𝑗)𝑠𝑡𝑒𝑝 𝑘−1 𝑖=𝑚 𝑗=g 𝑖=1 𝑗=1 → 0 (3-18) Constraints: { 𝑆𝑖𝑗 ≥ 0 ∑(𝑆𝑖𝑗)𝑠𝑡𝑒𝑝 0 𝑖=𝑚 𝑗=g 𝑖=1 𝑗=1 = 0 (3-19) 15 In which: • m : number of rows in TCT circuit, • g : number of photovoltaic panels; • (𝑀𝐼𝑚𝑖𝑛 )𝑠𝑡𝑒𝑝 𝑘 : the number of opening and closing times in the session of restructuring stepk 3.5.2.2. Balancing the number of opening and closing times of the switching matrix During the restructuring, the photovoltaic panels that are frequently shaded will have the most position changes, resulting in an imbalance in the number of opening and closing times of different keys in the switching matrix. Therefore, the life of the matrix will depend on the life of the lock that closes and opens the most. So in many cases, the switching method with the least number of closing and opening times (calling the least number of closing and opening times as MImin) is not necessarily considered to be optimal, accordingly, it is necessary to choose the other switching method so that the key with the least number of opening and closing times is at the minimum level in order to balance the number of opening and closing of the switching matrix. The objective function on the most number of opening and closing times of the key is constant, still ensuring the minimum number of opening and closing times at each session of switching: { max 𝑖=1,𝑚 𝑗=1,g (𝑆𝑖𝑗)𝑠𝑡𝑒𝑝 𝑘 − max𝑖=1,𝑚 𝑗=1,g (𝑆𝑖𝑗)𝑠𝑡𝑒𝑝 𝑘−1 → 0 𝑀𝐼𝑠𝑡𝑒𝑝 𝑘 = ∑(𝑆𝑖𝑗)𝑠𝑡𝑒𝑝 𝑘 𝑖=𝑚 𝑗=g 𝑖=1 𝑗=1 −∑(𝑆𝑖𝑗)𝑠𝑡𝑒𝑝 𝑘−1 𝑖=𝑚 𝑗=g 𝑖=1 𝑗=1 → (𝑀𝐼𝑚𝑖𝑛 )𝑠𝑡𝑒𝑝 𝑘 (3-20) Constraints: Same as the equation (3-19). In which: • m : number of rows in TCT circuit, • g : number of photovoltaic panels; • 𝑀𝐼𝑠𝑡𝑒𝑝 𝑘 : the number of opening and closing times in the session of restructuring stepk • (𝑀𝐼𝑚𝑖𝑛)𝑠𝑡𝑒𝑝 𝑘: the minimum number of opening and closing times in the session of restructuring stepk (according to the objective function ( )) 3.5.3 Seeking configuration with the least number of switching times applying MAA (CT1) In a study published in 2015 (CT1), the MAA algorithm was applied in finding the configuration so as to achieve the least number of unlocking times from the initial connection configuration to the optimal connection configuration in each session of restructuring to solve the problem proposed in section 3.5.2.1. Considering the example of dynamic planning algorithm in section 3.4.2.2 on finding the irradiance equalization configuration, we get the results as shown in Figure 3-5 and Figure 3-13. There are 16 panels subject to position movement (maximum movement). 16 3.5.3.1. Application of the MAA algorithm The method of application is performed by the following steps: Step 1: - Consider m rows in the original G matrix as m workers respectively. - Consider m rows in the outcome G_OP matrix as m tasks respectively. - Cost C matrix is built on the principle that: Cij is the number of elements presenting in row i of the G matrix without presenting in row j of the G_OP matrix. Step 2: Apply the MAA algorithm to find the smallest total cost from the C matrix We have: 𝑧𝑝 =∑∑𝐶𝑖𝑗𝑥𝑖𝑗 𝑚 𝑗=1 𝑚 𝑖=1 = 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 (3-22) with 𝑥𝑖𝑗 = 1 when arranging the worker i with the task j. Step 3: Rearrange the position of rows in the G_OP matrix according to MAA results (the row i in the G matrix corresponds to the row j in the G_OP matrix when xij = 1). Rearrange the order of the elements in each row of the G_OP matrix corresponding to the G matrix. We get zp that represents the least number of solar panel position changes to convert the G initial connection matrix to the G_OP irradiance equalization matrix. So the number of key closing and opening times: 𝑀𝐼𝑚𝑖𝑛 = 2 × 𝑧𝑃 = 2 × ∑∑𝐶𝑖𝑗𝑥𝑖𝑗 𝑚 𝑗=1 𝑚 𝑖=1 = 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 (3-23) satisfy the objective function (3-18) in section 3.5.2.1. 3.5.3.2. Illustrative examples Consider the initial two matrices G and G_OP: 170 200 250 490 520 680 480 640 720 410 550 290 150 830 140 180 550 140 150 830 180 490 290 720 480 520 680 640 250 200 170 410 Matrix G Matrix G_OP 4 3 4 1 4 4 1 3 3 2 4 3 1 3 4 4 4 3 4 1 4 4 1 3 3 2 4 3 1 3 4 4 640 250 200 170 410 480 520 680 180 490 290 720 550 140 150 830 B1. Building cost matrix B2. Apply MAA B3. Rearrange the corresponding rows 3.5.3.3. Results evaluation Based on the position of elements of the matrix G and the matrix G_OP, a connection configuration is obtained in Figure 3-32. 17 (a) Initial configuration (b) Irradiance equalization configuration Figure 3-32. Example of irradiance equalization configuration It can be noticed that the position of 5 photovoltaic panels needs to be changed to change the initial connection configuration to the irradiance equalization configuration: 8, 10, 12, 16, 4, 11. Thus, after applying the MAA algorithm from the initial connection configuration G to the irradiance equalization connection configuration G_OP, the number of photovoltaic panels subject to change in position decreased from 16 to 5. For MAA algorithm with O(m3) complexity and m as the number of rows, in case of using 2.5GHz Intel Core i5 CPU, it only takes 0.122ms for the arrangement of 16 photovoltaic panels in each session of restructuring (CT1), thereby meeting real-time processing requirements. 3.5.4 Balancing the number of opening and closing times of switching matrix with the use of improved MAA (CT3) In the study published in (CT3), the improved MAA algorithm is proposed for the purpose of balancing the number of opening and closing times of the switching matrix, thereby extending the life of the switching matrix compared to the old method (section 3.5.3). 3.5.4.1. Proposal of improved MAA algorithm In case it is desirable to assign the worker u to do the task v, then assign the remaining (nM- 1) tasks to (nM-1) workers, the following method is proposed: Considering the C cost matrix in Figure 3-34. Worker Task 1 2 1 v 1 n 1 C11 1 C11 1 C11 1 2 C21 2 C21 2 C21 2 ... ... ... ... u Cu1 u Cu1 u Cu1 u ... ... ... ... nM 𝐶𝑛𝑀1 nM 𝐶𝑛𝑀1 nM 𝐶𝑛𝑀1 nM Hình 3-34. General C cost matrix 8 640 W/m2 1 170 W/m2 2 200 W/m2 3 250 W/m2 4 490 W/m2 5 520 W/m2 6 680 W/m2 7 480 W/m2 9 720 W/m2 10 410 W/m2 11 550 W/m2 13 150 W/m2 14 830 W/m2 15 140 W/m2 16 180 W/m2 + = 1110 W/m2+ + + + + = 2320 W/m2 + + = 1970 W/m2 + + + = 1300 W/m2 12 290 W/m2 + 1 170 W/m2 2 200 W/m2 3 250 W/m2 5 520 W/m2 6 680 W/m2 7 480 W/m2 9 720 W/m2 13 150 W/m2 14 830 W/m2 15 140 W/m2 + = 1670 W/m2 + + + = 1680 W/m2 = 1680 W/m2 + + + = 1670W/m2 12 290 W/m2 + 8 640 W/m2 10 410 W/m2 + + 16 180 W/m2 4 490 W/m2 + + 11 550 W/m2 18 Step 1: In the C cost matrix, create a matrix C’ by deleting all Cij values in row u and column v. Step 2: Apply the MAA algorithm (section 2.3.2) to find the minimum total cost with the C’ matrix consisting of the remaining elements (nM-1) x (nM-1). After obtaining result of MAA for the C’ matrix, create the result of the C matrix from the C’ matrix with the addition of Cuv. option. The results on the lowest cost vary as follows: 𝑧𝑃𝑛𝑒𝑤 = ∑ ∑ 𝐶𝑖𝑀𝑗𝑀𝑥𝑖𝑀𝑗𝑀 𝑛𝑀 𝑗𝑀=1 𝑗𝑀≠𝑣 𝑛𝑀 𝑖𝑀=1 𝑖𝑀≠𝑢 + 𝐶𝑢𝑣 → ∑ ∑ 𝐶𝑖𝑀𝑗𝑀𝑥𝑖𝑀𝑗𝑀 𝑛𝑀 𝑗𝑀=1 𝑛𝑀 𝑖𝑀=1 (3-24) 3.5.4.2. Improved MAA algorithm application method Step 1: - Consider m rows in the original G matrix as m workers respectively. - Consider m rows in the outcome G_OP matrix as m tasks respectively. - Cost C matrix is built on the principle that: Cij is the number of elements presenting in row i of the G matrix without presenting in row j of the G_OP matrix. Step 2: - Suppose we are considering the restructuring session stepk. - Find the maximum value of Sij in the closing and opening matrix Sstepk-1, Sij is the key of solar panel j in the G matrix. - Find the position of row u as the position of photovoltaic panel j in the G matrix. - Find the position of row v as the position of photovoltaic panel j in the G matrix. Apply the improved Munkres algorithm (item 3.5.4.1) to find the minimum total cost from the G matrix while assigning the worker u for the task v. We have: 𝑧𝑃𝑛𝑒𝑤 =∑∑𝐶𝑖𝑗𝑥𝑖𝑗 𝑚 𝑗=1 𝑗≠𝑣 𝑚 𝑖=1 𝑖≠𝑢 + 𝐶𝑢𝑣 →∑∑𝐶𝑖𝑗𝑥𝑖𝑗 𝑚 𝑗=1 𝑚 𝑖=1 (3-28) With: 𝑥𝑖𝑗 = 1 when arranging the worker i with the task j. In this case 𝑥𝑢𝑣 = 1. Step 3: Rearrange the position of rows in the G_OP matrix according to Munkres results (the row i in the G matrix corresponds to the row j in the G_OP matrix when xij = 1).. Rearrange the order of the elements in each row of the G_OP matrix corresponding to the G matrix. 3.5.4.3. Proof Considering at the generalized restructuring session k-1 The key with the most opening and closing times: 𝑚𝑎𝑥𝑆𝑠𝑡𝑒𝑝𝑘−1 = max 𝑖=1,𝑚 𝑗=1,g (𝑆𝑖𝑗)𝑠𝑡𝑒𝑝𝑘−1 (3-29) Considering at the generalized restructuring session k: 19 The improved MAA algorithm’s advantage is more remarkable than