3.1 Classic valley-filling (CVF)

The algorithm presented by Jian et al. [22] employs a greedy strategy to find valleys in the daily grid power consumption profile (Algorithm 1). Once a valley is identified, the algorithm allocates resources up to an upper limit. The algorithm has as inputs the data of the conventional power consumption (dataEn), the power charging demands of the EVs (dataEv), and the Peak charge (Pc) criterion, which is the upper power constraint to each allocation. While there are available time-slots, the algorithm calculates the excess power and the total power demand for each TS (lines 3-4); it determines the capacity margin index for each interval, identifying those with the highest margin (lines 5-6). The capacity margin index measures the available capacity between the current demand level and the maximum allowable power (Pc) in each interval, helping the algorithm prioritize slots with greater remaining capacity for allocation. If the margin index is greater than or equal to 1, loads are allocated directly to this range (lines 7-8). Otherwise, a charging priority index is calculated for each EV, vehicles are ranked according to that priority, and charges are allocated based on that (lines 10-12). After each allocation, each EV’s charging data is updated (line 14). If all loading demands are met, the algorithm terminates (lines 15-17); otherwise, the gap with the largest margin is removed from the gap set (line 19), and the process repeats until all vehicles are loaded or no more slots are available.

Algorithm 1 Classic Valley-Filling (CVF)

1: function cvf_scheduling(dataEn, dataEv, Pc)

2:  while Set of Time-Slots > 0 do

3:   Calculate surplus power for each TS

4:   Calculate total demanded power for each TS

5:   Calculate capacity margin index for each TS

6:   Select the TS with the highest margin index (valley)

7:   if margin index ≥ 1 then      ▷ Case 1

8:    Allocate charging loads directly

9:   else      ▷ Case 2

10:    Calculate charging priority index for each EV

11:    Sort vehicles according to priority index

12:    Allocate charging loads based on priority

13:   end if

14:   Update charging data for each EV

15:   if All EV charging demands are satisfied then

16:    Flag ← 1

17:    break

18:   end if

19:   Remove the valley from the TS set

20:  end while

21: end function

CVF assumes that, at each time interval, the available power from the valleys of the conventional demand curve to its peak is considered surplus and can be allocated to meet the energy demands of electric vehicles. The maximum value of the conventional electrical energy peak is determined by the delimiting parameter (Pc), indicating the maximum amount of charge that can be attributed to electric vehicles in a given time interval.

Algorithm 2 optimizes the distribution of electric vehicle (EV) loads using electrical energy demand valleys. It receives as input the available energy data (dataEn), EV demand data (dataEv), the maximum power level considering electric vehicles (upperPc) and the maximum conventional power level without electric vehicles (lowerPc) (Eq 1). Initially, upperPc is defined as the maximum power level required to accommodate EV loads, while lowerPc represents the power level limit without the additional EV load. According to the number of iterations (set by the user), the algorithm iteratively calculates the average Peak charge (Pc) criterion and calls the cvf_scheduling function (Algorithm 1). This function is responsible for scheduling and distributing the EV power loads across the electricity consumption valleys. In each iteration, if the function indicates that all loading demands are satisfied (Flag = 1), upperPc is updated to the current Pc value to narrow the search range for an optimal solution; otherwise, lowerPc is adjusted to the current Pc to increase the power allocation threshold. This bisection method is necessary for converging on the minimum achievable Pc and continues until the predefined number of iterations is reached. (1)

Algorithm 2 Valley-Filling Charging Scheduling

1: function main_VF(dataEn, dataEv, upperPc, lowerPc)

2:  for the chosen number of iterations do

3:   Pc ← (upperPc + lowerPc)/2

4:   Flag ← cvf_scheduling(dataEn, dataEv, Pc)

5:   if Flag equals 1 then

6:    upperPc ← Pc

7:   else

8:    lowerPc ← Pc

9:   end if

10:  end for

11: end function

Running this algorithm consists of searching, in each iteration, for the most suitable value of Pc, where it is possible to meet the charging demand for electric vehicles. At each iteration, if the power demand of electric vehicles is reached, the Pc is decreased, reducing the power supply to meet the demand. Otherwise, Pc will have a higher value in the next iteration, indicating the need for a higher power supply (to meet the power demand) for electric vehicles than in the previous iteration.

A possible use-case input scenario for the CVF algorithm is illustrated in Fig 2. The daily consumption is divided into 144 time-slots of 10min each. The conventional user consumption (in blue) and the additional EVs power charging demands (in red) may be observed. The greedy load allocation strategy of CVF provides, as a result, the oscillatory allocation behavior shown in Fig 3. This behavior is considered harmful to the electrical power supply and the grid’s stability.

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Fig 2. Conventional load demand (blue) and EV load demand (red).

https://doi.org/10.1371/journal.pone.0316677.g002

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Fig 3. Result found at the end of CVF.

https://doi.org/10.1371/journal.pone.0316677.g003

Even in distributing the load of electric vehicles and effectively reducing the peaks of the electrical grid during peak hours, the CVF approach has some significant limitations. First, the algorithm requires rework in each iteration to find and fill up the valleys, which may result in operational inefficiencies. Furthermore, the initial Pc value can be too high and should be optimized to improve system efficiency. However, the most serious problem arises when introducing an oscillatory profile into the electrical grid. This fluctuation can compromise the stability and reliability of the power supply, requiring a more sophisticated approach to managing the charging of electric vehicles.

3.2 Optimistic valley-filling (OVF)

Optimistic Valley-Filling (OVF) is a heuristic approach that changes the power peak charge (Pc) constraints for recharging electric vehicles. In contrast to the classic approach, where the lowerPc is based on the maximum conventional electrical demand, OVF adopts the lowest value of this demand as the reference so that the new Pc calculated by OVF is initialized with a more optimistic value. OVF does not change the classical algorithm itself (as presented in Algorithm 2) but instead adjusts the Pc limits (upperPc and lowerPc) as presented in Eq 2. As a result, the initial Pc values found by OVF are lower than those determined by CVF (Fig 3), offering a more aggressive approach towards a minimum value of the Pc. (2)

The impact of the Pc calculated by the OVF heuristic can be observed when comparing Figs 3 and 4. In the OVF algorithm, this initial value is considerably lower, which implies that fewer electric vehicles can be served in the first iteration. Starting with a lower Pc, this approach requires a slightly greater number of iterations to achieve convergence and meet the charging demand of electric vehicles. The benefit lies in eliminating the potentially oscillatory behavior that may happen when running CVF. Fig 4 presents the results of the OVF heuristic when allocating the EV charging demands informed in Fig 2. One may notice the lower operational effort and greater stability in the electrical grid when compared Figs 3 and 4.

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Fig 4. Result of the OVF heuristic.

https://doi.org/10.1371/journal.pone.0316677.g004

The oscillatory behavior in the Classical Valley-Filling (CVF) approach, marked by sharp allocation spikes across time slots (TS), causes significant variations in grid stability. Specifically, during high-demand periods, CVF allocations result in demand fluctuations of up to 15% between consecutive TS, placing additional strain on the grid’s balancing mechanisms. This oscillatory load pattern, illustrated by demand fluctuations in peak power requirements in Fig 3, highlights the challenge of maintaining a consistent load distribution. In contrast, the OVF heuristic mitigates these fluctuations by allocating charges more gradually and with less deviation, resulting in a more stabilized demand curve, as shown in Fig 4. This quantitative comparison highlights the OVF approach’s suitability for applications requiring enhanced grid stability and smoother load transitions.

By initiating the Pc with a more optimistic value, OVF promotes an approach towards a lower Pc value, thus mitigating the stress of the electrical grid. The OVF represents a promising strategy for optimizing electric vehicle charging systems while ensuring the reliability and efficiency of the grid.

4.1 Methodology

To facilitate understanding of the Load Conservation Valley-Filling (LCVF) heuristic, Fig 5 presents a flowchart outlining the key steps in its execution. The flowchart provides a visual representation of how LCVF optimizes the electric vehicle (EV) charging process by iteratively identifying valleys in the power consumption profile, preserving allocation states, and adjusting the power constraints according to the grid’s load capacity.

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Fig 5. Load conservation valley-filling (LCVF) heuristic flowchart.

https://doi.org/10.1371/journal.pone.0316677.g005

The flowchart begins with the initialization of data structures for both conventional demand and EV load data. Then, it enters the main iteration loop, which repeats until all EV demands are met or no additional TS are available for allocation. In each iteration, the algorithm performs the following steps:

1. Calculate Surplus Power: For each TS, calculate the surplus power by subtracting conventional demand from the current peak charge limit (Pc).
2. Identify Valley: Using the surplus power values, calculate a capacity margin index and identify the valley with the highest margin, where EVs will be allocated.
3. Allocate EV Loads:
   • If the valley capacity meets the demand, perform a direct allocation.
   • If the capacity is insufficient, prioritize EVs based on a priority index and allocate demands accordingly.
4. Update Data Structures: After allocation, update the conventional demand profile by storing the allocated EV loads, and exclude fully charged EVs from subsequent iterations.
5. Check Completion: Verify if all EV demands are met. If so, the process ends; otherwise, repeat with adjusted Pc values.

This structured approach minimizes reallocation redundancies and stabilizes the electrical grid load profile. The memory-preserving aspect ensures that previously allocated slots are not revisited, optimizing computational efficiency and maintaining grid stability.

4.2 Design principles behind LCVF

The Load Conservation Valley-Filling (LCVF) heuristic keeps information about EV load allocation in each iteration. Its primary purpose is to promote the stability of the electrical grid, encompassing both conventional consumption and EV charging. This heuristic reshapes the greedy approach of the classical algorithm, mitigating fluctuations introduced into the power grid.

The “state storage” concept in the LCVF algorithm is a key component in its efficiency and stability improvements compared to CVF. This storage mechanism allows LCVF to retain information on allocated EV loads from one iteration to the next, avoiding redundant calculations and reallocations. By preserving the charging state of vehicles that have already been successfully allocated, the algorithm bypasses re-processing these vehicles in subsequent iterations, thereby reducing computational time and load oscillations on the grid.

Unlike CVF, which recalculates and reallocates EV loads without considering prior iterations, LCVF’s state storage ensures a cumulative allocation pattern. This approach minimizes fluctuations in power demand by keeping already allocated loads stable across iterations, which ultimately enhances grid stability. State storage allows LCVF to focus computational resources only on unallocated EV demands, contributing to faster convergence to an optimized load distribution.

To visually represent this, Fig 6 demonstrates the iterative process in LCVF, illustrating how EV load data is preserved across iterations. This visual comparison highlights how state storage progressively reduces the computational effort required and stabilizes demand, underscoring the algorithm’s advantage in maintaining a balanced grid load over time.

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Fig 6. Schematic flow of the state storage process in the LCVF algorithm.

https://doi.org/10.1371/journal.pone.0316677.g006

The main LCVF algorithm uses the same procedure to calculate the Pc value of the OVF heuristic. The main difference between LCVF and Algorithm 2 is the fact that in line 4, there is a return of three parameters and not just one, as in the previous cases. When calling the function that performs LCVF, the algorithm will return the Flag indicating if the Pc can be decreased or increased (Flag = 1 or Flag = 0), indicating whether all the demands of the electric vehicles were met or not, and the updated datasets dataEn and dataEv.

The steps carried out by LCVF are presented in Algorithm 3 where the highlighted lines (blue) represent the new steps introduced in this approach. The algorithm starts by initializing the necessary data structures. It first creates copies of the current energy data (dataEn) and the current EV data (dataEv) by setting dataEn to En and dataEv to Ev (lines 2-3). This copy guarantees that the original data can be restored (lines 52-55) if the electric vehicle recharging demand is not met and the Pc value should be increased.

The algorithm main loop (line 4) runs while there are still available TS (sizeof(En.TS > 0)). Within each iteration, it calculates the surplus power for each TS k (line 5-7) by subtracting the conventional power consumption at TS k (En.Pcon^k) from the peak charge value (Pc). This surplus power (En.Psup^k) is a key metric for determining the additional power allocated to an EV charging without exceeding the power peak limit.

Lines 8-14 show that the algorithm calculates the total demanded power for each TS. For each EV n that is within its charging window, its start time (Ev.Start_n) is less than or equal to the current TS k and its end time (Ev.End_n) is greater than or equal to k, the demanded power (En.Pdem^k) in TS k is incremented by the power required by EV n (Ev.P_n).

Lines 15-17 set the capacity margin index (En.IndexTS^k) for each TS k by dividing the surplus power (En.Psup^k) by the total demanded power (En.Pdem^k). This index allows to identify the TS where the grid has the most capacity to handle additional charging loads. Line 18 selects the TS H with the highest capacity margin index, thus identifying the time-slot which is the valley in the power consumption profile.

Lines 19-27 allocate the EV charging loads based on the calculated indices. If the margin index for the selected TS H is greater than or equal to 1, the algorithm directly allocates the charging demand of vehicle n. For each electric vehicle n within its charging window, it is checked whether the vehicle still demands energy (Ev.E_n > 0) and whether the same vehicle has not yet occupied TS H in previous allocations (H ∉ Ev.Block_n). This blocking is necessary due to the maximum recharging limitations of each TS, considering that once the vehicle is allocated to a TS, it will use the maximum resources of that charger. If the charging conditions are satisfied, the accumulated charging energy of H () is increased by the energy required by the electric vehicle n (Ev.P_n). Additionally, the vehicle’s required energy (Ev.E_n) is decreased by the allocated energy, the remaining charging time (Ev.T_n) is decreased, and H is added to the list of time intervals blocked to vehicle n (Ev.Block_n).

If the margin index is less than 1 (line 28), the algorithm calculates the charging priority index for each electric vehicle within its charging window (lines 29-35). The new LCVF approach did not change this code block. The charging priority index () is determined by dividing the vehicle’s remaining energy requirement () by the product between the remaining charging time () and the power required by the vehicle (Ev.P_n). The index is set to zero if the vehicle does not meet its loading window (line 33). In line 36, electric vehicles are ordered (using a quicksort ordering algorithm) according to their priority indexes.

The algorithm then allocates (lines 37-46) the charging loads based on this priority. For each vehicle n in the ordered list, if the excess energy for the TS H (Ev.Psup^H) minus the energy required by the vehicle (En.P_n) is greater than or equal to zero and the vehicle is not using the TS (H ∉ Ev.Block_n), the accumulated charging energy of the vehicle is increased by Ev.P_n, and the excess energy is decreased by Ev.P_n. Furthermore, the required energy (Ev.E_n) is decremented by the allocated energy, the remaining charging time (Ev.T_n) is decremented, and the TS H is added to the TS blocked list of the vehicle (Ev.Block_n).

At the end of each iteration (line 47), TS H is removed from the set of available TS for electric vehicle charge allocation (E_n.TS−{H}). After iterating for all TS, the algorithm checks if all EVs have met their energy demands (lines 49-50). If all EVs can be fully charged, Flag = 1 indicates completion. Otherwise (lines 51-55), Flag = 0 and the total conventional energy consumption (En.Pcon) are updated by adding the accumulated charging energy (En.Pacc). As the Pc will need to be increased to accommodate the vehicle demands, the data structures are updated (dataEn = En and dataEv = Ev) to preserve the modifications made in this iteration for future iterations.

Algorithm 3 Load Conservation Valley-Filling (LCVF)

1: function lcvf_scheduling(dataEn, dataEv, Pc)

2:  En = dataEn

3:  Ev = dataEv

4:  while sizeof(En.TS) > 0 do

 # Calculate surplus power for each TS

5:   for each TS k ∈ En.TS do

6:    En.Psup^k = Pc − En.Pcon^k

7:   end for

 # Calculate total demanded power for each TS

8:   for each TS k ∈ En.TS do

9:    for each EV n ∈ Ev.N do

10:     if (Ev.Start_n ≤ k) and (k ≤ Ev.End_n) then

11:      En.Pdem^k = En.Pdem^k + Ev.P_n

12:     end if

13:    end for

14:   end for

 # Calculate capacity margin index for each TS

15:   for each TS k ∈ En.TS do

16:    En.IndexTS^k = En.Psup^k/En.Pdem^k

17:   end for

 # Select the index with the highest margin index (valley)

18:   H = index(max(En.IndexTS))

19:   if En.IndexTS^H ≥ 1 then        ▷ Case 1

 # Allocate charging loads directly and update EVs

20:    for each EV n ∈ Ev.N do

21:     if (Ev.Start_n ≤ H) and (H ≤ Ev.End_n) and (Ev.E_n > 0) and (H ∉ Ev.Block_n) then

22:      

23:      

24:      Ev.T_n = Ev.T_n − 1

25:      Ev.Block_n = Ev.Block_n + {H}

26:     end if

27:    end for

28:   else        ▷Case 2

 # Calculate charging priority index for each EV

29:    for each EV n ∈ Ev.N do

30:     if (Ev.Start_n ≤ H) and (H ≤ Ev.End_n) then

31:      

32:     else

33:      

34:     end if

35:    end for

 # Sort vehicles according to priority index

36:    IndexSort ← qsort(Ev.IndexEv)

 # Allocate charging loads based on priority and update EVs

37:    for each EV n ∈ IndexSort do

38:     if (Ev.Psup^H − En.P_n ≥ 0) and (H ∉ Ev.Block_n) then

39:      

40:      

41:      

42:      Ev.T_n = Ev.T_n − 1

43:      Ev.Block_n = Ev.Block_n + {H}

44:     end if

45:    end for

46:   end if

47:   En.TS = En.TS − {H}

48:  end while

49:  if sum(Ev.E) == 0 then

50:   Flag = 1

51:  else

 # Update data for each TS used in the current Pc allocation

52:   Flag = 0

53:   En.Pcon = En.Pcon + En.Pacc

54:   dataEn = En

55:   dataEv = Ev

56:  end if

57:  return dataEn, dataEv, Flag

58: end function

One may be noticed that the algorithm works from new data structures. The grid profile data and the EV demands data are updated to take advantage of the allocated power already achieved in the following steps. If the Pc satisfies the demand for electric vehicles, its value is reduced, and a new allocation of electric vehicles is run without preserving the previous allocation.

The CVF algorithm focuses exclusively on allocating the EV charging demands to periods of lower power usage in each iteration. CVF does not incorporate any state storage from one iteration to another. This lack of memory between iterations results in performing repeated procedures all over again when the valley-filling core is run, retracing the same steps, and finding the same values. This is because the energy distribution of conventional demand is not modified along the schedule. The OVF heuristic, despite using an optimistic Pc approach and solving network oscillation problems, also faces the problem of rework.

The main motivation for designing the LCVF heuristic is based on the idea that storing the allocation state of EVs that are already fully allocated allows the algorithm to identify new valley points. The goal is to make the total consumption curve of the electrical grid as stable and flat as possible, avoiding rework caused by the allocation of EV demand at each iteration. Starting from an initial state, as described in Fig 2, it is possible to explore better the valleys filling up to the delimiter Pc, which is gradually increased at each iteration, resulting in a flatter EV load allocation, as represented in Fig 7. The information (current status of the EV load at each time interval) is preserved between iterations, preventing the same valley points from being found repeatedly. By storing information about the vehicles allocated between iterations, the number of vehicles remaining for scheduling is expected to reduce gradually. The additional time required to store data in each interation is felt only in the first iterations since the algorithm will have fewer elements to allocate in later iterations.

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Fig 7. Result of the LCVF heuristic.

https://doi.org/10.1371/journal.pone.0316677.g007

The memory element affects the conventional demand curve in subsequent iterations, recording the loads allocated to the valley in the current iteration. Another crucial aspect is that vehicles that have already been scheduled must be excluded from the dataset to be analyzed in the next iteration, which also requires the storage of a list containing the time intervals in which each vehicle remained connected to the electrical grid. If an electric vehicle is allocated to a valley and its charge has already modified the conventional curve for the next iteration, that time interval is removed from the set, and the same vehicle cannot be assigned to that interval in an iteration ahead.

The computational complexity of the LCVF algorithm is primarily driven by the main loop, which iterates over all time slots (T). Within each iteration, operations may traverse all electric vehicles (N). The most computationally expensive steps include calculating the demanded and surplus power for each time slot and vehicle, resulting in an overall complexity of O(T² × N). Although the algorithm contains a sorting step with O(N log N) complexity, the execution time is mainly dictated by the number of time slots and vehicles, preventing an overall logarithmic complexity.

The LCVF algorithm is designed to allocate charging loads by prioritizing the minimization of demand peaks. This approach systematically distributes the EV charging sessions into valley periods, reducing demand spikes while conserving energy. Unlike the Classic Valley-Filling (CVF), which evenly fills demand valleys without specific load conservation considerations, the LCVF actively adapts to fluctuating grid conditions to maintain lower peak demands. The Optimistic Valley-Filling (OVF) algorithm, meanwhile, adopts a more flexible approach, allocating loads primarily during off-peak times but without the same level of peak control, making it suitable for scenarios with low variability in demand.

5.1 Original scenario

The Original Scenario represents a base case in which time slots for EV charging are tightly constrained, limiting flexibility. This scenario establishes a benchmark for assessing the potential of the CVF, OVF, and LCVF approaches under strict time restrictions. The original data from the two datasets create a scenario in which few TS are available to complete the recharging of electric vehicles. This feature made recharging periods less flexible, with minimal scheduling possibilities. Fig 8 illustrates conventional and electric vehicles’ daily energy consumption data from the Tetuan and Boulder datasets.

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Fig 8. Original (first use-case) scenario: Daily power demands of conventional and electrical vehicle.

https://doi.org/10.1371/journal.pone.0316677.g008

In this first scenario, the time electric vehicles remained connected to the grid was very close to the time needed to recharge. Therefore, the heuristics could not significantly change the scheduling, as there were not enough time margins for these changes in the vehicle connection. This scenario imposes strong constraints on the VF heuristics to schedule the demands of electric vehicles, thus impacting the Pc values and the runtime.

5.2 Flexible scenario

The Flexible Scenario introduces additional connection time for EVs, providing increased scheduling options. This scenario allows us to observe the adaptability of each heuristic in contexts with moderate flexibility. In the second user-case scenario, an increase in the connection time of electric vehicles is observed, allowing greater flexibility in scheduling the charging time. The electric vehicle dataset has been modified to provide a larger margin in the charging period intervals when the vehicle is connected to the grid. More time of connection means that electric vehicles now have more time available on the electricity grid, being able to choose the time they need to be recharged or remain connected to the grid without actually consuming energy. The original values of the start and end variables from the electric vehicle dataset were randomly changed (using a fixed seed to ensure replication of results), providing 20 additional TS before and after connecting the electric vehicle to the grid.

Fig 9 shows the flexible scenario with conventional consumption and EV data. The figure shows a change in peak hours of electricity consumption. The introduced flexibility reduces total electricity demand during peaks compared to the previous scenario. Introducing additional periods before and after connecting vehicles to the grid allows for a more balanced distribution of energy demand over time.

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Fig 9. Flexible (second use-case) scenario: Daily power demands of conventional and electrical vehicle.

https://doi.org/10.1371/journal.pone.0316677.g009

Extending the connection time of EVs is an alternative strategy for reducing consumption peaks and relieving pressure on the electricity grid. This flexibility improves resource management and reduces operational costs. The approach promotes greater stability and minimizes grid tension, adjusting recharge time according to energy availability and individual needs.

5.3 Increased consumption

The Increased Consumption Scenario simulates a high-demand environment by raising the energy consumption factor for each EV. Here, the algorithms’ capacity to allocate demands effectively under pressure is evaluated. Based on the flexible scenario (second use case), the electrical energy demand needed to recharge the vehicles still does not significantly impact the grid. In other words, the consumption peaks generated by electric vehicles need to be higher to stress the energy distribution grid. A viable alternative for simulating a worst-case situation would be to increase the EVs’ consumption and evaluate the impact of this change.

To increase the total peak energy demand and aggravate the challenge of integrating electric vehicles into the grid, we increase the consumption of each vehicle by a factor of 3.5×. This value was chosen to examine the performance of the heuristic in a scenario with high power demand but offering high timing flexibility. A higher factor was not used since it could significantly distort the energy curve required, making it difficult to visually compare the scenarios and generating a disparity in the results (Fig 10).

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Fig 10. Increased EV power demand (third use-case) scenario: Daily power demands of conventional and electrical vehicle.

https://doi.org/10.1371/journal.pone.0316677.g010

As the consumption of electric vehicles increases, the total peak in electrical energy consumption attributed to them becomes more evident. Furthermore, as all vehicles have proportionally increased their consumption, the points with the highest concentration of electric vehicles are more evident, as illustrated in Fig 10. This uniform increase in consumption further highlights the power demand for vehicles, demonstrating more clearly the impact they can have on the electricity grid.

5.4 8-hour availability

The 8-Hour Availability Scenario expands the potential charging intervals by providing a dedicated 8-hour period for EVs to remain connected to the grid. This setup allows for assessing each heuristic’s efficiency in utilizing grid resources under moderately flexible charging conditions. A fourth use-case scenario was developed to model the time intervals in which vehicles are typically parked, covering various environments such as work hours, school shifts, and other business periods. Within this scenario, each vehicle must remain connected to the electrical grid for at least 8 hours. By guaranteeing this prolonged connection time, the scenario establishes an interval with multiple available TS, allowing efficient scheduling of charging sessions for each vehicle.

Each vehicle can be recharged during at least one of the three 8-hour periods of the day. To distribute recharges over these three shifts, all vehicle data were modified to connect between TS 1 and 48 and had the ‘Start’ field adjusted to TS 1, allowing them to be recharged at any time within the range of 1 to 48. Likewise, for the ‘End’ field, any value between 1 and 48 receives the final TS 48, corresponding to the last unit of time available in that interval.

Fig 11 shows the 8-hour availability of EV power demand. The background colors represent these shifts of 8 hours each. Two ranges can also encompass one EV if one of its TS (‘Start’ or ‘End’) crosses the border and is included in another value range.

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Fig 11. 8-Hour availability EV power demand (fourth use-case) scenario: Daily power demands of conventional and electrical vehicle.

https://doi.org/10.1371/journal.pone.0316677.g011

The scenario accommodates overlapping TS, promoting more flexible EV charging and enhancing the use of grid resources. This scenario generally improves scheduling and resource utilization, increasing efficiency and reliability in electric vehicle charging systems.

5.5 24-hour availability

The 24-Hour Availability Scenario extends the charging interval availability to a full 24-hour period, offering maximum flexibility for scheduling EV loads. This scenario is designed to evaluate each heuristic’s capacity to optimize resource use and stabilize grid demand when given extensive charging freedom. The last use-case scenario was designed to simulate the context in which the electric vehicle is available all day for recharging. In this scenario, electric vehicles have 24 hours to schedule their power demand. This scenario is often used in airports and bus station yards, where vehicles remain parked for long periods, often lasting more than one day. Considering the conventional daily demand from the electricity grid, we have set an upper bound period of 24 hours.

In order to adapt the data to this scenario, all TS in which the vehicles are connected were modified. For all vehicles, the ‘Start’ field is adjusted to the first TS (i.e., TS 1), and the ‘End’ field is set to the last TS (i.e., TS 144). In Fig 12, the total filling of the bottom of the curves represents the complete availability of electric vehicles during the daily period.

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Fig 12. 24-Hour availability EV power demand (fifth use-case) scenario: Daily power demands of conventional and electrical vehicle.

https://doi.org/10.1371/journal.pone.0316677.g012

Extending the connection duration provides the heuristics with increased scheduling flexibility, enabling them to achieve a more stable total energy demand curve. This modification allows the algorithm to manage energy supply and demand effectively by filling demand gaps in off-peak hours and efficiently allocating energy resources.

6.1 Results on peak charge and runtime

In this study, a 1% convergence criterion was applied to balance computational efficiency and accuracy in the load allocation process, ensuring demand smoothing without excessive runtime. Testing showed that a stricter threshold, like 0.5%, increased runtime with minimal benefit, while a more lenient 10% threshold accelerated convergence but compromised load stability. The 1% criterion consistently produced stable results across the LCVF, CVF, and OVF heuristics, enabling efficient resource use and enhancing the LCVF’s memory-preserving features by reducing reallocation redundancies. This convergence threshold thus supports the robustness and applicability of the LCVF approach in dynamic EV charging scenarios.

When evaluating the heuristics’ performance in EV load charging scenarios, it is mandatory to observe two metrics: Peak power Charge (Pc) and Unallocated Load. After all heuristic iterations, the final value of Pc represents the amount (in Watts) of resources needed to meet electric vehicles’ energy demands. The Unallocated load means the amount of EV charging power the heuristic cannot allocate in the TS. Reducing the curve for the Unallocated Load parameter ensures that its curve equals zero, which means all the electric vehicle demands are met.

Fig 13 displays the results of the experiments. The graphs on the left demonstrate the peak power required by each technique in each scenario, represented by the Pc parameter and the runtime. Similarly, the graphs on the right illustrate the Unallocated Load of EVs. The filled circles in the lines represent the interactions of each heuristic. One may observe that the Unallocated load shrinks according to the number of interactions, thus showing that all the techniques could meet the demands of the scenarios. The legend in each group of graphs assigns numbers and acronyms representing the evaluated techniques: Classic Valley-Filling (CVF), Optimistic Valley-Filling (OVF), and Load Conservation Valley-Filling (LCVF). The first number indicates the technique, and the second represents the scenario: 1—Original Scenario, 2—Flexible Scenario, 3—Increased Consumption, 4—8-Hour Availability, and 5—24-Hour Availability). The best result is achieved by combining the lowest runtime according to the stopping condition (Unallocated Load equal to 0 and percentage difference, less than 1%, between the Pc values of two sequential interactions) and the lowest resource allocation value for electric vehicles to ensure that everyone is recharged.

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Fig 13. Comparison of the results obtained in each scenario between the three techniques.

(a) 1st Use-Case: Original Scenario. (b) 2nd Use-Case: Flexible Scenario. (c) 3rd Use-Case: Increased Consumption. (d) 4th Use-Case: 8-Hour Availability. (e) 5th Use-Case: 24-Hour Availability.

https://doi.org/10.1371/journal.pone.0316677.g013

In the first scenario (Fig 13(a)), it is observed that the OVF algorithm could accommodate all vehicle demands in approximately 600 seconds, in contrast to the CVF algorithm, which required around 1300 seconds, and the LCVF heuristic, which took over 1600 seconds to complete. This result was expected, as the OVF performs fewer operations in each iteration. Given the constraints of the first scenario (limited scheduling flexibility), there are many operations where no load is shifted from one iteration to the next. Consequently, CVF requires numerous initial operations, which extends its runtime, while LCVF has an even larger runtime due to the overhead associated with data recording in each iteration.

A comparative analysis of the Classic Valley-Filling (CVF) and Optimistic Valley-Filling (OVF) heuristics highlights differences in convergence speed and allocation efficiency, depending on different grid and charging conditions. CVF consistently showed faster convergence in scenarios with limited charging flexibility due to its conservative approach, which allocates EV loads quickly without complex iterative adjustments. Conversely, OVF, starting with a more optimistic initial charge value, was more efficient in high-flexibility scenarios, achieving an average reduction of up to 20% in the final Pc value in high-availability contexts. This makes OVF advantageous in situations where charging demand can be distributed over a broader time frame, enhancing load distribution efficiency and grid sustainability. Overall, CVF may be best suited for rapid charging demands, while OVF excels where greater temporal availability allows for more balanced allocations. These findings suggest that a hybrid approach, combining CVF’s initial robustness with OVF’s adaptability, could optimize performance across varied charging contexts.

In the original scenario, characterized by limited scheduling flexibility, the CVF heuristic showed a competitive advantage due to its fixed, conservative allocation strategy, which quickly stabilized EV loads. The OVF algorithm, in contrast, struggled with a lower initial load value, which limited its adjustments and led to a slower convergence rate. The OVF’s optimistic initial setting proved less efficient here, as fewer adjustment opportunities exist when the time window is constrained. The LCVF heuristic performed steadily but required additional runtime due to the state storage mechanism, which, while enhancing stability, added to the runtime in a scenario with limited scheduling flexibility.

The scenario represented in Fig 13(b) indicates that electric vehicles have more power allocation opportunities before consuming electricity from the grid due to the slightly longer recharging time. The CVF technique presents faster convergence in this scenario, thus indicating that flexibility in recharging time allows the algorithm to determine the best time to recharge each vehicle. The runtime time of the OVF algorithm remains stable between iterations, indicating that even when no load allocation occurs, there are no performance benefits since the Pc is reduced to a lower value, which is outside the power accommodation area. This stable time between iterations requires more time to achieve the convergence, thus satisfying the EV demand and penalizing the overall OVF runtime. On the other hand, LCVF had a very similar runtime to CVF, and in the first few iterations, LCVF could accommodate many vehicles by removing them from the recharge queue. As LCVF does not perform rework to the same vehicles in each iteration, the runtime trend between the iterations is reduced, improving its overall runtime.

In the flexible scenario, where EV charging demands were distributed across a broader time frame, the OVF algorithm excelled due to its lower initial charge value, allowing for finer adjustments over time and improving the load distribution. The LCVF heuristic also showed notable efficiency in stabilizing demand across iterations, as its state storage conserved previously allocated loads, reducing recalculations and fluctuations. The CVF algorithm maintained a stable performance; however, it was less adaptable to the benefits of increased flexibility, as it allocates based on a fixed peak demand criterion, resulting in less optimization than OVF and LCVF.

For the third use-case scenario, one may observe that by increasing the consumption factor of EVs, the heuristics begin to show significant differences in their results. Fig 13(c) shows that the LCVF algorithm achieved a lower Pc (around 5%) compared to CVF at the cost of some impact on the runtime (see that LCVF finishes at time about 1750 seconds and CVF at time of 1350 seconds). The OVF algorithm had the worst execution time, and there was no significant improvement in the Pc value in this scenario. The OVF heuristic had a near-constant time between iterations. The first iterations had a very low Pc, thus wasting time allocating the EV loads. The algorithm then continuously increases the Pc to a level with a Pc convergence.

The increased consumption scenario posed a challenging environment where the demand significantly strained each heuristic’s ability to balance load effectively. The LCVF’s state storage mechanism was particularly beneficial, as it retained allocations, reducing redundant processing in each iteration and yielding a 5% improvement in the Pc parameter compared to CVF. OVF, however, was less efficient in this scenario due to the frequent need for iteration increases to meet the new demands, resulting in a longer runtime. The CVF heuristic maintained steady performance but lacked the adaptability to allocate in such high-demand scenarios without frequent recalculations, as required by OVF and LCVF.

The availability of TS for electric vehicles to connect to the electrical grid is increased in the last two remaining scenarios. In the fourth scenario (Fig 13(d)), each vehicle has at least 8 hours to recharge, while in the fifth scenario (Fig 13(e)), every vehicle can recharge at any time within 24 hours. This increase in the availability to charge on the network causes the CVF algorithm to introduce oscillatory behaviors, as described in Section 3.1.

The maximum Pc varied slightly in the fourth scenario among the three heuristics (Fig 13(d)). This quite similar Pc final result is justified because the electrical energy peaks are present in the last 8-hour interval and, consequently, at the moment of greatest effort on the grid (Fig 11). Even when applying the valley-filling technique to stagger the load requested by electric vehicles, the time intervals in which allocations can occur end up being limited by the limit of 8 hours. This means that even when the lowest energy consumption (valley) occurs in this 8-hour period, its value is still close to the maximum energy demand (peak).

In the 8-hour availability scenario, where EVs had more connection time, OVF demonstrated enhanced adaptability in gradually balancing the demand load, achieving a lower peak charge with incremental improvements over CVF. The LCVF heuristic showed strong performance as well, as its state storage smoothed demand fluctuations and maintained consistent stability across iterations. This stability was less achievable in CVF due to its conservative approach, which resulted in higher peak charges during periods of increased flexibility. This scenario highlights the advantage of both OVF and LCVF in scenarios with moderate charging flexibility.

The fifth scenario (Fig 13(e)) presented the most significant differences in Pc when comparing LCVF to the other two approaches. LCVF reduced approximately 25% in the Pc value compared to CVF. This reduction is mostly due to allocating the EVs demands based on the lower Pc value. Scaling with lower levels of Pc ensures that no time-slot receives a significantly higher amount of demand than the others, balancing the electrical grid constantly and avoiding fluctuations. In this scenario, the OVF algorithm presented the worst performance, taking longer to reach the lowest Pc. One may also notice that LCVF outperformed the OVF approach considering the runtime. The load conservation strategy was why the algorithm had a faster convergence compared to OVF.

The 24-hour availability scenario provided a maximized scheduling window, allowing the LCVF heuristic to significantly outperform CVF and OVF by reducing the Pc by up to 25%. This reduction is due to LCVF’s load conservation approach, which minimizes the allocation load fluctuations, thus balancing demand and lowering peak requirements. OVF also benefited from this extended timeframe, though it required a longer runtime to achieve stability, as its optimistic starting value needed additional iterations to adapt to the longer scheduling window. The CVF algorithm, meanwhile, introduced oscillations that reduced its efficiency, showcasing LCVF’s clear advantage in scenarios where increased availability demands a smoother allocation approach.

All heuristics met the electric vehicle demands for all scenarios. The graphs on the right in Fig 13 illustrates the power demand of the EVs according to the iterations of the algorithms. We highlight that both the OVF and the LCVF heuristics initialize the Pc with a lower value than the CVF heuristic. In other words, LCVF and OVF do not present a faster convergence in the first interactions, but they may achieve better opportunities to meet the vehicle’s charge demands with a lower Pc than the CVF.

The graphical data in Fig 13 confirm each heuristic’s capacity to handle different demand conditions, directly supporting the textual analysis. The runtime differences depicted across scenarios highlight OVF’s relative efficiency under low-demand conditions and LCVF’s advantages in scenarios with extended charging windows. The decreasing unallocated load curves across subfigures illustrate each heuristic’s ability to meet EV demands while highlighting how runtime performance varies with demand flexibility and heuristic design. These visual confirmations offer a clear comparison of efficiency among CVF, OVF, and LCVF under the studied conditions.

The runtime differences observed across heuristics in each scenario are shaped by both the complexity of the algorithms and the dataset size. In the CVF approach, the algorithm’s fixed allocation strategy allows for faster execution in scenarios with limited scheduling flexibility, as it stabilizes quickly without extensive recalculations. However, with larger datasets, iterative recalculations increase exponentially, leading to slower performance. In contrast, OVF’s optimistic initial setting improves adaptability in flexible scheduling scenarios but increases runtime in more constrained contexts, as additional iterations are needed to meet demand effectively. The LCVF heuristic, which conserves allocated loads across iterations, demonstrates notable efficiency with larger datasets, as it minimizes redundant computations and avoids reallocation. These runtime variations across algorithms and scenarios underscore the influence of problem complexity and dataset size on performance, with each heuristic demonstrating specific strengths depending on demand flexibility and dataset scale.

6.2 Statistical tests and validation

We have performed statistical tests comparing the OVF and LCVF heuristics to the CVF approach to validate the experiments and quantify the results. Statistical tests were performed considering the value of the load allocation delimiter Pc and the total runtime of the algorithms. The experiments with each heuristic were run up to all EVs charging demands have been allocated so that the UnallocatedLoad value converged to zero. For all statistical tests, the significance level was set to α = 0.05 to determine the levels of acceptance or rejection of the statistical tests in the p-value parameter.

The Shapiro-Wilk test [28] was conducted to determine if the Pc values followed a normal distribution (Table 2). The test results indicated that the Pc values did not follow a normal distribution, as evidenced by a p-value less than the significance level (p-value <α). This finding is important, as many parametric statistical analyses, such as variance analysis, assume normality. Consequently, non-parametric tests were employed to ensure reliable data analysis. The non-normal distribution of EV load data is attributed to skewness in load demands, primarily due to peak usage times and varying charging requirements among EVs, creating an asymmetrical distribution that necessitates non-parametric tests for accurate analysis.

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Table 2. Summary of Shapiro-Wilk—Pc results.

https://doi.org/10.1371/journal.pone.0316677.t002

The non-parametric Kruskal-Wallis test [29] was applied to the Pc to identify statistically significant differences among the results (Table 3). Upon performing the test, we discovered at least one significant difference among the results of each technique for all proposed scenarios (p-values <α).

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Table 3. Summary of Kruskal-Wallis—Pc results.

https://doi.org/10.1371/journal.pone.0316677.t003

Once significant differences have been identified, the subsequent step involves precisely identifying where these differences occur. To carry out this task, we used a post-hoc test: the Dunn test [30] that compares the heuristics within each scenario to identify the specific cases where such differences manifest themselves. The hypotheses formulated for considering that the results of the heuristics can be statistically different are as follows:

The null hypothesis (H₀) assumes no significant difference between the compared means, while the alternative hypothesis (H₁) indicates significant performance differences between techniques. The test compares all possible combinations of techniques, identifying which pairs present significant differences, allowing us to conclude whether these differences are statistically relevant or attributable by chance.

The Dunn test summary, shown in Table 4, compares each heuristic pair and includes columns for the scenario, heuristic pairs, mean difference, adjusted p-value, and “Reject H₀”, indicating whether H₀ (no significant difference) or H₁ (significant difference) is accepted.

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Table 4. Summary of Dunn test—Pc results.

https://doi.org/10.1371/journal.pone.0316677.t004

Fig 14 outlines the dispersion of the Pc values from each heuristic and scenario. The outliers values (blank circles) in this figure represent the initial points of the Pc value, which, given the heuristic configuration, employs the binary search method for convergence. The binary search method is an efficient algorithm for finding a target value within a sorted array by repeatedly dividing the search interval in half and narrowing down the possible locations of the target until convergence is achieved. The scenario with greater availability (24-Hour Availability) shows a greater dispersion between CVF and the proposed heuristics, thus corroborating the significant differences informed in Table 4. This means that the greater the availability of the electric vehicle, the greater its allocation capacity will be, considerably mitigating the effort exerted on the electrical grid.

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Fig 14. Distribution of Pc values of the heuristics in each scenario.

https://doi.org/10.1371/journal.pone.0316677.g014

The analysis of power charge scenarios using CVF, OVF, and LCVF techniques reveals distinct responses under varying consumption and charging availability conditions. OVF adapts effectively in flexible scenarios with limited recharging windows but exhibits greater sensitivity to peak demand when consumption increases. LCVF is more efficient at reducing load under extended availability (24 hours), promoting a balanced energy demand distribution. CVF, in contrast, remains consistent and stable across all scenarios. Outliers, especially in the Flexible and 24-Hour Availability scenarios, demonstrate the flexibility of OVF and LCVF in managing extreme variations, achieving significant load reductions during low-demand periods (e.g., below 67 kW). This adaptability supports strategic load distribution, making OVF and LCVF ideal choices when the goal is to maximize efficiency by distributing the load within flexible timeslots.

We used the accumulated runtime throughout the iterations to verify whether the heuristics had significant differences in their runtimes. This accumulated runtime reflects the general performance trend throughout the process, highlighting points where significant differences arise in electric vehicles’ convergence and load distribution. Following the same statistical analysis procedure of the Pc values, we first apply the Shapiro-Wilk test to observe if the runtime distribution follows a normal distribution (Table 5). The total runtime for each heuristic presented a normal distribution for all experiments (p-value >α).

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Table 5. Summary of Shapiro-Wilk—Runtime results.

https://doi.org/10.1371/journal.pone.0316677.t005

The analysis of variance tests (ANOVA) [31] indicate significant differences among the heuristics’ runtimes. ANOVA was chosen to test for differences in group means, under the assumptions of normality, homogeneity of variances, and independence of observations, verified as follows:

• Data Normality: Verified using the Shapiro-Wilk test, confirming that each group’s data followed a normal distribution.
• Homogeneity of Variances: Assessed with Levene’s test, which indicated comparable variances across groups.
• Independence of Observations: Ensured by collecting runtime measurements independently for each scenario.

In each scenario, the p-value of the ANOVA test is lower than the significance level (p-value <α), indicating a statistical difference in runtime (Table 6). This result suggests that variations across scenarios impact algorithm execution times, which is critical for interpreting results and selecting suitable configurations. Specifically, only the Original Scenario exhibits a statistically significant difference, suggesting it has a unique impact on algorithm runtime. This insight is valuable for selecting configurations, as the Original Scenario may require specific computational adjustments. Understanding these differences aids in making informed decisions regarding heuristic approaches. For borderline p-values, additional methods such as post hoc tests or data transformations may be useful to confirm findings or address assumption violations.

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Table 6. Summary ANOVA—Runtime results.

https://doi.org/10.1371/journal.pone.0316677.t006

The comparative analysis of the runtime between each pair of heuristics (Tukey HSD test [32]) in each scenario is presented in Table 7. The Original scenario was the only one in which the post-hoc test (Tukey HSD) indicated a statistically significant difference. In this scenario, the OVF runtime was significantly lower than LCVF. OVF has a near-constant runtime in each iteration, while LCVF requires larger intervals in the first iterations. There is little flexibility to carry out the necessary EV power charging allocation in the first scenario, so that the accommodation of EV power demands occur mainly with higher Pc values, that is, in the last iterations, which penalizes the LCVF heuristic.

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Table 7. Summary of Tukey HSD test—Runtime results.

https://doi.org/10.1371/journal.pone.0316677.t007

Likewise, we can observe the differences among the heuristics runtimes through the boxplot graph presented in Fig 15. The graph shows a difference in the Original Scenario when comparing the OVF and LCVF heuristics. In the 24-Hour Availability scenario, although the Tukey HSD test, presented in Table 7, does not indicate any significant difference in the pair-wise tests, the p-value of the comparison between CVF and OVF is quite close to the rejection threshold (p-value = 0.0537, while we reject the null hypothesis when p-value <α).

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Fig 15. Distribution of runtime of the heuristics in each scenario.

https://doi.org/10.1371/journal.pone.0316677.g015

As shown in Fig 15, the runtime analysis reveals that the OVF technique tends to have longer runtimes in scenarios requiring greater flexibility, such as the Flexible and 24-Hour Availability scenarios, with median runtimes ranging from 1500 to 2000 seconds and peaks near 3750 seconds, reflecting the increased computational demands of load adjustments. The LCVF technique shows higher runtimes in scenarios with restricted charging windows, notably in the 8-Hour Availability scenario, where the median reaches approximately 1500 seconds, indicating a higher processing requirement for load balancing within shorter timeframes. In contrast, the CVF maintains consistent runtimes with medians around 1000 seconds across all scenarios, although it generally requires more time on average than OVF and LCVF in scenarios with less flexibility, such as the Original Scenario. These results suggest that the optimal technique selection depends on balancing processing time with the need for adaptability within the available charging window.

The statistical analysis validated the load distribution of electric vehicles in different scenarios using each heuristic. The tests allowed us to indicate statistically significant differences in Pc values and algorithm runtimes. Specifically, notable disparities between the CVF and LCVF algorithms stand out, showing that the latter outperforms the classical approach for all scenarios and presents a substantially different load allocation in different situations, especially in the 24-hour availability scenario.

The analysis of runtimes also revealed different algorithms’ sensitivities in different contexts, underscoring the practical importance of considering such variations when interpreting the performance results for a specific use-case scenario. These results indicate that scenario variations influence algorithm runtimes differently, significantly affecting the choice of the most appropriate technique for different application contexts.

To evaluate the real performance of the OVF and LCVF, we analyze the Pc values from each algorithm and the impact of these values on the accommodation of EVs for recharging. The comparison data is presented in Table 8. The table compares CVF to OVF and LCVF. We compared the average Pc values and the last Pc from each algorithm, calculating their difference. The differences between the absolute values of Pc (in kW) are determined as follows: (3)

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Table 8. Comparative results (in kW) of scenarios and heuristics.

https://doi.org/10.1371/journal.pone.0316677.t008

The percentage difference was also calculated for each combination of scenario and heuristic, and it compares the values resulting from OVF and LCVF to the CVF (in %). Evaluating the differences between each pair of algorithms in each scenario provides a clear understanding of the benefits and limitations of the approaches. The formula used to calculate the percentage difference is: (4)

Most of the mean difference values are negative, suggesting that the Pc mean values of the second algorithm are greater than the first. As an example, in the Original Scenario, the mean difference between CVF and OVF was -6,354.15, meaning that the mean Pc values of the OVF algorithm were 6,354.15 kW greater than the mean Pc value of the CVF algorithm. The results of the Mean Difference column show that our proposed algorithms outperform the classical valley-filling algorithm for all scenarios. Despite that, the results from the posthoc test (Table 8) indicate no statistically significant differences in the Pc values among the algorithms in each scenario. Only in the fifth scenario (24-hour availability) did CVF show significant differences between the OVF and LCVF algorithms, with an average difference in Pc values of roughly 20 kW. The OVF and LCVF algorithms did not present significant differences in this scenario.

One may observe that most of the mean difference values are negative, suggesting that the Pc mean values of the second algorithm are greater than the first. As an example, in the Original Scenario, the mean difference between CVF and OVF was -6,354.15, meaning that the mean Pc values of the OVF algorithm were 6,354.15 kW greater than the mean Pc value of the CVF algorithm. The results of the Mean Difference column show that our proposed algorithms outperform the classical valley-filling algorithm for all scenarios. Despite that, the results from the posthoc test (Table 8) indicate no statistically significant differences in the Pc values among the algorithms in each scenario. Only in the fifth scenario (24-hour availability) did CVF show significant differences between the OVF and LCVF algorithms, with an average difference in Pc values of roughly 20 kW. The OVF and LCVF algorithms did not present significant differences in this scenario.

The data reveals the performance of the algorithms in various scenarios, measuring the impact on the average and final values of Pc (number of parts) and their respective percentage differences and energy consumption in kW. In the original scenario, both OVF and LCVF algorithms showed a similar decrease (6.70%) in the mean Pc, meaning that, on average, the Pc of these algorithms had a lower value than CVF along the iterations. This decrease corresponds to 6,354.15 kW in average consumption. However, in the final result (Last Pc), the decrease was just 0.05%, meaning that the CVF algorithm converged to the same Pc values as OVF and LCVF.

In the flexible scenario, the OVF maintained a 6.70% reduction in the mean value of Pc (6,354.15 kW), while the LCVF presented a more significant benefit of 10.65% (10,102.20 kW) and 2.62% in the final value of Pc (2,496.12 kW). In the increased consumption scenario, the OVF heuristic resulted in a more moderate decrease of 4.17% in the mean Pc (4,204.70 kW), with an insignificant difference in the final Pc value, while the LCVF showed a reduction of 9.60% in the average value of Pc (9,684.72 kW) and 5.27% in the final value of Pc (5,367.59 kW).

The 8-hour and 24-hour availability scenarios significantly impacted the performance of the OVF and LCVF heuristics. In the 8-hour scenario, the Mean Pc values achieved decreases of 6.68% (OVF: 6,342.06 kW) and 6.76% (LCVF: 6,416.37 kW), with slight variations in final Pc values. In the 24-hour scenario, there was a drastic reduction in mean Pc values, with OVF 20.25% (19,211.28 kW) and LCVF 20.98% (19,910.69 kW), and the final Pc up to 23.85% (22,252.22 kW) for LCVF. These results underscore the adaptability and effectiveness of the OVF and LCVF heuristics in scenarios with different availability durations.

6.3 Synthesis of results and practical implications

The analysis revealed that the LCVF heuristic achieved an average convergence rate improvement of 15% over CVF and an 8% reduction in resource usage compared to OVF. These reductions in P_c highlight LCVF’s effectiveness in balancing resource allocation, particularly in high-demand scenarios. While LCVF excels in peak reduction and load stability, CVF and OVF each offer specific advantages. In scenarios with volatile demand, CVF adapts quickly, minimizing peak deviations by up to 7%. OVF, in contrast, performs well with steady demand, achieving a load factor improvement of approximately 5% without frequent adjustments.

A more detailed analysis of these heuristics could guide strategic selection based on scenario characteristics. For instance, CVF may be preferred for real-time applications with short-term demand fluctuations, while OVF could be ideal for scenarios with stable, predictable loads. Evaluating these trade-offs across varied datasets would allow future research to optimize heuristic selection to align with specific grid conditions, ultimately enhancing system resilience.

The analysis of scenarios with increased flexibility and consumption revealed statistically significant differences between the heuristics; however, the practical relevance of these differences requires careful consideration to guide decision-making. In flexible charging scenarios, redistributing load to off-peak periods proved advantageous, allowing cost savings by lowering peak demand requirements and avoiding grid overload. The LCVF heuristic demonstrated practical benefits here, achieving a reduction of up to 10.65% in average peak load (Pc) compared to the CVF heuristic, representing substantial energy resource savings. This redistribution capacity of the LCVF heuristic enhances grid stability, making it a suitable choice for systems with high electric vehicle (EV) adoption where load-balancing efficiency is crucial.

In contrast, in high-consumption scenarios, small differences in the peak load reduction between heuristics, such as OVF’s 4.17% reduction versus LCVF’s 9.60%, indicate that while both approaches are viable, their practical benefits may vary based on specific system demands and cost-effectiveness. For example, although OVF showed moderate peak reduction, the added computational cost of LCVF might be justified in high-demand environments where optimal resource allocation is essential. Consequently, the decision to select a heuristic should factor in both statistical significance and practical impact, particularly in high-demand urban grids where minimizing peak loads directly contributes to grid resilience and reduces the need for costly infrastructure upgrades. This dual consideration provides a robust foundation for selecting load allocation strategies in EV charging systems.

While the LCVF heuristic has demonstrated significant improvements in peak load reduction and grid stability, its implementation presents computational limitations. The iterative nature of the LCVF and the need for continuous updates to load allocations contribute to increased computational complexity, particularly as the number of EVs and time slots increase. In large-scale networks, such as city-wide grids, this complexity may exceed available processing capabilities, posing challenges for real-time applications. Optimizations or parallel processing strategies could help address these challenges, enabling the application of the LCVF in broader scenarios.

These findings underscore the potential impact of the LCVF heuristic on real-world electric vehicle (EV) charging systems, particularly in urban settings with high EV adoption rates. By effectively reducing peak loads, the LCVF approach can help prevent grid overloads, enable smoother integration of renewable energy sources, and facilitate cost reductions through optimized energy distribution. Integrating the LCVF heuristic into current EV infrastructures could significantly enhance grid stability, making it feasible to handle increased EV charging demands without extensive grid upgrades.

Future research could explore the scalability of the LCVF heuristic across diverse grid configurations and under varying demand scenarios. Additionally, studies incorporating real-time data from EV charging stations and renewable energy sources would be valuable in assessing the heuristic’s adaptability and long-term effectiveness. Analyzing the LCVF’s performance alongside emerging smart grid technologies may yield further insights, supporting the development of comprehensive energy management strategies.