2.1. The TCL Model

A TCL refers to a device or system that, through a thermostat, automatically adjusts its operation based on temperature changes. The purpose of such a load is to regulate and maintain a desired temperature within a specific range. TCLs are commonly used in heating, ventilation, and air conditioning (HVAC) systems, as well as various appliances and industrial processes, such as refrigerators and chemical reactors.

In the context of electrical systems, aggregations of TCLs can play a significant role in optimizing energy consumption, demand response, and grid stability. They can provide regulation services and mitigate power imbalances resulting from fluctuating distributed renewable generation.

The larger the safety band, the more flexibility the TCL will provide.

The model must also include several constraints that affect TCLs, such as short-cycling prevention and availability. Short-cycling prevention avoids the TCL to change its status too frequently, which could cause damage in the device components. The minimum status changing frequency is specified by τ_i. In addition, availability prevents the TCL from leaving the safety band and ensures that the TCL does not operate when the ambient temperature is above the safety band in heating appliances or below the safety band in cooling appliances.

2.2. The VB Model

A VB models the aggregated demand flexibility of a set of TCLs at each time instant k. The charging and discharging processes have different dynamics. Consequently, two different variables are introduced in the model for the evolution of energy and power magnitudes. Charging magnitudes are considered positive, while discharging magnitudes are considered negative.

2.3. The Demand Flexibility Control

The demand flexibility control developed for a VB operates in a cyclic process with two basic operations: aggregation of TCLs and priority-based dissagregation, as shown in Figure 1. The controller operates with a time step h. As stated before, this control system is used in this paper because of its precision, checking the status of TCLs in real time.

At aggregation of TCLs, all the variables describing the current situation of all TCLs are calculated, preventing their inner temperature from leaving the comfort band. Then, all the information of TCLs is gathered. As a result, a deviation power signal β^k is obtained, which is the difference between the power consumed by the VB, its expected base consumption and the amount of power to be switched on or off in the next time instant because of TCLs constraints. Then, β^k is compared with the system operator power signal r^k, which sets the power requirement. This calculates the regulation power signal e^k.

The priority-based dissagregation decides which TCLs must change their status from on to off, or vice versa, in order to cope with the regulation power signal e^k. The TCLs chosen first are the available ones that are far in time from being switched on or off due to device constraints.

Additional improvements in the demand flexibility controller of Martín-Crespo, Saludes-Rodil, and Baeyens [9] have been included in this article. First, the absolute error when tracking r^k is reduced, as the power difference between switching on or off the last TCL selected in priority-based dissagregation is checked. In addition, the number of $$ changes are examined, and thus, the activations of the short-cycle prevention constraint are reduced. Second, the controller allows to configure periods of time without demand management, which means that only aggregation of TCLs step is executed. This is the case when no r^k is required.

2.4. Demand-Side Participation in SEBM

Several European countries have opened their markets to consumers participation, such us United Kingdom, Belgium, Germany, or Sweden [3, 24]. Here, we focus on SEBM, managed by Red Eléctrica de España (REE). In these markets, active power bids can be either upward or downward (from the generation point of view). This is equivalent to a VB discharging, reducing consumption (negative flexibility) or charging, demanding more power (positive flexibility), respectively.

There are currently three SEBM: secondary regulation (SR), tertiary regulation (TR), and replacement reserves (RR) [25].

SR is an optional ancillary service whose purpose is to maintain the balance between generation and demand, correcting the unintentional deviations. Its temporal working horizon ranges from 30 s to 15 min.

TR is an optional ancillary service that, if subscribed to, is accompanied by the obligation to bid and is managed and compensated by market mechanisms. Its objective is to resolve the deviations between generation and consumption and the restitution of the secondary control reserve which has been used.

RR is an optional service managed and remunerated by market mechanisms. The objective is to resolve the deviations between generation and demand which could appear in the period between the end of one intraday market and the beginning of the next intraday market horizon.

All of them focus on maintaining the balance between generation and demand and operate in 15-min periods. The bids submitted to the markets consist mainly of the active power offered, the price, and some possible complex constraints, such as indivisibility. Once the bids are submitted, a matching algorithm is executed. The SEBM comply with the European Commission Regulation [26] and ENTSO-E [27] nomenclature. The main characteristics of the three SEBM are given in Table 1.

All the requirements that demand aggregators need to participate in SEBM are gathered in [29]. According to this, the minimum power bid must be 1 MW.

The method developed in this paper encourages demand-side participation because it provides demand aggregators with a rigourous tool to calculate how much power they could offer in electricity markets.

2.5. The VB Power Supply Probability Function

The control strategy explained previously allows a one-step ahead prediction of the VB parameters, as $$ and $$ are obtained before r^k. However, this forecasting horizon is not enough for sophisticated demand flexibility management, including participation in electricity balancing markets, where the demand flexibility must be accurately predicted before the actual provision of the service and for longer prediction horizons. The novel MC&ESB method presented in this paper solves this issue, as the predictions of the method can be calculated several hours in advance.

Predicting the flexibility of VBs requires studying how the temperatures of TCLs are expected to change over time. However, this is not an easy task. The evolution of TCL temperatures is influenced by errors in ambient temperature forecasting, model inaccuracy, and other disturbances related to the operation of the devices (e.g., when a refrigerator is opened). Representing these inaccuracies and perturbations requires the use of random models. The TCLs used in this work encapsulate all of this uncertainty in the perturbation parameter $$. Usually, $$ is considered to be distributed as a Gaussian random variable of zero expectation and constant variance σ² [30].

The randomness in the temperature evolution of TCLs is a consequence of random disturbances and uncertainty in the knowledge of the initial conditions. Consequently, the ability of a VB to respond to a given flexibility demand, that is, to satisfy a given power deviation from the reference consumption during a given period of time t, is a random variable characterized by a given probability distribution.

The VB is composed of a set of TCLs. The number of elements in this set, N_TCL, is considered large enough to achieve generality and reduce the influence of $$.

Let x be a real-valued variable representing a candidate deviation power that the VB could provide during a given time period t. In order to characterize the capability of VB to provide a power deviation during certain time period, a power supply probability function Φ(x) is defined.

Let S be a binary random variable representing whether a constant power can be successfully provided by the VB for a certain period of time. The binary random variable takes the value S = 1 if the VB successfully provides the demanded power. Otherwise, S = 0.

In spite of the random nature of the VBs system behavior, we can assume that the probability function Φ(x) is monotonic when the number of TCLs in the VB is large. In other words, the larger the absolute value of the demanded power to a VB for a given duration, the lower the probability of actually supplying it.

2.6. The MC&ESB Method for Estimating a VB Power Supply Probability Function

The monotonicity property of the power supply probability function Φ(x) can be used to estimate the maximum power (positive or negative) that a given VB could supply to an aggregator or other market actor with a probabilistic guarantee measure. This estimate is a prediction of the demand flexibility that can be provided by a TCL aggregate that is modeled as a VB.

To find the maximum flexible demand power that a VB can supply, a new method, named MC&ESB, is designed in this paper. It is based on the combination of Monte Carlo simulation techniques and extremum search using the bisection algorithm.

Monte Carlo simulation is a probabilistic method useful for estimating a value under uncertainty, especially for complex systems [31]. It consists of evaluating a function a large number of times, so it finally converges to the most probable solution (the mean value) despite of the existing uncertainty [32]. The accuracy of the method increases as the number of evaluations of the function increases [33].

We consider that a VB can provide the requested flexibility when the probability of supplying the aggregate power target for a given time period t is equal to 1 with a certain accuracy ε and confidence δ. A VB supplies the power target whenever |r^k| is less than or equal to $$ for positive flexibility or $$ for negative flexibility.

Let p be the probability that the VB provides a power target of value x, then, $$. This probability p is constant but unknown, but can be estimated by performing a sequence of experiments. The confidence measure of the estimate can also be obtained from experiments using Bayesian inference.

Using the above expression, we are interested in obtaining the number of trials to estimate the interval of power range [x_min, x_max] that can be supplied during the demand response event of duration t with a given confidence.

The above theorem allows us to estimate the probability of successfully supplying a given constant power during the demand response event with a measure of confidence. Let x_max be the maximum value of x, such that Ξ = N, then, p ≥ 1 − ε with probability greater that 1 − δ if no trial fails for a number of trials satisfying $$. In a similar way, let x_min be the minimum value of X such that Ξ = N, then, p ≥ 1 − ε with probability greater that 1 − δ if no trial fails for a number of trial satisfying $$.

In the Monte Carlo simulation stage of the MC&ESB method, the time evolution of the VB is simulated for each candidate value of power x. An estimate of the probability p is given by Equation (11), where N is obtained using Equation (14).

A schematic graphical description of the method is shown in Figure 2. The stage of the extremum search using the bisection algorithm in the MC&ESB method consists of searching for the maximum value of the power x in absolute value that the VB can supply with probability 1. The method starts by defining a search interval, which must be between [0, +∞], when searching for flexibility in charging and between [−∞, 0], when searching for flexibility in discharging. Next, the bisection algorithm computes a first value of x and the estimate of the probability p is computed by Equation (11). The bisection interval bounds a and b are updated at each iteration. The bisection algorithm terminates when |a − b| < γ, with γ being a given tolerance. The solution found x is the maximum flexible demand power that the VB can supply with probability greater than 1 − ε and confidence 1 − δ.

In this paper, the TCL and VB modeling and control, presented in Section 2, has been used as evaluation function for MC&ESB, but any other could be used as long as the effect of TCLs disturbances are considered.

3.1. Case Study 1: Estimation of VB Supply Probability Function

In this case study, the power supply probability function Φ is experimentally estimated along with confidence intervals for a certain VB composed of 3000 TCLs.

The TCLs participating in the VB are 1000 refrigerators, 1000 electric water heaters, and 1000 reversible heat pumps, which are considered as heating pumps in winter and cooling pumps in summer. Their characteristics are shown in Table 2. The location of the VB is Madrid, in Spain, and the season is summer. Thus, the forecast ambient temperature of every device i, $$, is considered constant and equal to 24°C for refrigerators and electric water heaters, whereas for reversible heat pumps is the temperature of August 10th at 15:00 UTC of the typical meteorological year, obtained with PVGIS [36].

The VB aims to modulate its consumption with a constant r^k, and the demand response event t is equal to 15 min, which is the time period in which SEBM operates.

The initial status of each devices, $$, has been randomized, the variance, σ², is set to 0.05, the minimum status changing frequency, τ_i, is 1, and the control time step, h, is 1 min (1/60 h).

The accuracy and confidence parameters ε and δ have been set to 0.02 and 0.005, respectively. Then, applying Theorem 1, the number of trials is N = 262. The MC&ESB method is applied to obtain x_min and x_max, where x_min (respectively, x_max) is the minimum (respectively, maximum) value of power that the VB can always supply for N = 262 experiments. Therefore, the probability of supplying power x for any x ∈ [x_min, x_max] is greater that 0.98 with confidence at least of 0.995. For this VB, x_min = −1138.3 kW and x_max = 5841.8 kW. By analogy, the MC&ESB method is also applied to obtain $$ and $$, where $$ (respectively, $$) is the limit value of power such that if $$ (respectively, $$), the VB can never supply the power x for N = 262 experiments. Therefore, the probability of not supplying power x for any $$ is greater that 0.98 with confidence at least of 0.995. For this VB, $$ kW and $$ kW. For $$ (respectively, $$), the probability function Φ and their bounds for a confidence of at least 0.995 are depicted in Figure 3. The curves have been obtained for 50 equally spaced supply powers x. At each power x, the probability is estimated as $$, where n is the number of experiments where the power x is successfully supplied and N = 262. The lower and upper bounds, $$ and $$, have been obtained using Equation (12), such that $$. The bounds ε₁ and ε₂ have been selected to be equal, whenever possible. However, this is not possible when $$ approaches 0 or 1. Consequently, nonsymmetric bounds are considered for these cases. The estimated demand probability function Φ and its confidence bounds can be used to characterize the capability of a VB. For example, for the VB of this study, we can state that it can supply a power x ∈ [−1138.3, 5841.8] during 15 min with probability greater than 0.98 and confidence 0.995. Moreover, we can also state from the data represented in Figure 3 that the VB can supply a power x = 5900 kW during 15 min with probability p ∈ [0.7021, 0.8322] and confidence 0.995. Thus, the knowledge of the power supply probability function Φ for a given VB is crucial to decide about participating in electrical markets.

3.2. Case Study 2: Demand Flexibility Prediction

The second case study aims to forecast the maximum flexibility power that the same VB of Case 1 can supply at different times of the year to participate in the different SEBM markets using the MC&ESB method. Three scenarios have been considered. In all of them, the simulation is divided into two periods.

The duration of the first period is the sum of the time taken by the algorithm to calculate the prediction (e.g., 5 min) and the minimum time required for the submission of bids in the corresponding SEBM. The duration of the second period corresponds to the demand management horizon, t equal to 15 min. The first period simulates the TCL evolution without any requested r^k power signal, while in the second period demand management is activated.

The MC&ESB method is configured to search for the maximum flexibility power with probability greater than 0.98 and confidence 0.995, as explained in Section 2.6. The remaining variables, parameters, and VB characteristics, if not mentioned below, are the same as in case study 1.

In Scenario 1, the aggregator wants the VB to participate in SR from 01:00 to 01:15 UTC on 7th February. At those hours, only refrigerators and electric water heaters operate. The requested flexibility power is positive because low power consumption is expected at night. The ambient temperature $$ is considered constant and equal to 22°C. The bounds of the bisection algorithm a and b are 0 and 5000, respectively.

Scenario 2 takes place in a summer afternoon. The VB is expected to participate in TR from 16:00 to 16:15 UTC on 19th July. In this case, all the reversible heat pumps operate as cooling pumps. The requested flexibility power is negative to achieve peak saving, as the power consumption on the grid is expected to be high. The ambient temperature $$ is considered constant and equal to 24 °C for refrigerators and electric water heaters, while that of the cooling pumps is obtained from the typical PVGIS weather year. The bounds of the bisection algorithm a and b are 0 and −3500, respectively.

Finally, Scenario 3 considers the morning of a winter day. The VB is intended to participate in RR from 10:00 to 10:15 UTC on 5th January. The requested flexibility power is negative, as in Scenario 2. The ambient temperature $$ is considered constant and equal to 22 °C for refrigerators and electric water heaters, while for heating pumps it is obtained from the typical PVGIS weather year. The bounds a and b are 0 and −3500, respectively.

The resulting flexibility power at each scenario is showed in Table 3. The highest amount of power flexibility is obtained in Scenario 1, followed by Scenario 2 and Scenario 3. Usually, the VB can provide more positive than negative power flexibility because TCLs take longer to increase or decrease their temperature when this change is not caused by an electromechanical element [9]. All calculated powers are above 1 MW in absolute value, which allows the aggregator to make bids in the SEBMs. These results demonstrate that the MC&ESB method is useful to calculate the availability of flexible power to participate in SEBMs and other similar markets in a short period of time, depending on the computational capabilities.

3.3. Case Study 3: Demand Flexibility Control

This case study shows how flexibility is managed once the aggregator’s offer has been accepted in the market and it is time to supply the energy. The flexibility power obtained in Scenario 2 of the previous case study (−1482.9 kW) must now be supplied to the grid. For this purpose, the demand flexibility control system discussed in Section 2.3 is activated. The VB has been simulated with the same conditions as in Scenario 2 during 45 min. During the first 30 min, the TCLs are not managed by the controller, while in the last 15 min the controller comes into operation to supply the requested power. The operation is shown in Figures 4 and 5.

The VB tracks the r^k power signal as soon as the controller is activated, that is, from the 30th min onwards. Before that, the TCLs consume the electrical power they need to satisfy the users’ comfort needs without any restriction. This comfort band is also maintained at all times during the 15 min of controller operation, but the coordination in the state of the TCLs makes it possible to supply the requested power flexibility. The VB consumption during the first 30 min was higher than the expected baseline consumption. For this reason, β^k is greater than 0 during this period. Deviations from the consumption baseline could occur in real time, so it is important to know the current state of the devices when managing flexibility.

Two aspects are worth noting. The first one is that β^k is always between $$ and $$ during controller operation, which means that the VB responds appropriately to the control signal r^k. The second one is illustrated in Figure 5 and refers to the fact that the absolute error of the controller is never larger than half of the maximum P_i of the TCLs participating in the VB, provided that r^k is always feasible.