Unit commitment in a hybrid diesel/wind/pumped-storage isolated power system considering the net demand intra-hourly variability

This paper presents a novel two-stage stochastic mixed-integer linear programming model for the generation scheduling of a hybrid diesel/wind/pumped-storage power system that considers the intra-hourly net demand variability during the ﬁrst stage. The model aims to minimise the scheduling cost, calculates the power system storage opportunity cost and considers that the start-up cost of thermal units depends on the time passed since the previous shutdown in a detailed manner for the ﬁrst stage (i.e. using integer variables) and in an approximate manner (relaxing the integrality constraints) for the second stage. To increase the model’s computational efﬁciency, a solution approach based on the Benders decomposition technique is applied. The model is implemented in the real power system of El Hierro island, an isolated hybrid diesel/wind/pumped-storage power system with a closed-loop pumped-storage plant and high wind power installed capacity. The schedules provided by the proposed model are compared to the ones provided by a similar model that does not consider the intra-hourly net demand variability, resulting in generation schedules with a signiﬁcantly lower wind energy curtailment. The results demonstrate the importance of considering the intra-hourly net demand variability in order to integrate more wind energy at a moderate extra cost.


Motivation
The operation of isolated power systems with a high share of variable renewable generation (VRG), is challenged by the VRG limited predictability and high variability, among others operational issues [1]. Previous works have reported that increasing the flexibility of the power system, known as the power system ability to adapt to expected or unexpected changes in the net demand (total system demand minus VRG) [2], allows to obtain higher VRG penetration values [3][4][5]. That flexibility increase can be achieved by means of physical assets like fast-acting generation resources, energy storage resources, or new transmission lines, or by the usage of more flexible power system operation optimisation models, for example, considering a sub-hourly time resolution for the unit commitment problem, that allows to better capture the variability of the VRG, the ramping capacity of the fast-acting resources etc. [2,6,7]. The uncertainty management in the unit commitment problem has been a very active research topic in the last years. Different approaches such as stochastic unit commitment, robust unit commitment, have been widely applied and summarised in recent review articles [8][9][10].
The potential impact that VRG variability has in a power system depends on various aspects including the nature of the renewable sources, the VRG installed capacity and the location of the VRG sources in the power system grid (aggregation effect etc.), among others [2]. Isolated power systems with a high wind power penetration will likely have to face severe VRG ramping events within the hour [11,12], which is the usual time resolution for the unit commitment problem [13]. Some authors have reported the potential impacts that the usage of traditional unit commitment models could have when applied in these power systems like the underestimation of the real cycling of the power system units, that is, the wear and tear costs [6], generation schedules that are not feasible when real time dispatch is applied [12] etc.

Literature review
Unit commitment models with sub-hourly time resolution provide more flexible generation schedules with a larger ability to cope with the variability of VRG. A few articles in the last years have proposed unit commitment models with sub-hourly time resolution. The authors of [11] propose a deterministic threestep methodology for the generation scheduling of a hydrothermal power system. The methodology comprises three optimisation models each with different planning horizon and time resolution that are executed in a sequential way: the first model is an hourly day-ahead unit commitment model that provides the slow power system units schedule; then a 3-h unit commitment model with a time resolution of 15 min provides the fast-acting units schedule and finally, an hour-ahead economic dispatch with a time resolution of 5 min is run. All optimisation models are executed on a rolling basis, daily for the first one, and every 15 and 5 min for the second and third ones respectively. This methodology is used to explore the operational benefits of pumped-storage in a power system with a high VRG share. Same methodology with different planning horizon and time resolution values is used in [14] to evaluate the impact of traditional large-scale energy storage operational practices in the Irish power system for different scenarios of VRG share. The authors of [15] present a two-step methodology: a stochastic day-ahead unit commitment model with a time resolution of 1 h for the commitment of the slow units and a deterministic hour-ahead unit commitment model with a time resolution of 5 min for the commitment of the fast units and the economic dispatch. The authors of [16] use a deterministic day-ahead unit commitment model with a time resolution of 30 min that is executed every 6 h to analyse the future wind energy curtailment in the Irish power system. A similar model with different time resolutions (5, 15, 30 and 60 min) is presented in [6]. The results obtained in [6] show that a smaller time resolution helps to obtain more realistic estimations of the total generation scheduling cost. Similarly to [6,16], the authors of [17] use a deterministic day-ahead unit commitment model with 15 min of time resolution to study the operational requirements of a thermal plant for the future British power system with an increasing VRG.
The authors of [7] compare to each other the results obtained with a stochastic day-ahead unit commitment model with a time resolution of 15 min and 1 h. The results show that generally the schedules obtained with a smaller time resolution are more conservative and less vulnerable to the wind forecast errors. The authors of [18] propose a deterministic unified unit commitment and economic dispatch model which is updated every 5 min. The model uses a variable planning horizon ranging from 12 to 36 h. The model utilises a variable time resolution of 5, 15, 30 and 60 min that depends on how long is left to the time period where the generation schedule is to be determined. The results of [18] show that the model proposed therein provides generation schedules with adequate capacity and ramping capability to face abrupt changes in VRG. In [19], the authors use a day-ahead unit commitment model with 1-h time resolution and a 4-h-ahead unit commitment model with 20-min time resolution for the generation scheduling of the Greek noninterconnected island systems. The results obtained in [19] show that the approach proposed therein provides better results in terms of VRG integration, conventional generation management and system operation cost.
The authors of [20] present a robust optimisation based dayahead unit commitment model that takes the sub-hourly wind power variations through a set of ramp constraints. The model is compared to a conventional unit commitment model without sub-hourly ramp constraints. The results presented in [20] demonstrate that neglecting the sub-hourly wind power variability might yield infeasible generation schedules with insufficient sub-hourly ramping capacity. A deterministic day-ahead unit commitment with continuous-time generation and ramping trajectories using a function space-based optimisation model is proposed in [21]. They compare the real-time operation using the schedules obtained with the proposed model and the ones given by a conventional day-ahead hourly unit commitment model. The results present a significant reduction of ramping scarcity events and a decrease in operational costs when applying the schedules given by the model of [21]. The authors of [22] present a similar approach to [21] but considering the uncertainty and storage, where a continuous-time stochastic multi-stage reserve unit commitment model is proposed. The results show a reduction of the operational cost and/or fewer real-time constraint violations when compared to a deterministic hourly day-ahead formulation.
It is important to note that the above-mentioned benefits of using a sub-hourly time resolution come at the expense of a very important increase in the required computational time for the resolution of such optimisation models. The authors of [23] report a computational time increase of one hundred times when considering a 10-min time resolution instead of an hourly one for a short-term deterministic unit commitment model. More than 24 h of computation are reported in [7] for a short-term stochastic unit commitment with a time resolution of 15 min, that makes it hardly applicable to the day to day power system operator's work.
In power systems with hydropower generation or pumpedstorage it is necessary to take into account the storage opportunity cost in the short-term generation scheduling. The storage opportunity cost is normally obtained as result of a longer term generation scheduling model and is considered as an input to the short-term generation scheduling model [24,25]. The authors of [26] suggest that the storage opportunity cost had better be endogenously calculated in the short-term generation scheduling. For this purpose, they propose to extend the planning horizon of the short-term scheduling model and a formulation with a lower level of detail along the planning horizon extension in order to counterbalance the increase of the computational time.

Literature review conclusions
Except [7], all above-mentioned articles suggesting the use of a sub-hourly time resolution to solve the unit commitment problem use a deterministic approach. None of the models presented in these articles endogenously calculates the storage opportunity cost as suggested in [26].

Objective and contribution
The objective of this paper is to present a novel two-stage stochastic mixed-integer linear programming (MILP) model for the unit commitment of a hybrid diesel/wind/pumped-storage isolated power system with a high wind power installed capacity. Nowadays, the importance of stochastic optimisation techniques and mixed-integer linear programming is rapidly growing in different applications (e.g. logistics [27], manufacturing [28] and energy efficiency [29]).
The main contribution of this model is that it calculates simultaneously the next-day generation schedule and the storage opportunity cost, while considering the intra-hourly net demand variability.
The proposed model considers: • the intra-hourly variability of VRG output in the first 24 h of the planning horizon, • the uncertainty of VRG output in the whole planning period, • hourly commitment decisions, • integer/relaxed variables to model the status of the system's units in the first 24 h/from hour 25 to the end of the planning horizon, • different start-up costs of the thermal generating units as a function of the time that the unit has remained off-line since the previous shut-down.
The model is solved by a method based on Benders decomposition technique in order to reduce the computational time making it tractable and practical for its daily use.
The work here presented is an expansion of the one presented in [25] but with an approach that gathers the main ideas and suggestions that have raised across the literature review and that have been previously exposed.

Paper organisation
The paper is organised as follows. Section 2 presents the mathematical formulation of the optimisation model. Section 3 describes the power system used as a case study. In Section 4 information about the optimisation model parameter settings is presented. Section 5 discusses the results obtained and finally, in Section 6 the main conclusions are summarised.

MATHEMATICAL FORMULATION OF THE MODEL
This section presents the formulation of the proposed two-stage stochastic MILP unit commitment model (hereinafter referred to as STOMIP model).
The STOMIP model is partially based on the optimisation models presented in [25], where a two-step methodology was proposed to solve the day-ahead generation scheduling of El Hierro power system: first a two-stage stochastic linear programming (LP) model was used to calculate the storage opportunity cost and second, a deterministic MILP-based unit commitment model was used to calculate the next-day generating schedule using as an input the storage opportunity cost obtained by the former.
The STOMIP model has two decision stages. The first stage (t ∈ T b ) corresponds to the first 24 h of the planning horizon. The second stage (t ∉ T b ) extends from hour 25 to the end of the planning horizon. A set s of hourly net demand scenarios has been used to consider the uncertainty of VRG output in the second stage of the model (see Figure 1). As can be seen in Figure 1, a single value of the system's net demand has been considered for each hour of the first stage. A set s 10 t of intra-hourly net demand profiles, each with a 10-min time resolution, has been used to consider the intra-hourly variability of VRG output (see Figure 2).
The STOMIP model aims to minimise the objective function (1) along the entire planning period. Fuel and maintenance costs of the diesel units are considered in the first summation term; start-up cost of diesel, Pelton and variable-speed pumping units are considered in the second summation term; the third term considers the start-up cost of the fixed-speed pumping units; the fourth and fifth summation terms include the wear and tear costs due to the power variation of the diesel units, the Pelton units and the variable-speed pumping units for the first and sec- The objective function is subject to constraints (2)- (20). Equation (2) guarantees that in every time step t and scenario s, the net demand is satisfied, being allowed the wind energy curtailment w c t,s for economic reasons [5].
All units i are able to start-up in a time much shorter than 1 h regardless of the time passed since the previous shut-down. It is important to note that the start-up decision variables have a resolution of 1 h for the whole planning period, despite that along the first stage a set of net demand scenarios s 10 t with a 10min time resolution are considered, consistently with the operational practice in most power systems. The selection of the unit's start-up type and status logic for diesel units, Pelton units and variable-speed pumping units are given by (3) and (4) respectively. As in [25], the considered start-up types of the diesel units refer to the different start-up costs as a function of the time the unit has remained off-line since the last shut-down. For the case of the fixed-speed pumping units, the start-up status is given by (5), since they can only operate at a single operating point and their start-up costs and ramps do not depend on the time the unit has remained off-line since the previous shut-down. Minimum on/off-line time of every unit i ∈ V is considered in (6) and (7).
As above-mentioned, the binary variables along the second stage (t ∉ T b ) are relaxed.The variables indicating the status of the diesel units, Pelton units and variable-speed pumping units, u i,t,s , are relaxed accordingly with the approach proposed in [30], by the inclusion of Equations (8) and (9). The usage of that formulation reduces the number of cases in which the power output of a unit is between zero and P i .
The hourly power output/input of the generating and pumping units is given by (10). The production cost curve of every diesel generating unit, as well as the power-discharge curve of every Pelton and variable-speed pumping units are piecewise linearised in q linear segments of equal maximum length p inc i,q and their power output/input above minimum is given by (11) and (12). Equation (13) gives the power output/input above minimum of each unit i ∈ V for each 10-min intervals, t 10 and scenario s 10 t . To satisfy the maximum ramp up/down rates along the first stage Equations (14) and (15) are used. The former guarantees that the maximum intra-hourly power variation is satisfied while the latter assures that the inter-hourly transition between the different scenarios s 10 t is feasible for two consecutive periods t and t +1. In the second stage, the ramp rates have been neglected since all units can vary their power output from minimum to maximum power within the hour and vice versa.
The intra/inter-hourly power variation of diesel, Pelton and variable-speed pumping units in the first and second stages are calculated in (16) and (17) respectively and considered in (1) to compute the corresponding wear and tear cost.
The hydraulic balance of the upper pond is given by (18) and its maximum water storage capacity is given by (19). Finally, a set of non-anticipativity constraints is included (20) to ensure that the decisions obtained in the first stage, the here and now decisions, are unique for all net demand stochastic scenarios s.
The model defined above is a large and complex model with a total of 467,248 constraints, 45,001 real variables and 360 binary variables. In addition, the decision variables of the first decision stage are subject to two different of non-anticipativity conditions: the value of the decision variables has to be unique for a period t ∈ T b regardless of the stochastic scenario s 10 t and the stochastic scenario s.

CASE STUDY
This section describes the power system used as a case study: Section 3.1 exposes the technical and cost parameters of the power system and Section 3.2 explains how the hourly and intrahourly net demand scenarios have been obtained.

Power system characteristics
The power system used as a case study corresponds to the one of El Hierro Island in Spain. The technical and cost parameters of the power system's units are the same as the ones used in [25], where the reader is referred for further details. El Hierro Island is a hybrid diesel/wind/pumped-storage power system with a closed-loop pumped-storage plant, that is, with no natural inflows in the upper pond. A simplified diagram of the power system considered is depicted in Figure 3, where it can be noted that a single-bus approach has been assumed in this paper similarly to [25]. The maintenance cost of the diesel units considered in (1) is given in (21) accordingly to [31], where the parameters M f i and M v i represent the maintenance cost per start-up and per MWh, respectively.
The hourly spinning reserve requirement is determined in [32] and is guaranteed by Equations (22) and (23). The parameter sr d t,s represents the expected inter-hourly increase in power demand between t and t +1 and sr w t,s represents the most likely wind power loss calculated from historical data. Equation (23) represents the loss of the largest spinning generating unit in period t .
Additionally to the power system unit characteristics presented in [25], Table 1 exposes the maximum ramp up/down rates for a 10-min period of every diesel, Pelton and variablespeed pumping units. The head-dependency of the powerdischarge curves have been neglected similarly to [25] since the maximum relative head variation is roughly 1.5% of the gross head.  The STOMIP model objective function (1) is subjected to the constraints (2-23) for the case study of this paper.

Net demand scenarios
The hourly net demand scenario tree used in this paper has the same structure (see Figure 1), length (2 weeks) and number of scenarios (9) as the one used in [25]. Please note that the total variable renewable generation in the El Hierro Island power system comprises only wind power generation so it can be referred indistinctly to as VRG or as wind power. From the analysis of historical load and wind power data with a 10-min resolution, we concluded that the intra-hourly variability of wind power is several orders of magnitude higher than the one of load demand.
To evaluate the intra-hourly wind power variability, an hourly wind power variability parameter, wpv t , has been defined as the sum of the distances, wpv t t 10 , of the 10-min wind power values to the hourly mean wind power for each period t accordingly to (24) and Figure 4.
The lag-1 autocorrelation of the wpv t parameter along the time range covered by the available historical data is 0.45. From this result, we chose to model the intra-hourly wind power variability as a time-independent variable. Several authors [33,34] have demonstrated that the wind power variability around an average value significantly varies as a function of the average value. Having this in mind, we have divided the wind power range of El Hierro's wind farm into forty segments, each with a length of 0.22 MW, and have grouped the historical data accordingly.
A k-means clustering approach has been used to obtain m mutually exclusive clusters of vectors of six 10-min distances (measured in MW) to the mean wind power for each wind power segment. Within each cluster we have selected the vector of six 10-min distances closest to the centroid of the cluster provided by the k-means algorithm. To calculate the adequate value of m, the silhouette criterion [35] was applied resulting that a value of m=3 exclusive clusters is enough to capture the intrahourly wind power variability for each wind power segment. As a result, for each hourly wind power segment, three hourly 10min wind power variability profiles are obtained. The k-means clustering approach has been widely used to obtain clusters on wind power data [36,37]. Prior to running the STOMIP model, the system's transmission system operator (TSO) must build a scenario tree of hourly net demand as explained in [25]. Then, in each hour of first stage of the model, the TSO must generate three 10-min net demand profiles by adding to the forecast hourly net demand the three 10-min wind power variability profiles obtained for the corresponding wind power segment.

MODEL PARAMETER SETTINGS
As it is well known, the target optimality gap and the computational time limit are the most widely used parameters to define the stopping criteria of MILP models. Section 4.1 summarises the results of the experiments we have carried out to find a suitable target optimality gap and computational time limit. Section 4.2 describes the solution method we proposed to solve the STOMIP model in a computational time that allows El Hierro system's operator to make use of it. It is important to remark that the results and discussions presented along this section have been obtained with a version of the STOMIP model, hereinafter referred to as STOMIP-h model, that does not consider the intra-hourly net demand variability of the first stage, that is, a less computationally demanding model.
The experiments summarised hereinafter have been performed with a computer with an Intel Xeon processor E5-2687@3.10 GHz and 64 GB RAM, using the CPLEX commercial solver.

Optimality gap
The STOMIP model has a particularity that forces us to set its target optimality gap with a special care. The STOMIP model has a planning horizon of 2 weeks. Its objective function is aimed to minimise the system's scheduling cost along the entire horizon. However, the model has been devised to be used by the system's TSO to compute the next-day generation schedule on a daily rolling horizon basis. Having in mind that next-day scheduling cost has small (≈1/14) relative weight in the objective function of the model, the usual target optimality gaps may not work well for the model. For this reason, a set of forty five problems each with different 2-week scenarios of hourly net demand (with the structure shown in Figure 1) and different initial upper pond volumes have been solved with the STOMIP-h model using different target optimality gaps (from 10 −2 to 10 −6 ). In order to make a fair comparison of the results obtained, the next-day scheduling cost obtained with all target gaps other than 10 −6 has been corrected as in [25]. Figure 5 shows the relative differences in absolute value between the next-day scheduling cost obtained with different target optimality gaps compared to that obtained with a target gap of 10 −6 . Please notice that the y-axis scale has been set to logarithmic for clarifying purposes. Please note as well that the next-day scheduling cost obtained with a target gap higher than 10 −6 is sometimes lower (better) than the one obtained with a target gap of 10 −6 . Hence our choice to show the relative differences in absolute values. The reason is that the objective function of the STOMIP-h model is aimed at minimising the scheduling cost in the entire planning horizon (2 weeks) but not only in the next day.
As it can be seen in Figure 5 the next-day scheduling cost obtained when setting a typical value of 1% for the target optimality gap, differs by several orders of magnitude from the one obtained with a target gap of 10 −6 , representing a very poor performance of the STOMIP-h model for the purpose it has been devised. As the target optimality gap decreases the scheduling costs are more similar to those obtained with a target gap of 10 −6 . However, it should be noted that the average computational time of the STOMIP-h model with a target gap of 10 −5 /10 −6 is, across the forty five cases

Benders decomposition
In order to reduce the computational time of the STOMIP model so as to make it useful for the day to day work of El Hierro's TSO, a solution approach based on the Benders decomposition technique is applied. As exposed in [38], the Benders decomposition technique has proved to be a successful approach for the resolution of unit commitment problems, particularly in the case of two-stage stochastic problems with no integer variables along the second stage. In addition, the problem is relatively easier to be solved when the value function of the second decision stage is convex with respect the first decision stage variables [39].
The STOMIP model has no integer variables in the second decision stage. Figure 6 depicts the system's cost in the second decision stage and the corresponding Benders cuts in one of the cases analysed in the paper. As can be deduced from the figure the value function of the second stage is convex with respect to the volume of the upper pond at the end of the first stage.
The approach we have used to solve the STOMIP iteratively and sequentially (one after the other) solves two single-stage optimisation problems (the first and second stage problems). The second stage problem (t ∉ T b ) is formulated as a stochastic LP problem and solved by the dual Simplex algorithm. The first stage problem (t ∈ T b ) is formulated as a stochastic MILP problem including the expected scheduling cost in the second stage (FC ) by means of Benders cuts and solved by a branchand-cut algorithm. The following pseudocode summarises the proposed Benders approach.
The solution procedure starts with the initialisation of the Benders iteration counter k, the convergence parameter and the objective function's value of the first stage problem OF k , as exposed in line 1 of the pseudocode. Pseudocode's lines 2-4 force the variables v 24,s to take the same initial value v * . Then the iterative procedure (lines 5-12) starts and continues while the value is higher than the tolerance , according to the steps below: 1. the second stage problem is solved; 2. then the k-th Benders cut is built according to (25) where FC represents the expected scheduling cost in t ∉ T b , which is approximated by a set of Benders cuts (see Figure 6); 3. the first stage problem (aimed to minimise the objective function (26)) is solved and the objective function's value OF is obtained; 4. iteration k finishes: the value of OF k is updated, the parameter is calculated according to line 10 of the pesudocode and a new iteration k+1 begins.
i,t,t 10 ,s,s 10 The solution procedure ends when ≤ that has been set with a value of 0.01%. It is interesting to note that a difference of 0.01% in the first decision stage objective function's between two consecutive iterations is of the same order of magnitude as the relative difference between the best integer and real solution provided by the STOMIP-h model, solved by a branch-and-cut algorithm with a target gap between 10 −5 and 10 −6 .
In order to evaluate the effectiveness of the proposed Benders approach, the same cases as in Section 4.1 have been  solved using three different target optimality gaps for the MILP problem (10 −3 , 10 −4 and 10 −5 ). Please note that a gap of 10 −5 in the MILP problem is approximately equivalent to a 10 −6 -10 −7 gap in the STOMIP-h model, and three different initialisation volumes v * (minimum and maximum upper pond volume, and initial volume v 0,s ). The time to convergence has been shorter using v * = v 0 in all cases analysed. Figure 7 shows the relative differences between the next-day scheduling cost obtained with a target gap of 10 −3 and 10 −4 with respect to the one obtained with a target gap of 10 −5 using v * =v 0,s . The dotted line represents a relative difference of 1%. The mean and the standard deviation of the computational time required to solve the forty five problems by using the proposed Benders approach and different target gaps with v * =v 0,s are exposed in Table 2, as well as the average number of Benders cuts needed. From the results presented in Figure 7 and Table 2, we have chosen to set a target optimality gap of 10 −4 in the MILP problem.

DISCUSSION OF RESULTS
This section presents the results obtained when solving the STOMIP model (i.e. considering the intra-hourly net demand variability) using a Benders-based approach.

Computational performance
The average computational time of the STOMIP model across the analysed cases is 27,383 s which is 2.52 times higher than the one of the STOMIP-h model, due to the consideration of the intra-hourly net demand variability. The average number of Benders cuts needed to solve the STOMIP model is 7.3, 2.15 times higher than the necessary cuts to solve the STOMIP-h model. For the case of the STOMIP model, an average computational time of 394 s has been required to solve the second stage LP problem while an average computational time of 3470 s has been required to solve the first stage MILP problem across the analysed cases. The average optimality gap of the solution of the first stage MILP problem is 0.91%. The size of the MILP problem significantly increases when the intra-hourly net demand variability is considered as shown in Table 3. From the above-mentioned results, it can be expected that the implementation of the model proposed in this paper for larger power systems will lead to an important increase of the computational time. A higher number of power system units has a direct impact on the amount of binary decision variables which lead to an exponential increase of the computational time needed to solve the optimisation problem. For this reason, in Section 6 has been included as one of the future research lines the relaxation of the hydraulic balance equation in order to reduce computational time.  In order to calculate the total next-day wind energy curtailment in which the generation schedules provided by the STOMIP-h model would result, we have checked in every t 10 and s 10 t of every hour t , if the generation schedule provided by the STOMIP-h model may cope with the intra-hourly net demand profiles used as input to the STOMIP model, strictly following the unit commitment status (on/off) provided by the STOMIP-h model. In every t 10 and s 10 t where the generation schedule obtained as a result of the STOMIP-h model cannot cope with the net demand without curtailing wind energy, we have calculated the amount of wind energy that would have to be curtailed. Then we have calculated at every t ∈ T b the average wind energy curtailment w c t,s across s 10 t , and have summed w c t,s across t ∈ T b . The resulting value is the wind energy curtailment due to hidden inflexibilities in the formulation of the STOMIP-h model, as referred to in [12], and amounts to 152.0 MWh across the forty five cases analysed in the paper. The other 136.7 MWh are not due to any hidden inflexibility in the formulation of the STOMIP-h model but rather are directly obtained as results of the model. Both the STOMIP-h and STOMIP models make such decisions for economic reasons as reported in many other papers [4,5].

Wind curtailment and hidden inflexibilities
The consideration of the intra-hourly variability of the net demand, by means of the proposed formulation and solution approach, allows reducing the wind energy curtailment in the power systems used as case study by 133.8 MWh in forty five cases, at a moderate extra cost of 66.7 € per additional MWh of wind energy integrated. The STOMIP model is able to integrate the 84.5% of the total available wind energy instead of the 71.2% that integrates the STOMIP-h model. Figure 9 depicts the maximum and minimum power output of the generation schedules provided by the STOMIP and STOMIP-h models, along with the 10-min net demand, in a case in which the STOMIP-h model have hidden curtailments in the 1 st -3 rd , 6 th , 8 th -10 th , 17 th -19 th , 22 nd -24 th hourly periods. The wind energy curtailed with STOMIP/STOMIP-h model in this case is 0.13/0.67 (0 not hidden + 0.67 hidden) MWh. These hidden wind curtailments are mentioned in [12] as one of the drawbacks caused by the hidden inflexibilities of traditional unit commitment formulations. It is important to note that the hourly net demand D t,s used as input to both models (STOMIP and STOMIP-h) is identical.
The above-mentioned hidden wind curtailments of the generation schedules provided by STOMIP-h are mainly due to the insufficient downward spinning reserves scheduled by the model. In the case depicted in the Figure 9 the total downward spinning reserve provided by the STOMIP/STOMIP-h models is of 19.7/15.5 MW. Across the forty five cases analysed in the paper the total downward spinning reserve provided by the STOMIP/STOMIP-h models is of 766.7/726.3 MW. The authors of the paper are at present working on the comparison of different existing criteria for the explicit consideration of a minimum downward spinning reserve to its implicit consideration through the use of intra-hourly net demand scenarios.

FIGURE 10
Next-day scheduling cost differences between STOMIP and STOMIP-h models.

Next-day scheduling cost
By contrast with the results given in the previous section, the next-day generation scheduling cost obtained using the STOMIP model, once the correction mentioned in Section 4.1 has been made, is in average 4.39% higher than the scheduling cost provided by the STOMIP-h model. The average next-day scheduling cost obtained with the STOMIP-h/STOMIP models are 4514/4713 € respectively. Figure 10 shows the next-day scheduling cost obtained with the STOMIP-h and STOMIP models. As can be seen in the figure, there are no large differences in the scheduling costs obtained with both models.

CONCLUSIONS
This paper presented a novel two-stage stochastic MILP unit commitment model that simultaneously calculates the nextday generation schedule and the storage opportunity cost of a hybrid diesel/wind/pumped-storage power system, considering the system's net demand intra-hourly variability. The model is solved using an approach based on Benders decomposition technique in order to increase its computational efficiency. The model has been successfully implemented in the real power system of El Hierro island. The model has proven to be effective in the computation of the system's water value and the determination of the next-day generation schedule. Thanks to the use of a solution approach based on the Benders decomposition technique the computational burden of the model is moderate and therefore, the model could be used for the day to day TSO's work (the average computational time obtained is of 7.6 h across the cases analysed in this paper). The application of the model to other power systems is straightforward. The results provided by the proposed model have been compared to the ones provided by a similar model that does not consider the intra-hourly net demand variability, in a set of forty five cases. The results presented in the paper show that the generation schedules provided by the proposed model are able to integrate more wind energy than those obtained without considering the intra-hourly wind variability (84.5% vs. 71.2% of the available wind energy) with a moderately higher scheduling cost (4.39% in average). In summary, the paper shows the importance of considering the intra-hourly net demand variability to integrate more wind energy at a moderate extra cost and to obtain more reliable generation schedules.
As future work, the authors are planning to work on the relaxation of the hydraulic balance constraint (18) in order to decrease the computational time needed to solve the model and make it thus applicable to large power systems. At present, the authors are working on the upgrade of the STOMIP model for the consideration of a battery energy storage system, and on the comparison of different existing criteria for the explicit consideration of a minimum downward spinning reserve to its implicit consideration through the use of intra-hourly net demand scenarios. Finally, the implementation of the proposed model in power systems with thermal generation units requiring more than 1 h to start-up is another planned future research.

Indices and Sets
t, T hourly periods of the planning horizon T b hourly periods corresponding to the first stage [1,24] t 10 10-min intervals of the hourly period t ∈ T b , running from one to six l, L start-up type, from one (hottest) to N L (coldest) q, Q q linear segments of the production cost curve of unit i ∈ V , from one to ten i, I generating and pumping units of El Hierro power system (i ′ is a mirror index of i) s, S stochastic scenarios of the hourly net demand (s ′ is a mirror index of s) s 10 t stochastic scenarios of 10-min net demand for the period t ∈ T b (s 10 ′ t is a mirror index of s 10 t ) G t diesel generating units