On the Comparison of Stochastic Model Predictive Control Strategies Applied to a Hydrogen-based Microgrid

On the Comparison of Stochastic Model Predictive Control Strategies Applied to a Hydrogen-based Microgrid P. Velardea,1, L. Valverdeb, J. M. Maestrea, C. Ocampo-Martinezc, C. Bordonsa aSystem Engineering and Automation Department, School of Engineering, Universidad de Sevilla, 41092 Seville, Spain. bAICIA, School of Engineering, University of Seville, 41092 Seville, Spain. cAutomatic Control Department, Universidad Polit´ ecnica de Catalunya, Institut de Rob` otica i Inform` atica Industrial (CSIC-UPC), 08028 Barcelona, Spain. Abstract In this paper, a performance comparison among three well-known stochastic model predictive control approaches, namely, multi-scenario, tree-based, and chance-constrained model predictive control is presented. To this end, three predictive controllers have been designed and implemented in a real renewable-hydrogen-based microgrid. The experimental set-up includes a PEM electrolyzer, lead-acid batteries, and a PEM fuel cell as main equipment. The real experimental results show significant differences from the plant components, mainly in terms of use of energy, for each implemented technique. Effectiveness, performance, advantages, and disadvantages of these techniques are extensively discussed and analyzed to give some valid criteria when selecting an appropriate stochastic predictive controller. Keywords: Hydrogen storage, Microgrid, Model predictive control, Stochastic processes, Supply and demand. 1. Introduction A microgrid is a network of electric generation that may take advantage of several renewable energy sources: solar panels, wind mini-generators, micro-turbines, fuel cells, among others, to meet the consumer demand by working together with the centralized grid or autonomously [1]. In a microgrid, the energy is generated only at certain times, being necessary to provide continuous service to meet the demand at any time of the day. Challenges arise from the natural intermittency of renewable energy sources and the requirements to satisfy the user energy demand [2]. Thereby, storage Email addresses: [email protected] (P. Velarde), [email protected] (L. Valverde), [email protected] (J. M. Maestre), [email protected] (C. Ocampo-Martinez), [email protected] (C. Bordons) 1Corresponding author Preprint submitted to Journal of L A T EX Templates January 19, 2017 Nomenclature Symbols ez Electrolyzer EEnergy (Wh) fc Fuel cells EExpected value grid Grid FCumulative distribution function H2Hydrogen JCost function net Net power KNumber of scenarios res Renewable energy source MHL Metal hydrides level (%) ref Reference NPrediction horizon Superscripts PPower (W) ∗Working point PProbability bMeasured value RReduced number of scenarios Acronyms SOC State of charge (%) ARMA Autoregressive-moving-average aiWeight for state variables CC Chance-constrained biWeight for input variables cdf cumulative distributed function uInput variables IID Independent and identically distributed xState variables KPIs Key performance indicators ωDisturbances MPC Model predictive control ∆PRate power (Ws−1) MS Multiple-scenarios δxRisk of constraint violation PEM Proton exchange membrane Subscripts PF Perfect forecast batt Batteries SNEN Spanish national electricity network dem Demand UPG Utility power grid TB Tree-based devices become very important in the operation of this type of systems. Among wellestablished energy storage technologies, there are batteries, super-capacitors, conventional capacitors, etc. In this work, we focus on the use of hydrogen as an energy vector for energy storage. Hydrogen, combined with other renewable energy sources, is a safe and viable option to mitigate the problems associated with hydrocarbon combustion because the entire system can be developed as an efficient, clean, and sustainable energy source, as mentioned in [3]. The hydrogen is converted into electrical energy by using fuel cells; the reverse process, i.e., the transformation of electric energy into hydrogen, is conducted by electrolysis [4], or ethanol reforming [5], among other techniques. The control problem in a microgrid is to satisfy the electricity demand under economical and optimal conditions despite the uncertainties and disturbances that might appear in the processes. Taking into account that there are mathematical models available that represent the main dynamics and the load of these systems [6], and that the control problem here requires the simultaneous handling of constraints, delays, and disturbances, model predictive control (MPC) emerges as a solution to this problem. MPC is a control strategy widely used in industry for solving problems considering constraints on the manipulated and controlled variables, delays, nonlinearities, etc. Very succinctly, the main idea of MPC is to obtain a control signal by solving at each time instant an optimization problem in a finite prediction horizon based on the system model. Only the first component of the control signal is implemented in the current time step, and the problem is solved in the next time instant in a receding horizon strategy. Due to its versatility, MPC has become one of the most popular techniques in industrial control applications; see e.g. [7]. From using the MPC approach, different variants of this technique have been applied to achieve an economical and optimal efficiency in energy management of a microgrid, see, e.g., [2, 8–13]. The potential benefits of 2 the application of an MPC controller are discussed to solve this control problem in an economical manner. A review of the approaches applied for controlling smart grids have been presented in [14, 15] and references therein. Uncertainty in the load and generation profiles has been mainly addressed indirectly in the dispatch problem by using the MPC approach [14]. The classical formulation of MPC does not allow considering systems with uncertainties although some MPC schemes have been proposed to ensure stability and compliance with constraints in the presence of disturbances [16]. One way to address this problem is by means of the conservative min-max approach, e.g., in [17] control of microgrids using this technique is shown. A less conservative approach is the stochastic one, which is based on the design of predictive controllers for dynamical systems subject to disturbances and/or uncertainty in terms of the probability that a certain solution is feasible [18], mainly because it is not strictly possible to speak about guaranteed feasibility in this context. Due to the increasing importance that uncertainty plays in the power dispatch of smartgrids, stochastic MPC can be used to deal with the uncertainty in the energy demand and the renewable generation. At this point, the question of which method provides the best performance to manage the inherent uncertainties in microgrids arises. Although each method provides a different solution to the same problem, a comparison between the available techniques has been missed in the literature up to date. This is indeed the main contribution of this paper. Three popular stochastic MPC techniques have been implemented and tested to obtain experimental evidence in a real comparison framework of their suitability. The criteria to assess the controller performance is based on specific key performance indicators (KPIs) defined in this work. Also, guidelines for controller design, tuning, and real-time implementation in a hydrogen-based microgrid are also provided. Finally, the experimental results obtained highlight advantages and weaknesses when coping with disturbances and uncertainties within the closed loop of a hydrogen-based microgrid. Therefore, in this paper, three different stochastic-programming-based MPC techniques are used to deal with the uncertainty of the power demand and power generation. In the first place, multiple-scenario MPC (MS-MPC) is considered, which consists in calculating a single control sequence that takes into account different possible evolutions of the process disturbances. Hence, the control sequence calculated has a certain degree of robustness against the potential realizations of the uncertainties. This approach is used for example in [19] for water systems and in [20, 21] within the context of the control of smart grids. One of its advantages is that it is possible to calculate bounds on the probability of constraint violation as a function of the number of scenarios considered [22]. An alternative to model the uncertainty that is faced by this type of systems is to use rooted trees. The rationale behind this approach is that uncertainty spreads with time, i.e., it is possible to predict –more accurately– both the energy demand and energy production by a renewable source in a short horizon than in a large one. For this reason, the possible evolutions of the disturbances can be confined to a tree. In the tree, there is a bifurcation point whenever the disturbances branch into two possible trajectories. Consequently, the outcome, the so-called tree-based MPC (TB-MPC), is a rooted tree of control actions. This approach is used for example in [23] for a semi-batch reactor example, in [24] for the energy management of a renewable hydrogen-based microgrid, 3 and in [25] in the context of water systems. Finally, chance-constrained MPC (CC-MPC) is also considered in this work. CCMPC uses an explicit probabilistic modeling of the system disturbances to calculate explicit bounds on the system constraint satisfaction. For instance, [26] presents a chanceconstrained two-stage stochastic program for unit commitment with uncertain wind power output and [27] shows an autoregressive-moving-average (ARMA) type prediction model for the underlying uncertainties (load/generation) into chance-constrained finite-horizon optimal control. An application of this technique in the context of the drinking water network of the city of Barcelona is reported in [28]. In addition, [29] shows a comparison between MS-MPC, TB-MPC, and CC-MPC approaches applied to drinking water networks via simulation. Further, this subject has drawn significant interest; a stochastic optimization model implemented in the context of the control of microgrids can be seen in [30–33] and references therein. The remainder of the paper is organized as follows. First, a description of the microgrid, its linear model and constraints are shown in Section 2. Section 3 presents the optimization problem and the formulation of the considered stochastic MPC techniques. The comparative study and the analysis drawn from the experimental results are presented in Section 4. Finally, in Section 5, the main conclusions and future research lines are proposed. 2. Case Study Description The microgrid under study is the lab-scale microgrid called HyLab [34]. The microgrid test bench used in this study is an experimental platform specifically designed for testing control strategies. HyLab is composed of a modular system equipped with various components that allow experimentation and simulation of several types of renewable energy sources. In the Figure 1, a picture of the experimental Hylab platform is shown. PEM Electrolyzer PEM Fuel cell Battery bank Hydrogen storage Electronic load and Power source Figure 1: Experimental HyLab Plant. 4 The system consists of a solar field, emulated by an electronic power source, which produces electricity to supply the electronic load. Any excess of power can be either stored in a battery bank or derived to the electrolyzer. If the power obtained from renewable energy is not enough, both the fuel cell and the battery bank can support the load, which is emulated electronically. This type of hybrid storage operation allows implementing strategies in separated times scales: the battery can either absorb or contribute to balance small amounts of energy in fast transient periods while the hydrogen path complements larger variations [35]. The microgrid can work either connected to the utility network or as an isolated system. The Hydrogen Path is composed of three subsystems: the electrolyzer, which is proton exchange membrane (PEM) type [36], for producing hydrogen; a metal-hydride hydrogen storage tank; and finally a PEM fuel cell [37, 38] that provides power to the loads/batteries. It is important to notice that both subsystems –electrolyzer and fuel cell– cannot work simultaneously. DC/DC power converters are used as power interfaces that allow energy transfer between different distributed generation units. The equipment is connected to 48 VDC bus that is held by the battery bank. Table 1 presents the nominal values of the HyLab equipment. Table 1: HyLab equipment. Equipment Nominal Value Electronic power source 6 kW Electronic load source 2.5 kW PEM fuel cell 1.2 kW PEM electrolyzer 0.23 Nm3h−1@5barg 1 kW Metal hydrides tank 7 Nm3 5 bar Battery bank C120 = 367 Ah DC/DC converters 1.5 kW,1 kW 2.1. Microgrid linear model and constraints As it can be inferred, behind the experimental setup there is a set of complex nonlinear subsystems. The detailed description of sub-models and the physical equations are out of the scope of this paper. The complete non-linear model of the plant, its simulation, and validation are presented in [39]. Remark 1. To apply linear MPC techniques is required to find a linear model of the system around a working point (x∗,u∗). The identification process for obtaining the linear model of the plant is developed in [9]. The continuous linear system was discretized using Tustin’s method with a sampling time of 30 s. Also, the working point is given by u∗= [0 kW,1.75 kW]Tand x∗= [50 %,50 %]T. 5 The linear discrete-time model of the plant consists of two input variables, PH2(k) and Pgrid(k), which are measured in kilowatts (kW). Here, PH2(k)represents the power of the electrolyzer and the power of the fuel cell: when it is greater than zero, the PEM fuel cell is working (Pfc(k)), and when PH2(k)is negative, it indicates that the electrolyzer is operating (Pez(k)). Both the electronic load and the electronic power source can either deliver or absorb power from the utility power grid (UPG). The connection with the electric network is “virtual”, since it is emulated by the source and electronic load. Moreover, Pgrid(k)represents the power of UPG, which is positive when the power is imported by the microgrid from the UPG, and it is negative when exporting power to the UPG. The system is subject to uncertainties from the power produced by a renewable energy source, in this case, it is the power from the solar field, (Pres(k)) and the power demanded by the consumers (Pdem(k)); the difference between them can be considered as disturbances (Pnet(k)) to the system. Moreover, the plant counts with an additional variable, the power of the batteries (Pbatt), which is controlled indirectly, resulting of the power balance. The states are given by the state of charge of the batteries (SOC(k)), and the metal hydrides level (MHL(k)) of the storage tank, both measured in percentage (%). A scheme of the power variables is shown in Figure 2. The discrete-time linear model of the plant, for each time instant k∈Z+, around a working point (u∗, x∗), can be written as x(k+ 1) = Ax(k) + Bu(k) + Dω(k),(1a) that is, x(k+ 1) = x(k) + 8.1360 5.958 −15.2886 0 u(k) + 5.958 0ω(k).(1b) In this model, u(k) = [PH2(k), Pgrid(k)]Trepresents the vector of manipulated variables, x(k) = [SOC(k), MHL(k)]Tis the state vector of the system and ω(k) = Pnet(k)∈Rndrepresents the system disturbance, where nd= 1. Pez Pfc Pres PH2 Pgrid Pdem MHL SOC HyLab Plant + Controller Renewable Source (Electronic Power Source) Electrical Demand (Electronic Load) PEM Electrolyzer PEM Fuel cell Hydrogen storage UPG Bank of Batteries Figure 2: HyLab variables scheme. 6 The system is subject to constraints that avoid equipment damage and guarantee its safe operation. In particular, the Hydrogen Path –both the electrolyzer and the fuel cell– has constraints for limiting the values of PH2(k)since its power capacity is limited to 0.9kW; this value reflects some conservatism and it ensures that the hydrogen path does not work at its nominal value to protect the equipment. In this way, a longer lifespan is expected. Also, the Hydrogen Path has a dead zone between −0.1kW and 0.1kW that ensures a minimum production of power from both the electrolyzer and the fuel cell. The constraints for Pgrid(k)correspond to physical limitations of the electronic units. Furthermore, it is necessary to include constraints on their incremental signals ∆PH2(k)and ∆Pgrid(k), to guarantee the physical safety of the equipment. These constraints are mathematically expressed as follows: −0.9kW ≤PH2(k)≤0.9kW,(2a) −2.5kW ≤Pgrid(k)≤2kW,(2b) −20 Ws−1≤∆PH2(k)≤20 Ws−1,(2c) −2.5kWs−1≤∆Pgrid(k)≤2kWs−1.(2d) Overall constraints have to be considered as hard constraints, since the equipment lifespan could be drastically reduced. Both the battery bank and the metal hydrides storage tank have limited capacity to prevent any plant damage by overcharge or undercharge. Constraints on SOC(k)guarantee suitable voltage levels in the 48 VDC bus. Also, they protect the battery bank of strong load voltage variations. These state constraints are written as 40 % ≤SOC(k)≤90 %,(3a) 10 % ≤MHL(k)≤90 %.(3b) The input constraints given by (2) can be properly rewritten as u(k)∈ U ⊆ Rnu,(4) with nu= 2, while the state constraints defined by (3) are expressed as x(k)∈ X ⊆ Rnx,(5) with nx= 2. Furthermore, the total power delivered to the load, in order to satisfy the consumer demand, must satisfy the energy balance Pdem(k) = PH2(k)−Pbatt(k) + Pgrid(k) + Pres(k).(6) 3. Stochastic MPC formulation for hydrogen-based microgrids MPC is a strategy based on the explicit use of a dynamical model of the plant to predict the state/output evolution of the process in future time instants along a prediction horizon N[7]. The set of future control signals is calculated by the optimization of 7 a criterion or objective function. Only the control signal calculated for the time instant kis applied to the process, whereas the others are withdrawn. One of the advantages of MPC over other control methods includes the easy extension to the multivariable case. The optimization problem to be solved at each time instant kis formulated as min {u(k),...,u(k+N−1)} N−1 X i=0 J(x(k+i), u(k+i)),(7) subject to x(i+ 1) = Ax(i) + Bu(i) + Dω(i),(8a) x(0) = x(k),(8b) x(i+ 1) ∈ X,(8c) u(i)∈ U,∀i∈ZN−1 0,(8d) where A∈Rnx×nx,B∈Rnx×nu, and D∈Rnx×ndare the system matrices defined in (1). The multi-objective cost function to be minimized is given by J(x(k), u(k)) = a1(SOC(k)−SOCref )2 +a2(MHL(k)−MHLref )2(9) +b1P2 H2(k) + b2P2 grid(k). The sequence of inputs that must be applied to the plant along the horizon is denoted by {u(k), ..., u(k+N−1)}. Note that only u(k)is actually applied. Here, SOCref = 65% and MHLref = 40% are the references given for the state of charge of the batteries and the metal hydride level, respectively. The tuning of the cost function weights seeks for a soft tracking of the output variables towards the given references and an efficient use of the energy. More specifically, the controller is designed such that the batteries are the first way of energy storage. If there exists a big difference between the demanded energy and the produced energy by the renewable sources, it proceeds to the production of hydrogen. These prioritization weights ai,bihave been adjusted by trial and error approach carried out on simulation tests reported in previous works with this plant, see, e.g. [4, 8, 9]. In this sense, they have been established as a1=a2= 10,b1= 5000, and b2= 8000. As can be seen, the weight associated with the hydrogen production is lower than the weight related to the power of the grid in order to minimize the power interchange with the UPG. The weights associated with the outputs take low values compared with the others to give flexibility to the smart grid. However, these values can been modified in the multi-objective function (9) for tracking the reference. In this work, the energy management is the main objective, therefore the weights associated with the hydrogen path and the grid are higher than those associated with the outputs of the system. In this paper, MPC is used to satisfy the power demanded by local consumers. As mentioned before, both the demand and the generation of energy show a stochastic behavior. Therefore, the use of stochastic MPC techniques that account for the uncertainty is well suited for this context. 8 Next, the description of the stochastic MPC techniques designed and implemented is presented. 3.1. Multiple-scenarios MPC approach (MS-MPC) The optimization based on scenarios provides an intuitive way to approximate the solution to the stochastic optimization problem. In order to design the MS-MPC, it is required to know several scenarios with possible evolutions of the energy demand and generation. The scenario forecasts can be obtained either from historical data or by introducing a random scenario generation. The idea behind this approach is that a general control sequence that optimizes all the considered scenarios is calculated, obtaining in this way a certain robustness against the different possible evolutions of the disturbances. The scenario-based approach is computationally efficient since its solution is based on a deterministic convex optimization, even when the original problem is not [40]. One advantage of this approach does not assume a prior knowledge of the statistical properties that characterize the uncertainty (e.g., a certain probability function) as generally required in stochastic optimization. The main idea for optimization with a finite number of scenarios is to consider the same system for each one of the known disturbance realizations. The problem consists in solving min {u(k),...,u(k+N−1)} K X j=1 N−1 X i=0 J(xj(k+i), u(k+i))!,(10) subject to xj(i+ 1) = Axj(i) + Bu(i) + Dωj(i),(11a) xj(0) = x(k),(11b) ωj(i) = bωj(i),(11c) xj(i+ 1) ∈ X,(11d) u(i)∈ U ∀i∈ZN−1 0,∀j∈ZK 1,(11e) where K∈Z+is the finite number of scenarios considered and bωj(k)is the disturbance forecast for scenarioj∈ZK 1. Due to the stochastic nature of the disturbances, the number of scenarios considered Kdeserves special attention to ensure compliance with the state constraints with a certain confidence degree, i.e., P[xj(i+ 1) ∈ X]>1−δx, where P[·]denotes the probability operator and δx∈(0,1) is the risk acceptability level of constraint violation for the states. The number of scenarios needed to achieve this goal can be calculated as a function of δx, the number of variables in the optimization problem (z), and a quite small confidence level (β≤10−6), as indicated in [41] K≥ z+ 1 + ln( 1 β) + q2(z+ 1) ln( 1 β) δx .(12) 9 ance in the system. As it has been shown through experimental tests, there are clear differences in the way each controller manages the power signals of the electrolyzer and the fuel cell. Figure 4(d) shows the grid power signal generated by applying the stochastic MPC controllers. From the point of view of the network operators (DSO3/TSO4), the use of the UPG is minimized with the CC-MPC approach. In this manner, the impact in the electrical system generated by the renewable sources present in the microgrid is reduced. On the other hand, for the consumer point of view, it might be convenient not to force the equipment to a deep duty cycle and take advantage of the grid to smooth the power profiles. Figure 4(e) shows the evolution of the SOC and MHL for each proposed controllers along the test period. In general, for all the implemented controllers, the batteries are discharged until the fuel cell turns off at the first time, and then they raise their charge level lower than 85% for MS-MPC and CC-MPC controllers. Regarding TB-MPC controller, it holds a charge level around 75% for a longer period compared with the other ones. Then, the SOC starts to decrease again for all controllers under study. The MHL presents a minor variation, and it reduces its level below 40% until the renewable source can contribute with power to the load. After this, the MHL seeks to track its reference. Figure 4(f) shows the comparison among the different powers delivered to the load by applying the controllers. As seen, the demand is satisfied by the power from the microgrid for all the controllers as imposed in their design. Notice that, in some situations, using the “elasticity” of the consumer; it might be possible to momentarily unbalance the power demand to satisfy other microgrid objectives [45]. Nevertheless, demand response is out of the scope of this paper. In order to quantitatively assess the performances of these three stochastic approaches that have been implemented in the HyLab microgrid laboratory, several KPIs have been defined as follows: •KPI1defines the final cumulative cost given by (9) (in cost units). •KPI2is the computational time to solve the optimization problem (in s). •KPI3counts the average unmet demand with respect to the overall power demand (in %). •KPI4is the time that the fuel cell is operating (in hours). •KPI5is the time that the electrolyzer is operating (in hours). •KPI6indicates the final value of SOC (in %). •KPI7indicates the final value of MHL (in %). 3DSO: Distributed System Operator 4TSO: Transmission System Operator 16 Table 3 summarizes the numerical results of the KPIs. As it can be seen the highest value for KPI1is obtained when using MS-MPC. The rationale behind this value is that the controller optimizes a sequence of control actions valid for the most favorable scenarios as well as the least favorable ones. In this sense, an over-conservative control action is carried out. This issue can be relaxed by calculating a tree of control actions that is subject to non-anticipatory equality constraints. In this way, the control actions are calculated in a closed-loop fashion, i.e., the controller can adapt the future control actions to the evolution of the disturbances. As can be seen, TB-MPC reduces its cumulative cost by increasing the number of control variables involved into the optimization problem. Hence, its computational time is the biggest within this comparative study. Regarding CC-MPC, it has the lowest cumulative cost without increasing the number of control variables. For reference purposes, the final cumulative cost for an MPC with a perfect forecast (PF-MPC), obtained via simulation, is 2.05 ×1012. The computational time comparison is provided by KPI2. Table 3: Comparison of the MS-MPC, TB-MPC, and CC-MPC controllers applied to HyLab microgrid by means of KPIi, i = 1, ..., 7. Controller KPI1KPI2KPI3KPI4KPI5KPI6KPI7 (cost units) (s) (%) (hours) (hours) (%) (%) MS-MPC 3.89 ×1012 7.76 0.12 3.26 2.90 63.53 42.91 TB-MPC 2.75 ×1012 18.15 0.11 2.63 2.45 62.51 41.58 CC-MPC 2.44 ×1012 1.04 0 3.70 2.97 63.71 43.85 The three tested controllers are able to meet the overall demand in a satisfactory way, as indicated by the performance comparison given by KPI3. At this point, we must remark that the three tested controllers solve their optimization problems faster than the sampling time. Therefore, it is possible to select the approach that has the best performance in terms of the demand satisfaction and the use of the hydrogen path. The comparisons between the proposed controllers regarding the time when both the fuel cell and the electrolyzer are operating are given by KPI4and KPI5, respectively. In this sense, TB-MPC shows the lowest time for the fuel cell and the electrolyzer. It offers a larger conservatism when working with the hydrogen path, which is obtained at the expense of a higher computational time since TB-MPC meets the current demand and reformulates its disturbance tree at each time step. Notice that the main difference is at the time that the hydrogen path is working. The final values of SOC and MHL, which present similar values for the three controllers, are around 63% and 42%, respectively. These values are provided by KPI6 and KPI7. Table 4 presents a comparison among the total energy produced by the fuel cell (Efc), the electrolyzer (Eez), the batteries (Ebatt), and the grid (Egrid) during the test period. The negative sign in Egrid indicates that the amount of energy sold to UPG is greater than the energy purchased. The total energy of the batteries indicates the difference between the stored energy and the delivered energy to the load: the negative 17 value means that the stored energy predominates over the delivered energy. Table 4: Energy produced by the fuel cell, electrolyzer, batteries, and grid during the test period by applying the proposed stochastic MPC controllers. Controller Efc Eez Ebatt Egrid (Wh) (Wh) (Wh) (Wh) MS-MPC 302 481 −62.2−418 TB-MPC 261 217 −110.23 −661 CC-MPC 348 642 −43.09 −268 Notice that the absolute value of the energy amounts are taken to achieve a reliable comparison in terms of energy consumption for each component of the system. In this sense, CC-MPC has better performance regarding energy efficiency. CC-MPC achieves less exchange with UPG, and the batteries provide enough power to supply the load. Also, both the fuel cell and electrolyzer use energy in a wider range when compared to the MS-MPC and TB-MPC approaches. Note also that TB-MPC and MSMPC handled more cautiously hydrogen energy from the path while performing more exchanges with the UPG, specially TB-MPC. Another KPI to compare the performance of the controllers for energy management in a smartgrid is the number of start-ups for both equipment, the fuel cell and the electrolyzer. From the results obtained from the experimental setup, the number of start-ups is the same for all the controllers. However, it is a major factor that could reduce the lifespan of the hydrogen path. Finally, Table 5 shows the range of values of each variable obtained during the experiments by applying the proposed approaches. As seen, the control actions satisfy the constraints given by (2) and (3). Table 5: Range of values for the states and control inputs obtained during the test period by applying the proposed stochastic MPC controllers. Variable MS-MPC TB-MPC CC-MPC SOC (%) [57.61,83.73] [57.59,75.30] [48.82,84.42] MHL (%) [38.51,50] [39.07,50] [37.06,50] Pfc(W) [100,268.13] [100,259.69] [100,250.44] Pez(W) [100,432.9] [100,202.13] [100,584.94] Pgrid(W) [−529.4,320.3] [−705.02,314.0] [−312.8,117.5] In order to extend the comparative analysis to general results and taking into account that the experimental setup of the plant is limited, the non-linear simulation model developed in [39] is used to compare the controllers in other situations and the same circumstances. This simulation model replicates the main dynamics of the 18 real plant with enough accuracy. An additional case study for testing the three stochastic MPC controllers and a PF-MPC controller is introduced to enhance the results and obtain conclusions. Figure 5 shows the evolution of the signals by applying the three stochastic MPC controllers and a PF-MPC controller for a cloudy day in the simulation model of the HyLab microgrid. All controllers present the same evolution to satisfy the demand. The fuel cells are turned on when the power from the renewable sources is not enough to meet the electric demand. Hence, SOC and MHL decrease gradually to supply power to the load. For this particular day, the microgrid imports energy power from the UPG. Given that the excess of renewable energy production over the demand is not enough, the batteries are charged, and the electrolyzer stays off. To compare the behavior of these MPC controllers, Table 6 shows the results from aforementioned KPIs. The results obtained from the comparison are similar to the previous experimental case study. As expected, the lowest value of KPI1is presented by standard MPC controller with perfect information; this value gives a target for the comparison. In this sense, CC-MPC controller results in a lower cumulative cost as well as the computational time compared with MS-MPC and TB-MPC controllers. The electrical demand is satisfied by all controllers. Regarding KPI3, MS-MPC controller uses the hydrogen path longer than the other two approaches. Finally, KPI6shows very similar values for all controllers, the battery SOC is reduced until its lower constrained level. The lowest value of KPI7is presented by CC-MPC controller because this controller delivers a bigger amount of energy from the fuel cell. Finally, the electrolyzer stays off over the simulation period; therefore KPI5is zero for all controllers. Table 6: Comparison of the MS-MPC, TB-MPC, CC-MPC, and PF-MPC controllers applied to the simulation model of HyLab microgrid for a cloudy day by means of KPIi, i = 1, ..., 7. Controller KPI1KPI2KPI3KPI4KPI6KPI7 (cost units) (s) (%) (hours) (%) (%) MS-MPC 6.24 ×1012 7.76 0.10 6.00 40.56 20.43 TB-MPC 5.33 ×1012 18.20 0.11 5.67 40.64 23.46 CC-MPC 4.22 ×1012 1.04 0.10 5.68 40.46 14.08 PF-MPC 4.05 ×1012 0.98 0 5.90 40.01 19.77 Table 7 compares the stochastic MPC controllers regarding energy for a cloudy day via simulation. The CC-MPC controller results in higher energy consumption from the fuel cell. The energy from the renewable sources is not enough at the time to turn on the electrolyzer for storing energy as hydrogen. The batteries are used to provide energy to the load; both, MS-MPC and TB-MPC controllers, show a similar use of the energy of the batteries. A remarkable difference is shown in the energy exchanged with the grid, in this case, the TB-MPC controller presents the highest value. All in all, Table 8 shows priority factors for each one of the proposed stochastic 19 (a) Energy generated by solar panels Pres, demand of energy Pden, and Pnet corresponding to a cloudy day. (b) Battery power. (c) Fuel cell power and Electrolyzer power. (e) Battery SOC and MHL. (f) Electric power provided by the microgrid compared with the consumer demand. (d) Grid power. Figure 5: Simulation results for a cloudy day applying the proposed stochastic MPC approaches and a standard PF-MPC. MPC controllers based on the overall analysis at the time of selecting one of them. 20 Table 7: Energy produced by the fuel cell, electrolyzer, batteries, and grid during the test period for the simulation model by applying the proposed stochastic MPC controllers. Controller Efc Eez Ebatt Egrid (Wh) (Wh) (Wh) (Wh) MS-MPC 472 0 −285 688 TB-MPC 437 0 −287 721 CC-MPC 547 0 −295 604 Table 8: Priority factors for selecting one of the proposed stochastic MPC controllers. Priority MS-MPC TB-MPC CC-MPC Maximization of hydrogen path lifespan X Minimization of energy exchanged with the UPG X Cumulative cost X Computational burden X Demand satisfaction X X X Availability of historical data X X 5. Conclusions Three stochastic MPC schemes have been designed and applied to a microgrid based on hydrogen storage. Acting on the set-point power of the fuel cell, electrolyzer and grid, the controllers were able to regulate the hydrogen tank level and the state of charge of the battery bank to their desired values. In addition, the controllers consider constraints in both the manipulated variables and the system states for optimal performance and high functionality. Also, the system can deliver electric power from the fuel cell once it has been stored in the form of hydrogen to contribute further to the grid, to satisfy the energy demand under the influence of uncertainties both in the renewal generation and the consumer’s behavior. MS-MPC controller is generally over-conservative because it does not consider the controller capacity to adapt. It calculates a control series valid to all possible scenarios by means an open-loop formulation. However, it is possible to solve the optimization problem by using a control tree and increasing the number of optimization variables and the computational time. Regarding the control point of view, TB-MPC controller works in a closed-loop fashion to adapt the control actions to the expected evolution of the disturbances. Finally, CC-MPC controller formulates the optimization problem by taking into account the statistical features of the uncertainty without increasing the number of variables. The results obtained with the three presented versions of stochastic MPC controllers show their effectiveness in energy management under economic and optimal criteria. According to the results obtained and their evaluation by the KPIs, it can be said that CC-MPC controller relaxes the constraints of the optimization problem by 21 assuming a risk to offer better performance, resulting in a lower cost, less energy exchange with the network when compared to MS-MPC and TB-MPC controllers. This is also the approach with the lowest computational burden. The downside of this approach is that it requires a statical characterization of the disturbances. The TB-MPC approach, according to the results, provides a more moderate use of the hydrogen path, which could lead to a longer equipment lifespan. From the point of view of the user, the energy demand is fulfilled by increasing energy exchange with the network. The MS-MPC approach provides a certain robustness of the system, generating control actions able to cope with potential disturbances. This approach provides a trade-off between the time of use of the equipment and the satisfaction of the energy demand. 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