Optimization of battery/supercapacitor-based photovoltaic household-prosumers providing self-consumption and frequency containment reserve as influenced by temporal data granularity

1 Optimization of battery/supercapacitor-based PV householdprosumers providing self-consumption and frequency containment reserve as influenced by temporal data granularity J.C. Hernándeza* 1 , M. Gomez-Gonzalezb, F. Sanchez-Sutila, F. Juradob a Department of Electrical Engineering, University of Jaén, Campus Lagunillas s/n, Edificio A3, 23071 Jaén, Spain b Department of Electrical Engineering, University of Jaén, Escuela Politécnica Superior, 23700 Linares, Jaén, Spain Abstract Service complementarity between a frequency containment reserve and PV self-consumption can increase incomes for household-prosumers. Moreover, battery/supercapacitor-based hybrid energy storage systems (HESSs) play a major role. Fitting power and energy management improve HESS performance, and therefore increase the profitability of the asset. Furthermore, component sizing is critical. To achieve both targets, we developed a hybrid meta-heuristic optimization algorithm that deals with the management strategies and sizing. Accordingly, a four-dimensional, non-linear, non-convex, and mixed-integer optimization problem was formulated, and a cost function was minimized by combining the Haar wavelet (WT) transform and the teaching-learning-based optimization (TLBO) method. The algorithm has a flexible design, which is adapted in terms of a number of discrete states to suit input profiles defined according to different time discretizations. The effectiveness of the algorithm was proved by using different data granularity for a PV prosumer in Spain in various service scenarios. The simulations performed in this study reflected both technical and economic impacts. The results suggest that for optimization purposes, high-resolution data should be used to consider the full range of input fluctuations. However, these results largely depend on the service scenario setup. Indeed, in some scenarios, accurate results were obtained by using coarse-grained data, which entailed a lower computational burden. In contrast, in other scenarios, it was preferable to use data with a higher resolution. The optimal combination of services significantly increased the profitability of the asset. Keyword: PV power; frequency containment reserve; lifetime; hybrid energy storage system; multiobjective optimization; temporal data granularity. 1 * Corresponding author. Tel.: +34 953 212463; fax: : +34 953 212478 E-mail addresses: [email protected] (J.C. Hernandez), mggo[email protected] (M. Gomez-Gonzalez), [email protected] (F. Sanchez-Sutil), [email protected] (F. Jurado). 2 Nomenclature. List of symbols Symbol Description  Mother wavelet function =/ t f t tN Discretization time-step (sampling interval) [0.5 – 30 min] fcr,k fcr,k fcr,k fcr,k ( ) [ ( )] sl sl ee EE   Cumulated shortage(surplus) FCR energy supply vis-à-vis their nomination in the PV household-prosumer [MFCRU] at any period k [kWh] bt  MBU self-discharge coefficient [%/hour] sc  MSU self-discharge coefficient [%/hour] k  Set of inputs at the kth time interval, , , , pv, , , SISP,k LISP,k [ , , , , , , , , , , , , , , , , , ] sl k fcr k fcr k fcr-, k fcr+, k fcr,k fcr,k k hl k ev k fcr-, k fcr+, k RTPEP, k RTSEP,k RTPPAP, k RTSPAP, k RTPEUP, k RTSEUP, k E E E E p p p ,p                 ad  AC/DC converter efficiency [%] () ddb dds  Battery (SC) DC/DC converter efficiency [%] bt  Fraction of life consumed or cumulated aging rate of an MBU [p.u.] i  Ratio of operation and maintenance cost vs. initial investment cost for the ith component of the PV household-prosumer [%]  Scale factor , , and fcr k fcr k   Downwardand upwardFCR power availability factor at any time period k at the country level (base power: annual maximum FCR power) [p.u.] pv, k  Normalized PV power at any time period k of an MPVU (base power pv P ) [p.u.] ( ) RTPEP, k RTSEP,k  Real-time purchasing(selling) energy price at any time period k in the respective real-time pricing scenario [€/kWh] ( ) RTPPAP, k RTSPAP, k  Real-time purchasing(selling) power availability price at any time period k [€/kW] ( ) RTPEUP, k RTSEUP, k  Real-time purchasing(selling) energy utilization price at any time period k [€/kWh] SISP,k LISP,k ()  Real-time short(long) imbalance settlement price at any time period k [€/kWh] rat bt  Rated DOD of an MBU [%] rat sc  Rated DOD of an MSU [%] ad  Number of MADUs installed in the AC/DC converter ddb  Number of MDDBUs installed in the battery converter dds  Number of MDDSUs installed in the supercapacitor converter   2 ,k   ( )[ ( )] ii a n A n Approximation 2n Number of data points in the sampled signal ,, () EE a EE k CC Annual (at any time period k) net income from energy exchange of the PV household-prosumer with the grid ([€/year]), (€/period)   , a , FCR FCR k CC Annual (at any time period k) net income from FCR provision of the PV household-prosumer to the utility grid [€/year], ([€/period]] , a ( ) ii II CC  , , , , , } i ad ddb dds pv bt sc (Annual) investment cost of the ith minimum unit ([€/year]), [€/unit]    , & , , I a O M a Ra CC C   Total cost of {initial investment} (operation and maintenance) [replacement] for the PV householdProsumer(€/year) & , a & ( ) ii O M O M CC  , , , , , } i ad ddb dds pv bt sc (Annual) operation and maintenance cost of the ith minimum unit ([€/year]), [€/unit] , a ( ) ii RR CC   i bt (Annual) replacement cost of the ith minimum unit ([€/year]), [€/unit] d Nominal discount rate [%] ( )[ ( )] ii d n D n Detail DOD Depth of discharge [%] 3 ( ) [ ( )] fcr-, k fcr+, k fcr-, k fcr+, k ee EE Cumulated stored(released) FCR energy for the PV household-prosumer [MFCRU] at any period k [kWh] fg Grid frequency fs Sampling frequency g Expected annual inflation [%] ,ei H ith equality constraint ,ine i H ith in-equality constraint J Discrete-time cost and revenue objective function(or annual NPV) [€/year] 11 ( ) tt bt sc LE LE Beginning energy of MBU (MSU) [kWh] () NN tt tt bt sc LE LE End energy of MBU (MSU) [kWh] , lo bt k Le Lower energy of MBU achieved by the full upward regulation at any time period k [kWh] , up bt k Le Upper battery energy of MBU achieved by the full downward regulation at any time period k [kWh] max max () bt sc LE LE Maximum energy of MBU (MSU) [kWh] LCOE Levelized cost of energy for prosumer [€/kWh] min n Minimum number of decomposition levels b N Calendar lifetime Nine Numbers of inequality constraints t N Number of sampling intervals in a year bt rep NC Number of battery replacements during system lifetime i N   , , , , , i pv bt sc ad ddb dds ith minimum unit lifetime [year] max bt N  MBU lifetime according to maximum cumulated aging rate [year] p N PV household-prosumer lifetime [years] ad P Rated power of an MADU [kW] max max ( ) bt bt PP  Maximum charge(discharge) power for an MBU [kW] c P Contracted power from the electric mains [kW] ddb P Rated power of an MDDBU [kW] dds P Rated power of an MDDSU [kW] , ev k p EVCL at any time period k [kW] fcr P Prequalified power capacity of an MFCRU [kW] , , ( ) g k g k pp  Grid power output (input) at any time period k [kW] ( ) fcr-, k fcr+, k pp Downward-FCR (upward-FCR) power in the PV household-prosumer at any period k [kW] ( )[ ( )] fcrfcr+ p t p t Downward-FCR (upward-FCR) power in the PV household-prosumer on a 10-3 s basis [kW] , hl k p HCL at any time period k [kW] max max ( ) sc sc PP  Maximum charge(discharge) power for an MSU [kW] pv P Rated power of an MPVU [kW] , pv k p PV power for the PV household-prosumer at any period k [kW] & OM r Annual escalation rate of the operation and maintenance cost [%] pv r PV degradation rate [%] ()st Original signal () ESS SOC t SOC of the ESS t Time T Income tax rate [%] u Position factor   ,Wu  Wavelet coefficient SFCPMS control variables ( ) bt-, k bt+, k pp 2 Optimal charge(discharge) power reference for the battery at any period k [kW] ( ) sc-, k sc+, k pp Optimal charge(discharge) power reference for the SC at any period k [kW] ( ( )) kk tuu Discrete-time SFCPMS control variable vector at the kth time interval, [ , , , ] kbt-, k bt+, k sc-, k sc+, k u p p p p 1 2 Control variables are shown in bold letters 4 FCREMS control variables ( ) sc-, k sc+, k ee Optimal downward (upward) FCR energy reference for the SC at any period k [kWh] ( ( )) kk tvv Discrete-time FCREMS control variable vector at the kth time interval, [] ksc-, k sc+, k v e , e Sizing control variables w Sizing control variable vector, [ , , , ] fcrbt sc pv w= Ω Ω Ω Ω bt Ω Number of MBUs installed in the battery sc Ω Number of MSUs installed in the SC pv Ω Number of MPVU fcr Ω Number of MFCRU State variables , bt k Le Battery energy at any time period k [kWh] , sc k Le SC energy at any time period k [kWh] ( ( )) kk x x t Discrete-time state variable vectors at the kth time interval, , , [ , ] k bt k sc k x Le Le Subscripts a Annual app Approximation ad AC/DC converter bt- (bt+) To(from) the battery det Detail ddb DC/DC battery converter dds DC/DC supercapacitor converter EE Energy exchange with the grid ESS Energy storage system ev EVCL FCR FCR fcr+/fcrFCR to the grid (upward regulation /FCR from the grid (downward regulation) g- (g+) To(from) the utility grid hc HCL I Initial investment k Index of time period max Maximum min Minimum O&M Operation and maintenance ped PED ppd PPD pv PVG R Replacement sc- (sc+) To(from) the supercapacitor Superscripts 1 t At the beginning t N t At the end i Index of component of the PV household-prosumer lo Lower max Maximum min Minimum rat Rated up Upper Abbreviations AMPSO Adaptive modified particle swarm optimization CQGA Chaotic quantum genetic algorithm DE Differential evolutionary DP Dynamic programming DOD Depth of discharge ESS Energy storage system EV Electric vehicle 5 EVCL EV charging load GA Genetic algorithm GSA Gravitational search algorithm FCR Frequency containment reserve FCREMS FCR energy management strategy FSAPSO Fuzzy self-adaptive particle swarm optimization HCL Household consumption load HESS Hybrid energy storage system LCOE Levelized cost of energy MACO Multi-layer ant colony optimization MADU Minimum AC/DC converter unit MBU Minimum battery unit MDDBU Minimum DC/DC battery converter unit MDDSU Minimum DC/DC supercapacitor converter unit MFCRU Minimum FCR unit MINLP Mixed-integer nonlinear programming MPSO Modified particle swarm optimization MPVU Minimum PV unit MSU Minimum supercapacitor unit NPV Net present value PDF Probability density function PED Prosumer energy demand PPD Prosumer power demand PSO Particle swarm optimization PV Photovoltaic PVG PV generation QP Quadratic programming SC Supercapacitor SFC Self-consumption SFCPMS SFC power management strategy SO System operator SOC State of charge TLBO Teaching-learning-based optimization WT Wavelet transform WTTLBO Wavelet transform and teaching-learning-based optimization 1. Introduction The residential small-scale PV system market to increase self-consumption (SFC) of locally produced electricity has dynamically developed in recent years [ 1 - 234 ]. The declining prices of PV and batteries in combination with rising end-consumer electricity prices have made an increased SFC an attractive option for a PV owner [ 5 ]. Given the stochastic behavior of PV systems, this configuration is sustainable only if an integrated PV-storage solution is considered [ 6 - 78 ]. Frequency containment reserve (FCR) is an ancillary service provided by conventional generation to ensure transient stability in power systems. It helps mitigate the fluctuations in the supply and demand, caused by the variation of load or outputs from intermittent renewable resources [ 9 , 10 ]. Because of the fast penetration of renewable energy sources, the provision of FCR from small-medium sized renewable generation has recently come into the spotlight. Furthermore, the latest network codes and standards in the European Union [ 11 - 121314 ] advise or require small distributed generation to synthetically inherit ancillary service functions and then to provide them. Consequently, FCR provision from battery-based PV household-prosumers has been identified as one of the highest value services [ 15 - 16171819202122 ]. HESSs have recently been advocated as excellent candidates for FCR because of their extremely fast ramp rate [9,10, 23 ]. 6 Complementarity between FCR and SFC can be expected, as FCR is a service where power capacity is offered, whereas revenues from SFC are more driven by energy capacity. Within a residential context, some studies have already underlined that combining PV SFC with FCR can substantially increase profitability [15,18,19]. PV household-prosumers participating in SFC and/or FCR demand both high-energy and high-power densities [2324252627 ]. Therefore, HESSs, which combine the functionalities of supercapacitors (SCs) and batteries, are an effective design to extend battery lifetime, improve efficiency and reduce the sizing and operation cost [23-24252627]. A HESS uses the unique SC properties, which offer high power density. This permits the supply of the sharp variations in the demand profile and thus prevents damage to the battery. Meanwhile the battery is oriented to the provision of large time demand requirements. To effectively protect the battery in the HESS topology [23], component sizing and management strategies should be optimized [23, 28 - 29303132 ]. Moreover, the HESS cost strongly depends on SC hybridization, and excessively high costs are prohibitive to commercial acceptance. It is thus crucial to design methods to optimize sizing and management strategies to fit each prosumer application. In the literature, various optimization models and frameworks that aim to provide the optimal sizing of battery/SC-based HESSs have also been investigated for electric vehicle (EV) technology [282930, 33 , 34 ] and microgrids [ 35 - 363738 ]. However, there are few studies on household-prosumers [ 39 , 40 ] and none on ancillary services. Existing studies merely optimize battery-based energy storage systems (ESSs), e.g. for household-prosumers in [15, 41 - 4243 ], and ancillary services in [15, 44 , 45 ]. The most frequently used optimization techniques [ 46 , 47 ] for optimally sizing previous ESSs are directed search based methods and heuristic methods. Directed search-based methods [44,42] include mixed-integer nonlinear programming (MINLP) [38], dynamic programming (DP) [30,43], and Pontryagin's minimum principle method [28,29]. In contrast, heuristic methods comprise neural networks [36] and population-based algorithms. Regarding the population-based heuristic algorithms, the two most important groups are evolutionary algorithms (e.g., genetic algorithm [GA] [33,37,41,45], and simulated annealing [35,39]) and swarm intelligence-based algorithms (e.g., teaching-learningbased optimization (TLBO) [17] and particle swarm optimization [PSO] [35-3940]). In order to signify the economic/performance role of systems that include at least two power or energy sources, an efficient power or energy management strategy is needed. The power or energy splitting among different sources can significantly influence overall dynamic performance, including equivalent consumption, system component lifetime, and overall cost. The optimal management problem is to find a power or energy sequence over time for each independent source as a control variable which minimizes some predefined objective function. In this regard, the literature on EVs [28,29,31,32-3334, 48 -55], 49505152 microgrids 5354555657 [15,35,38, 58 - 596061626364656667686970 ], household-prosumers [41,43, 71 ], and ancillary service applications [44,45, 72 - 737475767778 ], support that power or energy management strategies can be divided into three categories. 7 The first category involves an engineering-intuition control, which determines control laws based on rules [31,35,38,55,58,59,73,74,78], or predictive models based on artificial neural networks [77], fuzzy logic [33,56,60], wavelet transform (WT) [56,60,61], among others. This type of strategies are based on a set of simplified assumptions that are easy to implement and computationally fast. However, they are problem specific and may not provide the optimal global solution for complex engineering optimization problems. In comparison, the second and third categories include the optimization methods most commonly used because they can find the globally optimal solution based on a finite-horizon optimization. More specifically, the second category entails theoretical methods such as DP [30-3132,34,43,48-4950,54,72,76]. Nonetheless, significant work has been done for other theoretical methods, such as the interior point optimizer method [75], robust optimization [73], CPLEX-optimizer [71], convex programming [31], and heuristic methods. These heuristic methods include gravitational search algorithm (GSA) [64], multi-layer ant colony optimization (MACO) [65], GA [41,48,69,70], chaotic quantum GA (CQGA) [62], PSO [51,52], adaptive modified PSO (AMPSO) [66], TLBO [17], and artificial neural networks [53]. In addition, some authors solved their proposed model by means of hybrid optimization methods, such as fuzzy self-adaptive PSO (FSAPSO) [67], quadratic programming (QP) together with PSO [63], differential evolutionary (DE) and modified PSO (MPSO) algorithms [68]. The third category includes Pontryagin's minimum principle. The most efficient optimization method used to solve power or energy management strategies is DP, which is regarded as the absolute optimization method. The method of Pontryagin's minimum principle is nearly good as that of DP but it has the advantage of being more flexible [28,29,31,54,56,57]. The heuristic optimization methods can effectively explore the global optimal region. Furthermore, these methods are less dependent on problem formulations and have the optimizing capability of black-box systems. Of these methods, TLBO [17] outperforms population-based heuristic algorithms. Since the TLBO algorithm was introduced in 2011, several modifications have been proposed by researchers to avoid being trapped by local minima. For example, references [ 79 , 80 ] proposed new variants for improving its performance. In this context, given the large computational cost of DP [ 81 ] and to balance the advantages of the of rule-based and theoretical control strategies applied to power or energy management, this paper proposes a new hybrid meta-heuristic control framework based on the Haar WT algorithm (predictive model) and the TLBO heuristic method (theoretical model) that deals with the management strategies as well as sizing. Most of the previously mentioned power strategies have been used to deal with systems of double power sources (e.g., main source and battery or SC). However, when there are three sources (e.g., main source, battery, and SC), the two-dimensional power management optimization was only assessed in EV applications [28,29,49,50,57]. In that case, the optimization problem included one degree of freedom and two control variables, and associated state variables, in the power management design process. 8 Some of these articles coincide that component sizing and power management are two problems usually coupled together, and thus should be studied in greater depth. In an effort to investigate interactions, two-dimensional optimization methods have become an important research focus. Thus, references [28,29,30,32] propose an integrated optimization approach to determine the best system component sizing and power management. Data temporal granularity of input series (i.e., profiles of household consumption load [HCL] and PV generation [PVG]) and thus the selected time-step resolution for state and control variables have a crucial impact on the results of the optimization algorithms. Consequently, the component sizing such as the battery in PV prosumers [8, 82 ] changed because of inputs that are known to fluctuate at a temporal high-resolution (i.e., interval 0.01−10 Hz [23]). As longer time-step resolutions were envisaged, the input dynamics became increasingly ill-suited, resulting in the underestimation of battery aging [ 83 ] and an overestimation of sizing [82] and SFC ratios [8,23]. This meant that it was necessary to discretize the state and control variables at a fine level to provide an accurate picture of such changes in the level of ESS. Nonetheless, there is a trade-off between the computational burden and accuracy, which are determined by this discrete time-step [ 84 ]. To date, a major shortcoming is the fact that such inputs have not been adjusted to temporal highresolution. Most of the examples mentioned, such as [4,31,38-39404142,71,73] used hourly or even 30-min time-steps [31], which did not provide accurate results. Moreover, quite a few papers evaluated the bias from the use of coarse-grained data when only calculating the household performance. The resulting picture is ambiguous. Reference [ 85 ] concluded that five-minute time-steps provided a good balance between accuracy and the burden of data size, whereas [82] showed that using hourly data led to large biases compared to one-minute data. However, coarser data could be sufficient for household aggregation [ 86 ]. A shorter fluctuation at the granularity of just 4-seconds was investigated in [ 87 ]. Reference [ 88 ] found the impact of time-resolution was significantly influenced by the system configurations. Still another shortcoming is the way to explicitly include the most temporal details of a full time data series. Thus, most studies used time slices, more specifically, a reduced timespan chosen to characterize key aspects of temporal variability, for example, covering weekdays and weekends, different times of day, and different seasons. In this way, the time horizon envisaged was typically restricted to minutes [28-2930,32,49,50,57], a few hours [31,71,74] or a few days [42,73], which can be extremely overoptimistic when predicting cost. While the current trend is to use more temporally granular data sets in household applications, the influence of temporal granularity has yet to be analyzed using a comprehensive and high-resolution data set. Furthermore, the application of HESSs for providing FCR and PV SFC in household-prosumers is relatively new and has a high potential market. Therefore, to the best of our knowledge, the complementary study of services such as FCR and PV SFC in household-prosumer applications on the 9 basis of a joint optimization of power and energy management and component sizing has not as yet been reported in the literature. A more in-depth investigation of this problem is thus needed. For this purpose, we have thus expanded our previous work on battery-based PV householdprosumers [17], and have explicitly adopted not only the power management and sizing [28,29] but also the energy management required for FCR provision and application to an HESS. This was carried out by means of a hybrid meta-heuristic optimization algorithm which accounts for an increased number of optimization variables, especially applied to several service scenarios. This algorithm enables an accurate assessment of service complementarity, especially when different temporal data granularity is considered. This study addressed the following research questions:  How can time resolution be efficiently reduced by balancing the accuracy of the optimal sizing and management with the computational load?  Does service complementarity affect component sizing, and mainly the HESS?  What impact do optimal management strategies have on the effective cycle of the battery to extend its lifespan? To answer these questions, this research developed a novel high-resolution domestic multi-powerenergy model with several key features that have never been previously combined. This new model includes the following features: (i) separation of multi-energy services from multi-power inputs; (ii) accurate representation of HESS (e.g. battery degradation model) and converter efficiencies; (iii) high granularity. Furthermore, a hybrid meta-heuristic optimization algorithm based on WT and TLBO (WTTLBO) was used to obtain the best solution in the four-dimensional, non-linear, non-convex, and mixed-integer optimization problem. This problem deals with power and energy management and component sizing. The algorithm is designed so that it can be flexibly adapted in terms of the number of discrete states to suit the input profiles defined in different time discretizations. Its main advantage is that it provides a globally optimal solution, with a relatively non-excessive computational load stemming from the very high number of state and control variables. A case study using different data granularity for a PV prosumer in Spain showed the optimal results for several service scenarios. These results provide valuable insights into the influence of data granularity and service complementarity. The ultimate goal of this research was to determine the optimal granularity and the economic feasibility of providing complementary services. The remainder of the paper is organized as follows. Section 2 explains the design and modelling of the PV household-prosumer, and Section 3 quantitatively describes the economic modelling. The power and energy management are then discussed in Section 4, Section 5 deals with the optimization methodology, after which, the input data are provided in Section 6. The simulation and case studies in Section 7 are followed by the concluding remarks in Section 8. 16 3.2. Operation and maintenance cost The annual present value of the operation and maintenance cost for ith each component of the PV prosumer’s facility, &, , i O M a C is:        & , & 111 , , , , , , 1p N ii pp ii O M a O M i pp KK C C T i ad ddb dds pv bt sc NK       (29) where:   & 1 1 1 ii p O M g Kr d    (30)   & , , , , , , ii O M j I C C i ad ddb dds pv bt sc     (31)   & , & , a , , , , , , i O M a O M i C C i ad ddb dds pv bt sc  (32) The operation and maintenance cost for the PV prosumer’s facility is:   & , & , , , , , , , i O M a O M a i C C i ad ddb dds pv bt sc  (33) 3.3. Replacement cost The battery will eventually need to be replaced because of degradation during usage. The number of battery replacements during the lifetime of the facility, bt rep NC is: max min( , ) bt bt PV household prosumer p rep battery bt Lifetime N NC Integer Integer Lifetime N N           (34) The battery may reach the end of its useful life in two ways. It may either reach a maximum number of years from the time of its installation bt N (calendar lifetime), or have a maximum cumulated aging rate max () bt bt   at year max bt N  (cycling lifetime), before reaching its maximum number of years max ( ). bt bt NN   Let us define , Ra C as the net present cost of the battery replacement, namely, its replacement battery cost minus the revenues at the end of the facility lifetime due to the remaining battery lifespan:            max max max min , , , , a , a min , 11 min , 1 1 bt p repbat bt bt p bt bt bt s N N s N NC bt bt bt R a R a I p rep bt I sN s N N s gg C C C N NC N N C d d                       (35) 3.4. Net income from energy exchange The incomes associated with purchasing/selling electricity from/to the grid are computed on the assumption that different prices apply to energy purchasing/selling RTPEP, RTSEP, ( , ) kk  on a spot intraday market. Therefore, the net income is split between a cost due to energy purchasing and a revenue due to 17 energy selling. Every kth time the model activates a power transaction , , ( , ) g k g k pp  and the net income, , , EE k C is calculated as: RTPEP, , , , RTSEP, , , 0 0 k t g k g k EE k k t g k g k pp Cpp                 (36) As price on the real-time energy market is assumed to be constant during 2019, the present worth of the net income on an annual basis is:         , , , 1 1 1 1 1 1 1 1+ p ptt N n nN k N k N EE a EE k EE k n n k k pp qq g C C C N N q d                (37) where: 1 1 g qd   (38) 3.5. Net income from FCR provision The PV household-prosumer is paid for FCR power availability (downward-FCR , fcr-, k p upwardFCR ) fcr+, k p [ 97 ]. In other words, the remuneration is proportional to FCR power and the tendering period. Moreover, it is paid for the actual amount of FCR energy supplied () fcr+, k e or consumed () fcr-, k e [97]. Nonetheless, when the SOC in the HESS is too low or too high, it may fail to supply or absorb the amount of FCR energy requested by the system operator (SO) during the time interval. In this case, the prosumer receives a penalty that is proportional to the shortage of FCR energy () sfcr,k e at a short imbalance settlement price ( ). SISP,k  When there is a surplus, it receives the long imbalance settlement price () LISP,k  [ 98 ]. We then compute the net income from the FCR provision , () FCR k C with 80% of the aggregator's gain [15] over discretized values of power and energy buying/selling (first and second term) and long or short positions (third term):     , , , , , 0.8* ( ) ls FCR k RTPPAP k fcr-, k RTSPAP k fcr+, k RTSEUP k fcr+, k RTPEUP k fcr-, k LISP,k fcr,k SISP,k fcr,k C p p e e ee                    (39) The present worth of the net income from FCR provision on an annual basis is:         , , , 1 1 1 1 1 1 1 1+ p ptt N n nN NN FCR a FCR k FCR k n n k k pp qq g C C C N N q d                (40) 4. Power and energy management The battery and SC hybridization when managing power and energy can only take advantage of the characteristics of each source if effective power and energy management strategies are modelled. The power management strategy (or energy strategy) should set two objectives: (i) the power (or energy)- 18 splitting objective on a multi-time scale and (ii), the constraint objective of the feedback SOC. Accordingly, this paper models an integrated strategy based on a Harr WT and theoretical TLBO method for power and energy management. The three power sources are involved are the AC grid, battery, and SC. Therefore, one degree of freedom and two state control variables model the SFCPMS design. In contrast, two energy sources in the HESS involve one degree of freedom and one state variable in the FCREMS design. The basic principle of the WT-based power/energy management strategy is to supply a transient signal of the prosumer power demand (PPD) or prosumer energy demand (PED) from the SC, while supplying the corresponding base signal from the battery. In the WT, an original signal can be decomposed into localized contributions characterized by a scalable modulated time window of varying size. Each contribution represents a portion of the signal with a different frequency. By employing the long windows at low frequencies and short windows at high frequencies, the WT is capable of simultaneously comprehending the time and frequency information without concealing the details of the signal [61]. The discrete WT decomposes a discretized signal into different resolution levels as follows:     2 1 , ( ) ( ) , 2 , 2 , , jj tu W u s t dt u k k                (41) Of the different kinds of wavelet, the Haar wavelet has the shortest filter length in the time domain and is the most popular WT type [61]. Meanwhile, the function of extracting transients can still be well implemented without degradation. More details can be found in [97, 99 ]. The three-level Haar WT decomposition and reconstruction are applied in this study on the original discretized signals () i sn (PPD and PED profiles i.e., , , , ,, ppd k ped k ped k p e e  ) to model the first objective of the integrated power (or energy) management, Fig. 3. Thus, the Haar-WT based strategy uses a lowpass filter 0()lz and a high-pass filter 0()hz in the wavelet decomposition structure. The output of the jth high-pass filter is the detail part ,() ji dn and the output of the jth low-pass filter is the approximation part ,() ji an . Similarly, the synthesis filter bank consists of a low-pass and high-pass synthesis filter ( ( ), ( )). 11 l z h z The down-sampling and up-sampling method is employed in the decomposition and synthesis processes, respectively. The data size is reduced by half in down-sampling operations while it doubles in up-sampling operations [56]. 19 Fig. 3. Decomposition and reconstruction based on the Harr Wavelet. The transient signal of the ith PPD profile (or PED profile) () i Dn is reconstructed by combining the three decomposed detail parts while matching the number of samples of each detail part with upsampling. Meanwhile, the base signal of the ith PPD profile (or PED profile) () i An is reconstructed from the approximation part achieved after the final decomposition. By using this Harr WT-based strategy that decomposes the high-frequency and low-frequency components, the independent PPD (or PED) signal can take advantage of the three power sources (or two energy sources). Adopting only a high-frequency and low-frequency splitting strategy for the PPD (or PED) profile may not be sufficient to regulate the SOCs on the HESS. The battery and SC should have enough power and charge to supply the required PPD and PED and it should also have enough capacity to recover them. The majority of the studies implemented the SOC control through a semi-empirical strategy [55,58-5960]. Although this was simple, it did not optimize the power signal. In order to rationally adjust the initial power (or energy) references generated by the Harr WT for the battery and SC, optimal SOC control power (or energy) schedules, computed from the theoretical TLBO method, determined the optimal power (or energy) references of the integrated strategy, such that the SOCs in the battery and SC were recovered at close to their optimal values. 5. Optimization methodology 5.1. Optimization problem approach The optimization of the component sizing, SFCPMS, and FCREMS for PV household-prosumers is a constrained optimization problem. The discrete-time cost and revenue objective function to be minimized can be formulated as:   1 , , , , , t N k k k k k Mininize J F x k   u v w (42) h0(z) 2 l0(z) 2 h0(z) 2 l0(z) 2 h0(z) 2 l0(z) 2 si (n) Level 1 Level 2 Level 3 D1,i (n) 2 D2,i (n) 4 D3,i (n) 8 A3,i (n) 8 Di (n) Ai(n) d1,i (n) d2,i (n) d3,i (n) a3,i (n) a1(n) a2(n) Wavelet coefficients Three-level Haar WT decomposition h1(z) 2 l1(z) 2 h1(z) 2 l1(z) 2 h1(z) 2 l1(z) 2 si (n) Three-level Haar WT reconstruction 2 Downsample 2 Upsample Sample size Reconstruction for PMS and EMS 20 where, , , , , , k k k k xk  u v w are the vectors of state variables, SFCPMS control variables, FCREMS control variables, sizing control variables, and inputs, respectively. k represents a time interval. The prosumer state model is formulated in the discrete-time domain as the set of difference equations (5) and (8). These equations relate the state variable transition from the k-1th to kth state and can be written as follows: 1 ( , , , ), 1,..., k k k k t x f x k k N  u v ,w (43) The initial and final conditions for the state variable vector in (14) can be written as follows: 11 ( ) ( ) tt NN x x t x x t   (44) The vectors of the state variables , , ( [ , ] ) k bt k sc k x Le Le and control variables   ( , , = kk u v w = [ , , , ], [ ], [ , , , ] ) fcr   bt-, k bt+, k sc-, k sc+, k sc-, k sc+, k bt sc pv p p p p e , e Ω Ω Ω Ω ) are explicitly constrained in their domains by the equality and in-equality constraints (hard constraints). The equality constraint ,() ei H in (9) can be written as follows: ,( , ) 0, 1 e i k k Hi  u (45) The in-equality constraints ,() ine i H in (10)-(13) and (15)-(26) can be written as follows: ,( , , , , ) 0, 1,2,..., ine i k k k k ine H x i N  u v w (46) 5.2. Detailed optimization problem formulatation The optimization problem applied to PV household-prosumers is complex since there is a trade-off between different power and energy sources as well as component sizing. Three independent power sources, the AC grid, battery and SC should work simultaneously to meet the PPD. Meanwhile, two independent energy sources, the battery and SC, should fulfil the PED. Accordingly, a HESS power cooperative operation (SFCPMS) is necessary to balance the power inputs and outputs. Furthermore, a HESS energy cooperative operation (FCREMS) is required to balance the energy inputs and outputs. Hence, the discrete-time optimization for the PV household-prosumer with a HESS is a fourdimensional, non-linear, non-convex, and mixed-integer optimization problem. The optimization problem is defined by the vectors of state variables () k x and control variables ( [ , , ]) kk u v w , an objective function from (42), a discrete-time state-space model (43), and the corresponding constraints (45) and (46). Regarding the SFCPMS optimization, the control variables are the charge (or discharge) power reference for the battery and SC [ , , , ] . k    bt-, k bt+, k sc-, k sc+, k u p p p p Regarding the FCREMS optimization, the control variables are the downward (or upward) FCR energy reference for the SC [ ] . k    sc-, k sc+, k v e , e The state variables are battery and SC energy , , [ , ] . k bt k sc k x Le Le   Regarding the sizing optimization the control variables are the number of MBUs, MSUs, MPVUs, and MFCRUs [ , , , ] . fcr  bt sc pv w= Ω Ω Ω Ω 21 The optimization goal is to find the best sizing []w together with the optimal SFCPMS [] k u and FCREMS [] k v for different power and energy sources that must result in the lowest cost. Therefore, the objective function (42) is expressed in economic terms, and corresponds to the addition of terms expressed as:       , & , , , , 11 1 , , , , 1 p tt N NN k k k I a O M a R a EE k FCR k kk p qq F x k C C C C C Nq               uv (47) The terms in the objective function include all the costs and revenues involved in the project lifetime. They comprise the initial investment cost , , Ia C (27), operation and maintenance cost &, , O M a C (33), replacement cost , , Ra C (35), net income from energy exchange at the kth interval , , EE k C (37), and net income from FCR provision at the kth interval , , FCR k C (40), calculated in Section 3. LCOE for the PV prosumer is computed as follows [5]:   ,, 1 t N ev k hl k k J LCOE pp     (48) The search space contains all possibilities such as the following: max max max 0, 0, 0, , pv bt sc                   pv bt sc Ω , Ω ,Ω max 0, fcr    fcr Ω, min max min max , , , , bt bt sc sc p p p p          bt,k sc,k pp ,fcr k ad dds e 0,       s+,k e and ,. ad dds fcr k 0, e        s-,k ee 5.3. Proposed optimization algorithm Fitting power and energy management in the battery/supercapacitor-based PV household-prosumer is a key issue to improve HESS performance. Furthermore, component sizing is critical to increase the profitability of the asset. Therefore, the optimization in this research involves a four-dimensional, nonlinear, non-convex, and mixed-integer optimization problem with quite a large number of constraints, considering the discretized equations at every sampled interval. Integer values in the domain of solutions has been modelled (some of control variables discretized). The nonlinearity of the problem, the nonconvex functions in the objective function, and the excessive number of constraints pose additional challenges with multiple local optima. In this context, this paper has proposed a hybrid meta-heuristic optimization algorithm based on the WT algorithm and the TLBO heuristic method that deals with the management strategies as well as sizing. A more detailed description of the Haar WT algorithm can be found in [62]. The TLBO algorithm proposed was based on [17], but it used a variant for improving its performance. Thus, in teaching phase, a perturbed scheme was applied to prevent that current best solution from being trapped in local minima. Whereas a global crossover strategy was incorporated into the learning phase, which aimed at balancing local and global searching effectively [79]. 22 6. Set of inputs Experimental data of a residential facility from Jaen, a city in southern Spain, were acquired by smart meters at different granularities from 30 to 0.5-min time intervals (see [23] for details). Data included HCL profiles as well as EVCL and PVG profiles from an EV charging system and a 2.97-kWp PV system, respectively. In order to better compare results, the raw input data were downscaled to Spainaverage profiles. Namely, the PV annually yielded 1400-kWh/kWp. Meanwhile, 5450-kWh and 2440kWh represented the annual average Spanish-household electricity consumption [23, 100 ] and the EV charging electricity for the annual average driving distance in Spain of 13559 km, respectively [100]. Although the PPD data were collected at a granularity ranging from 30 to 0.5-min (30, 20, 10, 5, 2.5, 1, and 0.5-min) over the course of 2019, running the analysis at a 0.5-min time-step for 365 days was not computationally feasible. In order to cluster a representative sample of days, we chose 10 days of each month [ 101 ]. Fig. 4a1 and b1 show the PVG and HCL profiles for different granularities for one day in September whereas Fig. 4a2 and b2 represent the annual probability density functions (PDFs). These figures highlight the twofold nature of the HCL, the continuity of roughly 0.45-kW base-load and the intermittent spikes of up to a 5-kW peak-load. The level of variability at a 0.5-min compared to a 30min time resolution was significant as the peak power increased more than 270%. Using coarser temporal granularity ranging from 0.5 to 30-min substantially affected the PDF shape for the PVG and HCL. The shape was skewed more frequently near those hours with lower power, something that removed many of the extremes. The extreme ends of the PDF were of potential interest as they represented periods of very low or very high consumption/generation. Since Spain has a liberalized electricity market, some electricity marketers now offer real-time pricing tariffs [17,23]. Thus, the electricity price model used a 1-hour discretized time resolution and computed the price from the day-ahead wholesale market of electricity [17] with a 10% benefit added for the electricity marketer plus the access toll of the current Spanish tariff 2.0 [ 102 ] plus taxes. Regarding economic FCR data, it was assumed that the prosumer was a price-taker in the European FCR market (joint tender of German, Dutch, Belgian, Swiss, French, and Austrian SOs) [17, 103 ]. This market consists of two main elements; payments for availability and utilization as well as payments for downward or upward regulation. To assess the FCR power of a PV household-prosumer, hourly data of the upward and downward FCR power availability factor at the country level (in Spain) were used [17]. Moreover, for modelling the different cumulated FCR energies in Section 2.1, we considered the frequency data which the PDF for 2019 shows in Fig. 5. These data were provided by the Spanish SO at a specific point in the synchronous area of continental Europe (Hernani, Spain) [17]. As can be observed, high-frequency components were quite damped due to the effect of the system inertia and FCR provision. Some distinct peaks were visible in the medium range, which were mainly 23 because of the way the market was operated. Furthermore, Fig. 4c1 shows the cumulated stored FCR energy for one day in September, and Fig. 4c2 reflects the corresponding annual PDF. As granularity decreased, there was also increasingly less probability that the prosumer would be requested to ramp down power capacity to 0.5 kW. Fig. 4. Input data for different time discretizations: (a) PVG of an MPVU: (b) HCL; (c) cumulated stored FCR energy for an MFCRU. (-1) profile for one representative day in September; (-2) annual PDF. Fig. 5. PDF of the frequency signal at Hernani, Spain. 24 7. Results Based on the formulation in Section 5, the overall system in Fig. 1, was modelled in the MATLAB programming environment to solve the optimization problem. The results obtained are presented in this section. This environment worked in a grid computation system which included 680 cores, 2 TB RAM, 6.2 TFlops of computing power, 13 TB of storage, and 32 calculation nodes with DDR at 20Gbit/s with a Sun Grid. Given this framework, interesting conclusions were obtained regarding the granularity of data when PV household-prosumers deal with different scenarios of service complementarity. The performance of the optimization algorithm was firstly compared for different data granularity. After that, the accuracy and convergence of the proposed optimization algorithm was subsequently compared with other wellknown algorithms. The next step was to focus on the best sizing results. The optimal results for the SFCPMS and FCREMS on the HESS were then added. Finally, the battery aging in different scenarios and data granularity were compared. The simulations considered a time horizon of one year (ten days in each month) and seven resolutions for the granularity of data, from a half-minute to 30 minutes. This time horizon allowed for the daily, weekly, and annual cyclicality of PPD and PED profiles to be accurately assessed. The referential technical values and factors assumed for the cost model are summarized in Table 1. The parameters of the frequency control module for FCR provision came from [9,10]. The battery aging parameters are given in [17,103]. 104105106107108 Table 1. Technical and financial parameters applied in the optimization. Parameter Value Parameter Value Parameter Value ( )[ ]ad ddb dds C –Power specific capital cost for the AC/DC converter (battery converter) [SC converter]– 100 €/kW [4,33,99] i N   , i bt sc –ith minimum unit lifetime– 20 years T –Income tax rate– 0% [17] bt C –Energy specific capital cost for the battery– 300 €/kWh [4,6,15,16,99,104] , ( ), [ ] ad ddb dds P P P –Rated power of the minimum unit of AC/DC converter (DC/DC battery converter) [DC/DC SC converter]– 1 kW/unit t  –Discretization time-step– 30, 20, 10, 5, 2.5, 1, and 0.5min pv C –Power specific capital cost for PV– 1000 €/kWp [4,99] max max () bt bt PP   –Maximum discharge(charge) power for the minimum battery unit– C-rate =1 rat bt  –Rated DOD of the minimum battery unit– 80% sc C –Energy specific capital cost for the SC– 10000 €/kWh [4,104] c P –Contracted power from the electric mains– 5 kW rat sc  –Rated DOD of the minimum SC unit– 100% d –Nominal discount rate– 3.5% [104] fcr P –Prequalified power capacity of the minimum FCR unit– 0.5 kW/unit max bt  –Maximum number of battery units installed– 25 units g –Expected annual inflation – 1.4% [105] pv P –Rated power of the minimum PV unit– 1 kW/unit max fcr  –Maximum number of FCR units installed– 1 unit max bt LE –Maximum energy of the mininum battery unit– 1 kWh & i OM r   , , , , , i pv bt sc ad ddb dds –Annual escalation rate of the operation and maintenance cost of the ith minimum unit– 1.4%, 2%, 0.3%, 0.3%, 0.3 % [99,106] max pv  –Maximum number of FCR units installed– 20 units max sc LE –Maximum energy of the mininum SC unit– 0.1 kWh pv r –PV degradation rate– 0.5% [107,108] max sc  –Maximum number of SC units installed– 100 units i N   , , , i pv ad ddb dds –ith minimum unit lifetime– 25 years [4] t –Time interval– 10-3-s 25 7.1. Performance comparison of the optimization algorithm The optimal solution found by the algorithm proposed did not yield exactly the same result each time it was run. In order not to confuse differences in solutions stemming from data granularity with differences in solutions due to the nature of the algorithm, each calculation was repeated 30 times. The performance comparison of the algorithm was validated on the PV household-prosumer with HCL, PVG, HESS, and FCR (hereafter defined as scenario #4A). Fig. 6 shows the LCOE results. Also shown are the normalized computation times. When data was defined on a finer granularity, the algorithm converged to the optimal fitness value after lower iterations. The higher the population size was, the lower the LCOE. This outcome was more significant for coarser data granularity. Nonetheless, as the population size got higher, the reduction in LCOE became smaller, which was possibly because a higher number of constraints had to be satisfied. The finer data model imposed additional security and dispatch requirements on the prosumer at each sub-period. As reflected in the standard deviation, the ranges of near optimal solutions decreased when higher resolution data were used. This indicated that many solutions gave approximately the same objective function, which became flatter around the optimum. Computational cost was also an important issue, as shown in Fig. 6. The computational time was normalized by the reported time for a 30-min granularity and a population of 1000, which was 31 min 12 s on average. As can be observed, the computational load rapidly (and critically) increased with each increase of time resolution. The size of sub-30-min models nonlinearly increased the difficulty of solving the resulting optimization problem. If the search space granularity was reduced, an exhaustive search converged proportionally faster, but at the expense of a less accurate approximation of the true solution. The rise in computational time was most noticeable for scenario #4A, as compared to the others (see Section 7.3). This was because the model had more complex constraints with respect to time when storage, FCR and SCF were included. Fig. 6. Best solution for various population-size values and time granularity. 32 Fig. 10. Results for the optimal SFCPMS at scenario #1A for three samples of data granularity. Fig. 11. Results for the optimal SFCPMS at scenario #2A for three samples of data granularity. Fig. 12. Results for the optimal SFCPMS at scenario #2B for three samples of data granularity. Fig. 13 shows the optimal results for the FCREMS from the PED profiles ,, (ped k fcrk ee  and ,, ) ped k fcr+ k ee  in scenario #3. This includes the optimal upward and downward FCR energy references for SC ( , ) sc+,k sc-,k ee along with the initial energy references, namely, the achieved upward and downward detail energies of the PED profiles after WT decomposition ,, ( , ). det k det k ee  As reflected in the 30-min data granularity, note that detail energy for SC assisted battery with a downward energy peak of 0.022 33 kWh versus 0.032 kWh of cumulated stored FCR energy (time 1.77). There was also a delivered upward energy peak of 0.064 kWh versus 0.095 kWh of cumulated released FCR energy (time 1.83). The FCREMS not only controlled the SOCs in the HESS, but also shifted the energy burden on the battery to the SC. This smart energy splitting was observed around the time of 1.83 where the upward PED reached the peak and the FCREMS gave priority to assisting the battery rather than to tracking SC SOC, thus generating negative power to discharge the SC. In practical terms, the FCR energy references for SC were the control variables to be determined. Thus, SC supplied or absorbed the peak FCR energy regulation. Its SOC thus fluctuated over a wide range. The battery followed a slow and fairly stable energy reference from the upcoming upward regulation throughout these days, and thus, the battery SOC decreased at a relatively low rate. Regarding the three snapshots of data granularity, the finer the interval resolution, the closer the outliers in the optimal FCR energy references for SC, came the detail part. Large biases in battery energy were thus observed. Fig. 13. Results for the optimal FCREMS at scenario #3 for three samples of data granularity. The comparison of the SFCPMS results in scenarios #2A,B (Figs. 11 and 12) and #4A,B (Fig. 13a and 15) indicated that although scenario #2B or #4B included two loads (EVCL and HCL) in comparison to #2A or #4A, respectively, the speed of changes in EVCL was relatively low. As a result, the WT power splitting algorithm generated detail power references that were equivalent to the SC. Generally, there was a minor difference in the resulting battery SOC from the additional FCR provision in relation to scenarios #2A,B. This means that shallow cycling from the FCR provision had almost no effect on the macro cycles, and thus on the battery SOC. However, this FCR provision had more influence on the SC. The SC operated less often and therefore its SOC underwent lower changes in scenario #4 because of a more continuous and lower power requirement and a larger SC sizing. Moreover, in scenario #4, the optimal power reference for the battery moved closer to the approximation part, and the grid thus operated less often. A closer examination of the approximation part in scenarios #2A,B showed that they 34 were different from scenarios #4A,B, because of their different PPD profiles mainly during nocturnal hours. The reason for this was the EVCL. The comparison of the FCREMS results in scenario #3 (Fig. 13 ) and #4A (Fig. 14) disclosed that a greater burden was supported by the SC in #4A. This can be explained by its higher optimal sizing. The battery barely absorbed and delivered power at the coarser temporal resolution. (a) (b) Fig. 14. Results for the optimal SFCPMS (a) and FCREMS (b) at scenario #4A for three samples of data granularity. Fig. 15. Results for the optimal SFCPMS at scenario #4B for three samples of data granularity. 35 7.5. Analysis of battery capacity degradation. This section studies the influence of data granularity on the battery capacity degradation by using the degradation model in Section 2.4.1 that considered the DOD and C-rate. Fig. 16 depicts the statistical distributions for the C-rate and DOD over a year for all planned service scenarios and three samples of data granularity. These distributions reflected how the SFCPMS and FCREMS controlled the battery cycling. Insofar as the 30-min data granularity in scenario #3, the C-rate showed a distinctly high peak at zero and a wide peak around -0.149 pu, whose range was up to -0.876 and 1.303 pu. The larger battery sizing in scenarios #2B and #4B as compared to #2A and #4A, respectively, resulted in C-rate distributions with peaks at lower levels. Scenario #2A had a distinct peak at 0.002 pu and a narrow and low peak at around -0.027 pu. In contrast, scenario #2B had a higher peak at around zero, but the main distribution was in the narrower interval of -0.15–0.206 pu. This difference of distributions could be explained by the more stable PPD in scenario #2B. Concerning the DOD at each cycle, typical behavior from the PV SFC (scenarios #2 and #4) was observed with large DODs (e.g. daily cycling). However, scenario #3 resulted mostly in lower DODs, typically below 8.25% because battery power underwent significantly higher polarity reversals. The larger PPD in scenario #2B versus #2A increased the DOD values. In the three snapshots of data granularity, the C-rate had a dissimilar range and average peak-to-mean ratio. However, the DOD demonstrated a much higher ‘peak-to-mean’ ratio. More specifically, the finer interval resolution was, the outliers in the C-rate usually slowly decreased, and the distribution tended to gradually approach a Gaussian distribution, except for scenario #4B. In contrast, the finer resolution led to a strong decrease in the DOD. Scenarios #2A and #4A Scenarios #2B and #4B Scenario #3 Fig. 16. Distributions for C-rate and cycle DOD throughout a year. For most planned scenarios, the algorithm proposed optimally sized the battery so as to adjust its expected lifetime to about 20 years, which avoided the replacement cost. This resulted in a diverse 36 battery sizing for different PPD and PED profiles but in the same battery capacity degradation. In order to solely highlight the influence of the DOD at each cycle and the C-rate variables on battery capacity degradation for each scenario with a specific battery sizing, the battery capacity degradation was normalized by a ratio, the discharging energy cycled versus the battery capacity. Fig. 17 thus displays this normalized capacity degradation and the DOD at each cycle during a 9-day snapshot. The multiplicity of depth cycles was due to the SFCPMS and FCREMS. As can be observed, when the interval resolution was finer, the normalized battery capacity degradation worsened. In this regard, the normalized degradation increase from largest to smallest was: 429% (#3), 6.34% (#2B), 5.34% (#2A), 0.47% (#4B), 0.27% (#4A). Scenarios #2A and #4A Scenarios #2B and #4B Scenario #3 Fig. 17. Profile of normalized battery capacity degradation and cycle DOD. 8. Conclusions Increasing interest in services such as FCR and PV SFC for household-prosumers may in the near future significantly change domestic PPD and PED profiles. To understand the impact of such services, it is necessary to fully appreciate the sizing and management aspects of such prosumers. This type of holistic optimization treatment is necessary to fully capture the effects of coincidence in various service demands within and across dwellings. To that end, this paper has presented and discussed high-resolution power and energy models as well as a hybrid meta-heuristic optimization algorithm based on WTTLBO that dealt with management strategies and sizing in a four-dimensional optimization problem. The design of the algorithm was flexibly adapted in terms of number of discrete states to suit the input profiles as defined on different time discretization levels. The proposed optimization algorithm was illustrated in MATLAB by means of the case study of PV household-prosumers in Spain that provided examples of different data granularity. Additionally, real data from FCR and energy market were used. 37 The influence of data granularity on the best solutions for sizing and its financial profitability was first discussed. Initially, for finer data granularity, the optimization results indicated that there was a cost increase ranging from 0.6% to 354.7%. There was also a drop or rise in optimal sizing levels of -42.1% to 28.2% for PV; -60.2% to 28.9% for the battery; and -4642.5% to -463.6% for the SC, depending on the service scenarios. Nonetheless, excluding scenario #3, the increase in costs did not exceed 8.3%. Without storage (scenario #1A), optimal PV sizing noticeably fell to 42.1% when time resolutions were increased. The availability of storage had an impact on the results with respect to the data granularity since the smoothing of the demand and production fluctuations could take place by means of storage. Thus, in scenario #2A, PV sizing behaved similarly to the no-storage solution, whereas optimal HESS sizing largely decreased, particularly the SC. However, in scenario #2B, the larger PPD reduced and reversed the difference in the PV and battery sizing change with increasing time resolutions. On the other hand, the additional FCR provision did not substantially change the behavior in the PV, battery and SC sizing except for the finest data granularity. Therefore, there are no specific guidelines with respect to the influence the data granularity on sizing. The results suggest that for profitability purposes, it is not necessary to use the finest data granularity except for scenarios #3 and #4A. In fact, the high-resolution data showed that many solutions had a similar outcome, such that even near-optimal solutions were able to produce satisfactory outcomes. However, when evaluating the sizing rather than the NPV, accuracy can be gained by using the finest resolution data. Furthermore, when the objective is not to optimize average prosumer performance (cost effectiveness) but rather to increase component performance (e.g. battery aging), the short-term fluctuations should be suitably represented in the optimization process. In any case, when 0.5-min data rather than 5-min data were used, the NPV increase was very small and the change in sizing was somewhat limited. Consequently, considering the computational burden and limits to modelling flexibility that accompany the use of high-resolution data, it is not advisable to use data with time-steps smaller than five minutes for optimization purposes. This provides a sizing close to that of the finest resolution. The results obtained in several service scenarios highlighted that the combination of FCR and SFC was clearly complementary. Furthermore, the incomes and savings obtained, thanks to the incorporation of PV, HESS, and FCR provision were higher than the amortization costs. The optimal results for the SFCPMS and FCREMS showed that the SC successfully supplied and absorbed short-duration and peak power from the PPD for the PV SFC, as well as the corresponding short-duration and peak energy from the PED for the FCR. The battery followed the fairly stable power of the daily macro cycles of the PPD together with the slow change of the PED required by FCR energy regulation. The AC grid started to work only under high and continuous PPD conditions. Regarding data granularity, it can be concluded that when interval resolution was finer, the biases in battery were larger. As a result, battery degradation was significantly increased per normalized kWh 38 delivered up to 6.34% for finer data granularity (excluding scenario #3). Battery aging thus played a crucial role in prosumer profitability because of the replacement cost. The fact that the data sample was from a single PV household-prosumer in Spain is a limitation of this study. However, this data sample was sufficient to achieve our primary objective, which was to demonstrate the usefulness of this optimization methodology and to examine the impact that temporal granularity and complementarity of services can have on the optimal results. In conclusion, hopefully, this study will lead to future work and discussion in the research community regarding the provision of complementary services by residential PV prosumers and encourage similar work on datasets from other multi-family residential buildings. Acknowledgements This research was funded by the Agencia Estatal de Investigación, Spain (AEI) and the Fondo Europeo de Desarrollo Regional (FEDER) aimed at the Challenges of Society (Grant No. ENE 201783860-R ‘‘Nuevos servicios de red para microredes renovables inteligentes. Contribución a la generación distribuida residencial”). References [ 1 ] T. Kaschu, P. Jochem, W. Fichtner, Solar energy storage in German households: profitability, load changes and flexibility, Energ Policy 98 (2016) 520–32. [ 2 ] M. Talaat, M.A. Farahat, M.H. Elkholy, Renewable power integration: Experimental and simulation study to investigate the ability of integrating wave, solar and wind energies, Energy 170 (2019) 668–82. [ 3 ] M. Bianchi, L. Branchini, C.Ferrari, F. Melino, Optimal sizing of grid-independent hybrid photovoltaicbattery power systems for household sector, Appl. Energ. 136 (2014) 805–16. [ 4 ] D.N. Luta, A.K. 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