Predicer: abstract stochastic optimisation model framework for multi-market operation

First, here are listed selected studies concerning stochastic optimisation models related to single actor energy market operation in terms of methodology. The authors of Fleten and Kristoffersen (2007) present a stochastic optimisation model concerning day-ahead spot-market bidding for hydro-power system. This paper provides the theoretical background for combining physical production and market bidding and provides method for creating bidding curves based on arbitrary price levels. The authors of Akbari et al. (2019) present a model structure and a case study for stochastic programming-based optimal bidding concerning an energy system including compressed air energy storage and thermal units. The results indicate that the stochastic optimisation method is efficient for optimal bidding in the example case. The authors of Campos et al. (2016) examine a methodology to obtain an optimised bidding curve for the secondary reserve market including an application to the Spanish secondary reserve market. In addition, this paper lists a variety of optimisation models related to reserve market modelling. The authors of Silva et al. (2022) present a multistage stochastic modelling approach to day-ahead, balancing market and intra-day markets based on linear optimisation. This modelling approach provides the optimal bidding strategy for a virtual power plant with wind power, solar photovoltaics (PV), and energy storages. The results presented in this paper indicate that participating in multiple markets increases profits by 10% in the case study. This paper also lists a variety of optimisation models related to optimal bidding of VREs in the electricity markets. The authors of Xiao et al. (2020) present a stochastic bi-level optimisation model to be utilised in strategic bidding in energy and regulation markets aggregated prosumers. A case study based on Australian prosumers indicate that aggregated strategic bidding decreases operation costs by 7.5%. The authors of Wong and Fuller (2007) examine stochastic optimisation modelling as a tool for pricing in day-ahead energy and reserve markets. Finally, the authors of Parvar and Nazaripouya (2022) use another approach to model PV and battery based multi-market operation under uncertain pricing by utilising robust optimisation which minimises the cost function over a worst-case scenario.

The papers examined above, however, do not provide any general tools for the model user to construct freely any energy system for multi-market operation. The energy modelling community provides several modelling frameworks to construct arbitrary energy systems. TIMES (The Integrated Markal Efom System) model framework (see Loulou and Labriet 2007) for methodology details) provides a technology-rich economic model generator for multi-regional and multi-sector energy systems, in which the user can define the model structure by setting input data. In Finland, the TIMES-VTT model, presented in Koljonen and Lehtilä (2012), has been used for several governmental studies in terms of national climate and energy strategy. However, as a partial equilibrium model and designed for sector analysis, the TIMES model framework cannot be utilised for single-actor multi-market optimisation purposes. The open-source FlexTool model framework presented in Taibi et al. (2018) is an investment and dispatch optimisation model designed for exploring flexibility issues concerning power systems with renewable energy. The FlexTool model structure can be set up by using input data and handles reserves from the system point of view. The open-source OMEGAlpes model generation tool is described in Hodencq et al. (2021). This district scale tool aims at making energy models understandable and easy to use. OMEGAlpes tool has vast library of use cases listed in Hodencq et al. (2021). The Open Energy Modelling Framework (oemof) is an open source modelling system described in Hilpert et al. (2018). This Python based flexible, generic and modular modelling framework has been used in several energy system research studies such as in Röder et al. (2020). Another data-driven energy system model framework is Backbone, documented in detail in Helistö et al. (2019). The open-source Backbone model framework is highly adaptable and has features such as stochastic parameters, multiple reserves and multiple energy sectors. Even though Backbone can be somewhat utilised to examine operation in energy markets from a single-actor point of view, reserve market operation is not handled in the Backbone model structure. The SpineOpt energy system modelling framework, introduced in Ihlemann et al. (2022), includes generic data structure, stochastic uncertainties and flexible temporal structure. These features of the SpineOpt model enable diverse implementations of energy systems.

While the papers and energy modelling frameworks presented above provide an important contribution to the research field, they either lack the detailed functionalities or abstract data structure required for modelling market and operational decision-making from the point of one actor on the markets. Predicer has been developed with these shortcomings in mind; a generic modelling framework capable of detailed energy multi-market decision-making and multi-asset operation from the perspective of one actor in a stochastic multi-scenario environment.

Our contribution in this paper is based on the presentation of the new open-source modelling framework which enables the creation of an arbitrary data-driven energy system and ability to optimise the multi-market interaction between the operator of this energy system and the energy and reserve market operators under uncertainty of required system data. Our new open-source tool Predicer, standing for Predictive Decider, is aimed at variable scale energy systems participating simultaneously in energy and reserve markets by creating bidding curves based on stochastic optimisation. This modelling tool supports multi-stage operation of a single-actor energy producer with easy data insertion and model topology creation. In terms of reserve market related operation, our new model framework provides a functional connection between different reserve market products and producing/consuming units participating in the reserve markets.

The Predicer model is implemented by using Julia programming language and specifically the JuMP optimisation package, and the current version uses the open-source HiGHS optimisation solver (Huangfu and Hall 2018), although the solver option can be easily modified within the model code. According to Weibezahn and Kendziorski (2019) a Julia JuMP based energy system model is significantly faster than Python based Pyomo optimisation modelling environment and at a similar level with the commercial GAMS modelling tool in terms of non-solving related speed. The current working version of the Predicer model source code is available in Github (Pursiheimo et al. 2022).

The basic features of the stochastic optimisation model framework Predicer are described in this chapter. It should be first noted that there are several important terms, used in the model definition, which are briefly clarified in Table 1. The abstract energy system model inside the Predicer basically consists of two elements, nodes and processes, which are linked to each other and have properties and parameter data attached. The main elements can be defined as follows:

The flow variables related to processes have topology set according to Fig. 3. Each flow variable is identified by the process it is attached to, as well as the source and the sink, which can be either a node or a process as illustrated in Fig. 3. In the unit type process, the combination of the source and the sink must be a combination of a node and a process, whereas in the case of the transfer process and the market process, both source and the sink must be nodes.

The participation of the Predicer model in the reserve markets (concerning multitude of different reserve market products) requires a complex modelling framework in order to bind together reserve markets, energy markets and properties of the used abstract energy system. Therefore, the reserve market handling must be examined in greater detail here. The abstract model in the Predicer handles the reserves as visualised in Fig. 4. There are three levels of reserve variables ensuring that reserve operation is distributed correctly to processes capable of reserve commitment. These reserve variables in accordance with Fig. 4 are as follows:

The link between reserve potential and final reserve product in an example with producer and consumer process and four different reserve products is illustrated in Fig. 5. It illustrates how up and down reserve potential directions relate to the load of producer and consumer processes. In the case of producer process, up direction represents operative reserve to increase load and down direction represents reserve to decrease load. In the case of consumer process, it is vice versa. It should be noticed that reserve potentials with different activation speeds are treated cumulatively. Also, Fig. 5 illustrates how symmetric reserve product (UP/DOWN) allocates aggregated reserve potential equally from both UP and DOWN reserve pools.

In the Predicer model, the ramp rates, representing limitations for the flexibility of the processes, are considered as illustrated in Fig. 6. Basically, the ramp rate parameter limits the alteration in process load between two timesteps, e.g., if the ramp rate of a process is 20%/h and production capacity is 1 MW, the load can alter 200 kW per hour and 50 kW per 15 min. However, there are factors affecting the modelling of the ramp rate:

The risk factor should be included in the stochastic optimisation model, and in Predicer, risk is involved by adding a Conditional Value at Risk (CVaR) element into the objective the model tries to minimise (seeKrokhmal et al. 2001; Fleten and Kristoffersen 2007 for CVaR related details). CVaR can be defined as the expected value of the scenario-based cost higher than (1−α) quantile of the scenario-based cost distribution. In the model, the expected value of the total cost of the system and the CVaR variable representing risk are weighed as the objective function to be minimised.

There are requirements for bid curves at each time step that affect the stochastic modelling structure and basically link the scenario paths to each other. Since bid curves are formed for each time step separately from scenario-based results (see Fig. 2), these price–volume pairs should be sorted in order to produce coherent bid curves. Therefore, the following bid curve requirements (ensuring that each bid curve is of increasing nature) are included in the model:

This bid curve requirement ensures that each scenario is not optimised independently, but there is a model-based link between each scenario, and therefore, for example the risk component is not trivial from the stochastic model point of view. These bid curve requirements apply to both energy and reserve markets.

Finally, it is important to note that multi-market optimisation tool is usually operated in cycles such as in the Fig. 8, that is, market bids are not sent to the operators simultaneously, but one market will close before another market. Therefore, closed markets are implemented in the model by setting the bid volume (and prices) to a fixed value and then proceeding to optimise the bid curve for other markets. In fact, bids for D-2 from day before current day-ahead optimisation must be taken into account by using fixed volumes, since they still affect the target day optimised. The Predicer model enables fixing market bids by setting time-series values in the input data.

The equations of the stochastic optimisation model are defined in this chapter. First, it should be noted that flow variables \({v}^{flow}\left(p,so,si,s,t\right)\) have index based on the process topology tuple \(\left\{p,so,si\right\}\in PRO\) and scenario (s) and time indices (t). This process topology index applies also to reserve potential variables \({v}^{res\_pot\_up}\left(rp,p,so,si,s,t\right)\) and \({v}^{res\_pot\_down}\left(rp,p,so,si,s,t\right)\), in accordance with principles from Fig. 3 and Fig. 4.

The balance of non-market or non-commodity nodes must be enforced by balance Eq. (1). The balance of nodes n includes the following components:

It should be noted that all the elements in the left side (points 1–5 above) of the balance Eq. (1) are represented in terms of power (MW) and storage state is represented in terms of energy (MWh) and therefore time step length dt must be used. Also, it is important to notice that the time step length can be shorter than one hour as long as the time frame is a full fraction of an hour. For example, if the time step is 15 min, the time step length dt used in the model equations is defined as 0.25.

Amount of energy charged to or discharged from node storage is limited by node specific parameters \({d}^{state\_max\_out}(n)\) and \({d}^{state\_max\_in}(n)\) in Eq. (3). Furthermore, the state of node storage is limited by storage size \({d}^{state\_max}(n)\) in Eq. (4).

Each process with reserve operation enabled subject to node n has reserve potential variables based on the operative load with respect to minimum and maximum limits of the process. Reserve potential is divided into up and down categories (rd) and also into categories based on the activation speed (rp). For example in the case of up direction, these process-based reserve potential variables \({v}^{res\_pot\_up}\left(rp,p,so,si,s,t\right)\) are aggregated for each node used in the reserve market as \({v}^{res\_up}\left(rp,n,s,t\right)\). See Eqs. (5)–(6) for node aggregation.

On the other side, the reserve products \({v}^{res\_final}\left(r,s,t\right)\) sold to the market are formed from the aggregated reserve potential pool(s) \({v}^{res\_up}\left(rp,n,s,t\right)\) and/or \({v}^{res\_down}\left(rp,n,s,t\right)\) with respective reserve directions and speeds, according to Eqs. (7)–(8). It should be noted that symmetrical reserve product (UP-DOWN) uses reserve potential equally from the UP and DOWN reserve potential pools.

There are six different cases concerning process flow properties depending on (1) online capability, (2) flow type (consumer or producer) and (3) reserve participation (these combinations are identified in Table 2). In terms of minimum and maximum constraints and ramp up and ramp down constraints, these cases require different equation formulations. It should be noted that since transfer processes cannot be online processes and cannot participate in the reserve market, they are only concerned subject to the Case 1 in Table 2.

The maximum operative limit equations concerning unit type processes are defined for six different cases of Table 2 in detail in Table 3 as Eqs. (9)–(14).

Furthermore, the minimum operative limit equations of unit type processes are defined in Table 4 for similar cases as in the maximum limit equation table.

In the model there are processes with capacity factor (e.g. variable renewable energy units) which have only output flow and volume of this flow \({v}^{flow}\left(p,so,si,s,t\right)\) depends on the capacity factor time series \({d}^{cf\_fix}(p,s,t)\). There are two types of CF processes: fixed flow (see Eq. 21) and upper limit flow (see Eq. 22).

The efficiency of the unit type process is modelled in two styles: fixed efficiency and piece-wise efficiency. The fixed efficiency \({d}^{eff}(p,s,t)\) defines how the sum of flows to the process relates to the sum of flows from the process. See Eq. (23).

The piece-wise efficiency relation is based on different efficiency values for different operative load intervals. In this measure, the production capacity is divided into operative slots (op) with minimum load value \({d}^{op\_min}(p,op)\) and maximum load value \({d}^{op\_max}(p,op)\) as stated in Eq. (24). Each slot has an input flow \({v}^{flow\_op\_in}\left(p,s,t,op\right)\) and output flow \({v}^{flow\_op\_out}\left(p,s,t,op\right)\) related via the efficiency Eq. (25). Also, for each operative slot there is binary variable \({v}^{flow\_op\_bin}\left(p,s,t,op\right)\) representing whether the slot is active (as stated in Eq. (28), sum of these binary variables must be 1). Actual input and output flows are calculated as sums of slot flows, see Eqs. (26)–(27).

Processes with online/offline status (\(p\in {P}^{online}\)) have binary online variables \({v}^{online}\left(p,s,t\right)\) which follow the dynamic Eq. (29). The status of online process is changed via binary start variable \({v}^{start}\left(p,s,t\right)\) and stop variables \({v}^{stop}\left(p,s,t\right)\). Furthermore, minimum length of the online and offline periods is constrained via Eqs. (30) and (31), so that \({v}^{online}\left(p,s,t\right)\) must be 1 for a period after unit is started and \({v}^{online}\left(p,s,t\right)\) must be 0 for a period after unit is stopped.

The ramp rate of a process defines how fast operative load of the process can change from time step to next. When assessing ramp rate constraints, reserves (and activation speeds of the reserves) and minimum load must be taken into account as illustrated in Figs. 6 and 7, using cases from Table 2. The ramp up equations are defined in Table 5.

Same applies to the ramp down constraints as stated in Table 6:

In the case of charging processes (\(p\in {P}^{cha}\)) participating as reserves (\(p\in {P}^{res}\)) the state of the storage must be considered when allocating reserve potential for these units. If the discharging process has reserve potential in UP direction, there must be adequate energy in the storage to cover this reserve potential (Eq. 44). Also, if the charging process has reserve potential in DOWN direction, there must be adequate empty space in the storage to cover this reserve potential (Eq. 45). The logical structure between reserve potential and storage state in the case of storage participating in the reserve market is illustrated in Fig. 9.

There are requirements for the charging processes in terms of input data: (1) there can be merely one input and one output, (2) efficiency must be 100% and (3) one of the nodes must be a state node. It should be noted that separate charging/discharging processes are necessary in the model only if these processes participate in the reserve market. If this is not the case, the node can just be defined as a state node without any separate charging processes (see the example case in the next chapter). In addition, it should be noted that these separate charging/discharging processes can be used in the storage modelling if the user needs to define ramp rates for charging or discharging.

There are two rules for scenario-based bidding curves in bid markets. Since bidding curve is formed for each time step and prices are scenario-based for these time steps, the volume sold to the market in scenario s1 must be lower than in scenario s2 if market price in scenario s1 is lower than in scenario s2. See Eqs. (47) and (49). Also, if the prices in scenarios are equal, the market volumes must be equal too. See Eqs. (48) and (50). This rule applies to both energy and reserve markets with bidding curves. However, in the case of energy market, the bid consists of energy flow from and to the market added with up and down balance market volumes. See Eq. (46).

In the case of reserve market, the bid is formed directly from the sold reserve product:

The model enables the construction of generic linear equality (Eq. 51) or inequality (Eq. 52) constraints concerning arbitrary flow variables (though from same time step and scenario):

The costs (or profits) of each scenario are calculated for the following items: commodities, energy markets, reserve markets, variable operation costs and start costs. Total cost \({vc}^{total}(s)\) for each scenario is calculated as sum of these items.

The risk element in the model is calculated by using Conditional Value at Risk (CVaR) by using total costs \({vc}^{total}(s)\) from each scenario:

The objective function of the stochastic optimisation problem to be minimised consists of weighted sum of expected total costs and CVaR element as follows:

In conclusion, the main principle of the Predicer model is to utilise mixed integer linear programming in order to minimise the objective function (64) so that all the model constraints (1)–(63) apply.