Real-world hydropower unit commitment: data and model pre-processing for infeasibilities

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Hydropower and hydroelectricity

Basics of hydropower

Hydropower denotes any form of power related to water. Though they are commonly used for one another – it will be the case in this thesis – energy and power are two different physically concepts: power is the instantaneous consumption or production of energy. Energy follows the conservation principle: it cannot be created nor lost but it is converted from one form to other forms or it is transferred from an object to other objects. The conservation principle relies on the abstraction of potential energy: it reflects the amount of tangible energy that can be released once forces have effect on the objects. Energy exists under several forms; to name a few: thermal, kinetic, electric, electromagnetic, chemical, nuclear.
In this thesis, we will focus on systems that transform kinetic energy of water naturally in motion into electricity: hydroelectricity. Other systems like waterwheels transform kinetic energy of water into another kind of kinetic energy. Steam engines within thermal plants also work with water in motion that has been previously heated; this process does not fall into the category of hydroelectricity.
The overall perpetual motion of water on earth is denoted water cycle. Its sketch is the following: water from clouds precipitates into rain or snow; rainfalls and snowmelt run off in streams or rivers; streams within drainage basins converge to lakes or larger water courses; water then flows into oceans; liquid water from all water bodies (oceans, lakes, etc.) evaporates into clouds. There is also motion of water in oceans: winds create waves, periodic gravitational forces of the moon cause tides, differences in temperature/pressure/salinity of water produces ebbs. Marine hydroelectricity relies on the use of such phenomena but we will not delve into the details of this technology.
In this thesis, we will focus on dam hydroelectricity. Dam – or storage or reservoir – hy-droelectricity relies on the possibility of retaining water in a reservoir thanks to a dam in order to release it to convert the kinetic energy of its discharge. Pump-storage hydroelectricity is a specific enhanced dam hydroelectric structure where the conversion is reversible: on top of the primary capacity of discharging water to produce electricity, we can use electricity to pump wa-ter from downstream to upstream; it is a transformation of electrical energy into gravitational potential energy. In addition to dam and pump-storage, another kind of hydroelectricity is de-noted run-of-the river. It relies on the conversion the kinetic energy of water running in a river into electric energy without control of the flow. Run-of-the-river hydroelectricity is somewhat a degenerate dam structure. A notable advantage of dam hydroelectricity is the capacity to store water and to decide when to use it. Pump-storage structures offer the capacity to indirectly store electricity.
To give an insight of the electrical power that can be generated, let us introduce a basic for-mula for the conventional power characteristic – or I/O function – of a single dam hydroelectric structure by considering the energy balance with elementary linear variations under simplifying assumptions.
Let V [m3] be the volume of an elementary water amount discharge downstream from the upstream reservoir; it is initially at a height h [m] from the turbine; its mass is V [kg], where density of water is [kg:m 3].
The weigh upon it is vertical, directed downwards and its norm is g V .
The work of the weight from the upstream reservoir to turbine (or the loss in gravitational potential energy) is gh V .
Not all the potential energy is transferred into electric energy, the power collected is damp-ened by an efficiency factor to account for those losses, hence the hydroelectric energy produced is gh V .
If it takes T [s] for that volume to go downstream, the average power transfer P is gh VT or P = ghQ
where Q = VT can be seen as the average flow.
The power characteristic P (Q; h) depends on the water flow Q and the relative water level h, which depends on the volume in the reservoir. Actually, the expression of the power charac-teristic is more complex; the interested reader can refer to [5] for a case of the Itaipu´ plant where variations in tailrace elevation, penstock head losses and turbine-generator efficiencies are considered in the power characteristic.
Commonly, several structures are built in a drainage basin to benefit from the several incom-ing streams and intermediate lakes; they are denoted multi-reservoir systems or valleys. The most elementary form is a series of cascaded reservoirs. More generically, multi-reservoir sys-tems have a dendritic form. Within a multi-reservoir system, the several structures are connected by water courses or pipes; for each structure, incoming flows include the water discharge from the upstream structures on top of precipitations and natural streams, outgoing flows feed the reservoirs of downstream structures. A description of the generic equipment of hydroelectric structures is given in the Appendix A.1.

Electricity technology comparison

When we compare energy forms, it is actually conversions that we compare: production of the energies of interest from a given primary energy – e.g., solar thermal energy versus solar electric energy –, consumption of the energies of interest for a given use, – e.g., electric cars versus fuel-based cars –, storage of an energy into some forms of potential energy of interest – e.g., electro-chemical batteries versus power-to-gas.
In order to highlight the specificities of hydroelectricity, let us compare it to a few other tech-nologies that produce electricity: nuclear power plants, conventional thermal power plants (coal, gas or biomass fired), solar (photovoltaic) power plants, wind power plants. This comparison is not universal and must be qualified according to existing devices, practices and infrastruc-tures that are context-dependent. Our comparison follows the enumeration of advantages and disadvantages presented in [77].
Hydroelectricity is a dispatchable, flexible, scarce, reversible energy. Electricity production is said to be dispatchable when it can be controlled (increased or decreased, switched on or off). Water, nuclear fuel, coal, gas, biomass are primary energy sources that can be stored. Control of water release for dam hydroelectricity (as described in Section 1.2) and of water steam for thermal plants (through the heat of nuclear reaction or fuel combustion) allows control of tur-bine spinning and electricity generation. Therefore, the productions of hydroelectric, nuclear, conventional thermal power plants are dispatchable. Conversely, windmills, solar plants and isolated run-of-the-river plants are not deemed controllable since their productions only depend on natural inputs over which there is no control. Such technologies are not totally uncontrol-lable as we can sometimes reduce their production outputs. Being able to dispatch electricity production is essential in order to match a varying demand, as we will see in Section 1.3. Also, operations of hydroelectric plants are quite flexible: they can be launched or shut down very quickly as technical specifications allow relatively fast variations. This property is not fully shared with thermal plants because of technical limitations on their operating domains for ex-ample. This property does not apply to solar and wind power plants as their operations is not even dispatchable. Flexibility can be regarded as a short-term dispatchability. However, water is a scarce resource. While thermal plants can be replenished on demand since fuel procurement is – theoretically – not limited, water levels in reservoirs depend on the inflows coming from seasonal precipitations. Scarcity is somewhat similar to long-run dispatchability. Additionally, pump-storage hydroelectricity allows to somewhat reverse electricity production by indirectly storing it to dispatch it later.
Hydroelectricity is a renewable, relatively clean and relatively cheap energy. Though there may be limited precipitations or droughts locally on the short run, there is no limited stock of flowing water on the long run as water keeps on cycling. Provided water cycles remain, hydroelectricity is renewable. In the future, climate change will however affect the regularity of water cycles – amounts and seasonality of precipitations and evaporations – as well as water needs – in case of floods, droughts or heatwaves. There is no greenhouse gas emission, waste nor residual pertaining to hydroelectricity. For this reason, water is sometimes denoted white coal. Water is a free fuel. Operations and maintenance costs of hydroelectric plants are somewhat low, while construction is much more capital-intensive [1]. The life of hydroelectric plants spans from 50 up to more 100 years, which is comparable if not longer than the lifespan other kinds of power plants.
A lot is at stake when deciding to build a hydroelectric plant and when deciding how to operate it. First of all, the construction of a hydroelectric plant necessitates the flooding of a drainage basin and causes considerable changes of the ecosystem. Indeed hydroelectricity is locally bound to drainage basins and water courses. The location constraints are somewhat less restrictive for other technologies. For example, thermal plants require water course prox-imity for cooling as well as transport accessibility for fuel replenishment and waste disposal. Secondly, building hydroelectric plants may require substantial changes in existing human ac-tivities: population moves, preclusion of waterway navigation. Storing water and controlling its discharge allows uses that are sometimes concurrent with generation of electricity: water sup-ply, flood control, irrigation, recreational activities, industrial uses, etc. The case of the Aswan dam across the Nile in Egypt and the renown ensuing relocation of the Abu Simbel archaeologi-cal sites (see illustrate how intricate the stakes can be. The Three Gorges Dam over the Yangtze river in China is another exam-ple where effects on the environment and population relocation were significant. (see https: // Thirdly, reliability of the structure is tantamount to ensure the safety of people living downstream; dam breaches are unfortunately not uncommon catastrophes, as we have witnessed in Brazil in November 2015 (see https: // Very simplisti-cally, fallouts from hydroelectric structures can be more critical than the ones related to thermal, wind and solar plants.

Electricity production management

Electricity production management aims at planning ahead which production levers to pull in order to satisfy demand in electricity as well as other requirements (as mentioned in 1.2.2: reli-ability, costs, scarce resources, etc.).
There is no unique framework to manage the production of electricity: levers and require-ments vary considerably according to the setting considered. In this section, we will describe some aspects of the electricity production management at EDF that remain relevant for other generating companies.

Specificities of electricity

The electric system is basically composed of two intertwined layers: the physical exchanges of electricity and the related financial trades. At one end, producers – also commonly referred to as generating companies (GenCos) – own or lease physical generating assets – that is to say power plants – to produce electricity, physically provide it on the grid and sell it to end users that consume it at the other end of the grid. For a generating company, the distribution of production assets according to their types is called the energy bundle – or energy mix. For the demand side, the aggregation of all electric consumptions at a given time is called the load. The grid is a network of nodes – sometimes denoted buses – connected by transmission lines. Buses correspond to injection points for generating companies or consumption points for end users. Safe and reliable operation of the electric system is ensured by Transmission System Operators (TSOs) and Distribution Network Operators (DNOs). The roles of and interactions between the major stakeholders involved in the electric system are presented in the Appendix A.2.
The electric system is demand-driven as the satisfaction of demand is essential: electricity is a good/service that is essential for human activity. It is interesting to note that electricity is an inevitable expense, thus its cheapness is also critical for the purchasing power of households and the bottom line of organizations. The satisfaction of the demand in electricity has to be instantaneous: end-users do not order it in advance nor are they willing to wait for a delayed delivery. The satisfaction of demand cannot be partial; for example, most electrical devices have technical specifications that forbid them to operate at half power. The satisfaction of demand is not flexible; a few end-users can curtail their consumption on demand, but, generally-speaking, there are less levers to control demand, compared to production.
The electric system is subject to global requirements. Not only satisfying demand is essential for its usage, it is also necessary for the reliability of the network. Indeed, the electrical network is somewhat fragile: a surge or a shortage of supply compared to demand can cause disturbances and even damages to production/transmission/end-users appliances. Sometimes the last resort is load shedding, that is temporary shutdowns for parts of the network. The equilibrium of supply of producers and demand of consumers is the primary requirement the TSO must pursue. Safety and reliability of the network is subject to more elaborate requirements (congestion and capacity of lines, power losses, frequency adjustment, etc.) that we will not explain further. On the grid, electricity a non-differentiable good, therefore the overall equilibrium is satisfied if each pro-ducer meets the demand of its own clients. Note also that, compared to the number of end-users, there are few power plants, mainly due to economies of scale. Those plants are owned/operated by few prominent players on the production side; their responsibilities in guaranteeing the equi-librium is therefore paramount. In case of mismatch in production and demand on the network, prominent players are bound to provide all the leeway they have on demand of the TSO in order to counter-balance the disequilibrium.
Production must be planned ahead in order to meet system requirements and satisfy a highly variable and uncertain demand. An electricity producer cannot rely on an inventory of past excess production because electricity is hardly storable at a large scale. Additionally, an electricity producer cannot resort to backlogs to match the demand of its consumers since consumption can-not be delayed. The usual stock equation, as derived in Lot-Sizing problems [76] for example, does not hold:
8t; productiont + stockt + backlogt+1 = demandt + stockt+1 + backlogt:
Instead, we have:
8t; productiont = demandt:
• productiont and demandt denote respectively the electricity production and the demand in electricity at time t,
• stockt and backlogt denote respectively the cumulative past surplus production and the cumulative unsatisfied demand at time t.
Another related implicit distinction with the usual stock equation is that the equilibrium must be satisfied in real-time and not over time windows. An electricity producer must coordinate the productions of the several plants that are part of its energy bundle (productiont; 8t). Due to the speed and magnitude of demand variations, production cannot track demand ex post. Indeed large real-time changes in the production of power plants are not most appropriate because of lack of technical flexibility, because they are economically prohibitive, and/or because they are restricted by the available resources. It is more sensible to anticipate by setting production levels ahead to meet forecast demand while making sure minor adjustments can be controlled on-the-go to meet real-time demand.

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Different stages of production management

We will now focus on the production management of electricity. Though not mentioned explic-itly previously, an electricity producer – as a company – aims at using efficiently its resources to make a profit while staying within a given threshold of risk. Those financial requirements are supplemented by specific requirements related to the electric system – supply-demand equilib-rium and stability of the network more generally – and related to complexity of the production process.
The payoff of a generating company is derived as follows: revenues come from the sales to end users and on wholesale markets, costs come from investment, production and buying on wholesale markets.
A generating company must cope with several unknowns or risks: the regulatory context is likely to change over the lifetimes of facilities; markets prices and depths are uncertain; demand is uncertain; production is affected by uncertain factors such as technical failures, maintenance durations, and water precipitations into reservoirs of hydroelectric plants.
Decisions relative to production management can be decomposed according to the usual strategic/tactical/operational planning paradigm [2]. Though the demarcations are fuzzy, this paradigm coincides with the chain of long-term/mid-term/short-term planning horizons.
Long-term planning deals with equipment investment decisions to ensure the energy bundle is adapted to offer enough capacity and maneuvrability to satisfy future demand. The installed production facilities are then considered frozen for the mid-term and the short-term planning horizons.
Mid-term planning deals with the design of management policies such as deciding on main-tenance campaigns for plants, stock policies – that are enforced either with guide-curves or through opportunity cost indicators – and hedging policies. Such policies then bind or at least influence the possible courses of actions for the short-term horizon.
Short-term planning deals with actual scheduling of power plants and is described in further details in Section 1.3.3.
In the very short term, adjusting the previously computed schedules to the realization of uncertainties is denoted rescheduling [62].

Short-term electricity production scheduling

Short-term scheduling deals with planning horizons ranging from one day to a couple of weeks and discretized in time periods themselves ranging from a few minutes to a few hours.
Scheduling is carried out for plants that are dispatchable: hydroelectric and thermal plants. When technically possible, non-dispatchable plants – windmills, solar plants and isolated run-of-the-river plants – can be disconnected from the grid which is most frequently economically irrelevant since their production is free. This kind of production is denoted inevitable and is not subject to scheduling. The contribution to the overall electricity provision can be estimated thanks to sunlight and wind forecasts.
At every time period, two kinds of decisions have be to made in a schedule of controllable plants: its commitment status – that is whether the plant is on or off – and its power level – that is how much power is produced. Scheduling commitment statuses of plants – or units within plants – is denoted unit commitment (UC). Scheduling power levels once commitment statuses are known is denoted economic dispatch (ED). Scheduling the two decisions together is more relevant and can also be denoted UC by extension.
Regarding power, we are only interested in the quantity produced and provided on the grid, and we disregard other power characteristics that are essential for transmission planning such as voltage, frequency and phase; indeed we assume such characteristics satisfy by default system requirements and they can be adjusted by the TSO to ensure system stability.
Three settings can be distinguished for unit commitment: security-constrained [33], price-maker bidding [24, 54] and price-taker bidding [68] scheduling. We will be interested in the former two and a description of the three is given in the Appendix A.3. Security-constrained unit commitment (SCUC) is centralized: all the power plants must be scheduled jointly so that the aggregation of their production levels meets the demand, thus ensuring the security of the system. Given market prices, a price-taker producer self-schedules – that is to say regardless of an external security constraint – its production with a view to maximizing its payoff. The aim of a price-taker producer in the self-scheduling setting is to maximize its payoff. Since there is no requirement binding the different power plants, nor do schedules for one plant affect payoff of other plants, the scheduling can be carried independently for each plant, in a decentralized fash-ion. This kind of scheduling problem is sometimes referred to as price-based unit commitment (PBUC).

Table of contents :

1 Introduction
1.1 Problem statement and contributions
1.1.1 Case study 1: a real-world hydropower unit commitment problem
1.1.2 Case study 2: a heuristic for MINLP
1.2 Hydropower and hydroelectricity
1.2.1 Basics of hydropower
1.2.2 Electricity technology comparison
1.3 Electricity production management
1.3.1 Specificities of electricity
1.3.2 Different stages of production management
1.3.3 Short-term electricity production scheduling
1.3.4 Short-term hydroelectricity production scheduling
1.4 Optimization for electricity production scheduling
1.4.1 State of the art of unit commitment
1.4.2 The unit commitment problem at EDF
1.4.3 Challenges
2 Real-world hydropower unit commitment: data and model pre-processing for infeasibilities
2.1 Model description
2.1.1 Notation
2.1.2 MILP formulation of the complete model
2.2 First computational tests
2.2.1 Test set and configuration
2.2.2 Test results
2.2.3 Problem statement
2.3 LP formulation of a simple model
2.4 Numerical errors
2.5 Scaling to deal with floating-point errors
2.5.1 Tolerances, scaling, and floating-point-based solvers
2.5.2 Computational tests
2.6 Corrective relaxation to deal with data errors
2.6.1 Corrective relaxations
2.6.2 Computational tests
2.7 Extensions of results to the complete model
2.8 Infeasibility classification
2.8.1 Approach
2.8.2 Computational results and interpretation
2.9 Target volume reformulation for feasibility recovery
2.9.1 Minimization of target volume deviations
2.9.2 Optimization of the original problem within deviations
2.9.3 Computational tests
2.10 Conclusion
3 A multiplicative weights update heuristic for mixed-integer non-linear programming – an application to a hydropower unit commitment problem
3.1 The MWU framework
3.1.1 Original MWU framework
3.1.2 The MWU approximation guarantee
3.1.3 The intuition of the MWU for MINLP
3.1.4 The MWU as a metaheuristic for MINLP
3.2 Pointwise reformulations
3.2.1 Concept and definition
3.2.2 Properties
3.3 MWU adaptation for pointwise reformulated MINLP
3.3.1 Sampling parameters
3.3.2 Solution and refinement
3.3.3 Computing the MWU costs
3.4 MWU for an NLP HUC
3.4.1 Model description
3.4.2 Pointwise reformulation
3.4.3 MWU adaptation for the HUC
3.5 Computational results
3.5.1 Test configuration
3.5.2 Comparative results on solution quality and CPU time
3.5.3 Sensitivity to instance size
3.5.4 Comparative results on primal integral
3.5.5 Sensitivity to varying initializations
3.5.6 Importance of the pointwise and refinement steps
3.6 Conclusion
4 Conclusion and perspectives
A Electricity production management
A.1 Hydroelectric structures
A.2 Stakeholders of the electric system
A.3 Unit-commitment settings
B Mathematical optimization
B.1 Brief background
B.2 Basic notions
B.2.1 Variables
B.2.2 Bounds and constraints
B.2.3 Objective function
B.2.4 Optimal solution
B.3 Problem types
B.3.1 Properties
B.3.2 Problems of interest
B.3.3 Complexity
B.4 Algorithms and solutions
B.4.1 Properties
B.4.2 Algorithms related to Mixed-Integer Linear Programming
B.5 Computations and solvers
B.5.1 Use cases
B.5.2 Advantages
B.5.3 Solvers of interests
B.5.4 Challenges
C Data 105
C.1 Parameter values for the NLP HUC
List of Tables


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