MILP formulation of the UC-problem with boundary conditions on the autonomous forecasting horizon

The intervals for determining the variables in the standard MILP-formulation of the UC-problem are analyzed. The need to take into account the state and load values in the pre-forecast and post-forecast time has been established. The suitability of the standard MILP-formulation of the UC-problem, problem with initial conditions, for solving dispatch control problems and the unsuitability of such a formulation for solving the generation capacity planning problems is shown. The standard MILP-formulation of the UC-problem on the cyclical forecasting interval is proposed. The suitability of the proposed MILP-formulation of the UC-problem for its application in the problem of planning the development of the generation capacity of power systems with a large share of RES is shown.


Introduction
The unit commitment (UC-) problem is probably the most important task in power system operation management [1][2][3].The improvement of the quality of such management is provided by the use of mathematical models of the optimal load of units.The vast majority of mathematical formulations of the problem of effective management of power systems based on the criterion of minimum operating costs have the form of a problem of mixed integer linear programming (MILP) [1,4].The constant improvement of mathematical formulations of the UC-problem of the optimal load of units is due to the need to ensure acceptable adequacy of model solutions.In modern formulations of the UC-problem, operational characteristics of various technologies of electricity production and network restrictions on its transmission and distribution are taken into account in quite detail.In addition, each new mathematical formulation of the UC-problem is examined for its compliance with the conditions for the existence of a single optimal solution and the possibility of finding it using existing methods and available algorithms [3,4].
Mathematical formulations of the UC problem are considered for a short-term forecast period lasting from one day to a week or a month, in some cases up to one year, with hourly reproduction of load modes of units, placement of power reserves on them, load of the transmission line network, etc. [1] The different formulations of the UC-problem have one thing in common: they are all developed on the basis of linear time, where there are past, present, and future periods.UCmodels establish a cause-and-effect relationship between the mode states of units in linear time and, as initial data, use available data on the state of units, the duration of their stay in such states, as well as on the volume of load of units and the amount of energy accumulated in Mathematical formulations of the UC-problem in the form of UC-constraints are used in the formulation of problems of planning the long-term development of power systems, especially power systems with a high level of penetration of renewable energy sources or extensive use of non-traditional energy storage technologies.Thus, operational costs for maneuvering, starting and stopping units, i.e. costs for ensuring the flexibility of power systems, are additionally taken into account in the formulation of the tasks of long-term development of power systems [5][6][7][8][9][10].
The planning of the development of power systems is carried out on the basis of optimization models of operating and capital costs for periods of time lasting several decades.In order to achieve the adequacy of the representation of the future load modes of power units in power system models, the costs of their start-up and power maneuvering are taken into account, which is extremely important in the conditions of penetration into the power system of a significant share of variable and unpredictable volumes of electricity production at the generating units of wind and solar power plants, as well as daily, weekly and seasonal irregularities in electricity consumption.
Usually, the simulation of load modes of power units is carried out in hourly, less often, half-hourly or fifteen-minute steps.Such detailing in time of load modes of power units is used for short periods of time lasting a day or a week and cannot be used for the entire multi-year planning period due to the excessive computational complexity of the mathematical problems that arise.Therefore, the model load modes of power units, determined on short-term forecast periods, are extended to monthly or seasonal periods, based on the assumption of periodicity of power unit load modes.
UC-models are used to simulate short-term load modes of power units.Such models were developed to solve the problems of dispatching control of the electric power system.UC-models are predictive models and require input data on the load of power units at the beginning of the forecast period, i.e., those that naturally characterize the current load mode of power units.The application of UC-models in the tasks of planning the development of power systems gives rise to two methodological problems: the first is the uncertainty of data regarding the load of power units at the beginning of each short-term planning period; the second is the non-periodicity of the model load modes of the power units.
These problems are solved below by formulating the power unit load model in the form of a UC-problem with boundary conditions.

Indices and sets
g ∈ G Thermal generators.l ∈ L g Piecewise production cost intervals for generator g: 1,. . .,L g .s ∈ S g Startup categories for generator g, from hottest (1) to coldest (S g ).

t ∈ T
Hourly time steps: 1,. . .,T .The dimension of the set G is determined by the number of power units in the power system.The dimensions of the sets L g and S g are determined by the degree of detailing of the technical and economic characteristics of each power unit with regard to fuel consumption in different modes of its start-up and load.The dimension of the set T is determined by the duration of the simulation period and the time steps of its presentation.Usually choose a daily or weekly duration of such a period with hourly steps of its presentation.

Parameters c l g
Cost coefficient for piecewise segment l for generator g ($/MWh).c u g Cost of generator g running and operating at minimum production P g ($/h).D(t) Load (demand) at time t (MW).R(t) Spinning reserve at time t (MW).P l g Maximum power for piecewise segment l for generator g (MW).

P g
Maximum power output for generator g (MW).Spinning reserves provided by generator g at time t (MW), ≥ 0.

MILP formulation of the UC problem with initial conditions
We will use a MILP UC formulation based on [11].We assume that the production cost is piecewise linear convex in p g (t), where L g is the number of piecewise intervals and P 0 g = P g is the start point of the first interval.Let generators have U T g > 1.We then formulate the UC-problem as follows: p g (t) + r g (t) ≤ (P g − P g )u g (t) Constraints (1b -1r) are standard in UC problem formulations with time-varying startup costs [1,11].We use typical one binary formulation for startup costs using only the status variable u.

MILP UC formulation with initial conditions
Let's analyze the areas of definition of those restrictions, the effect of which does not coincide with the areas of definition of the unknown variables used in them.
In the functional constraints (1d), unknown values w g (T + 1) of the functions w g (t) are used, which on the right go beyond the time domain of the definition of these functions.Therefore, the values of w g (T + 1) should be considered as parameters, and the indicated constraints should be represented as follows In the next constraints (1e-1f), unknown values p g (0) of the functions p g (t) are used, which on the left go beyond the time domain of the definition of these functions.Therefore, the values of p g (0) should be considered as parameters, and the indicated constraints should be represented in the following forms and In constraints (1i), unknown values u g (0) of the functions u g (t) are used, which on the left go beyond the time domain of the definition of these functions.Considering the values of u g (0) as parameters, we will present the indicated constraints in the form The area of definition of the u g (t) functions included in the constraints (1j) is shortened from the left by the value U T g .To extend the effect of the constraints (1j) to the entire area of definition of the u g (t) functions, the unknown values v g (0), v g (−1), . . ., v g (2 − U T g ) of the v g (t) functions should be considered as parameters, and the indicated constraints should be represented as follows Also, the area of definition of the u g (t) functions included in the constraints (1k) is shortened from the left by the value DT g .To extend the effect of the constraints (1k) to the entire area of definition of the u g (t) functions, the unknown values w g (0), w g (−1), . . ., w g (2 − DT g ) of the w g (t) functions should be considered as parameters, and the indicated constraints should be represented as follows Functional constraints (1m) contain values u g (0), u g (−1), . . ., u g (1−T C g ), which should also be considered parameters.
Thus, in the presence of values of  2), (1e-1f) to (3)(4), (1i) to ( 5), (1j) to ( 6) and (1k) to ( 7) can be considered as a problem with initial conditions.In tasks of dispatch management, namely in tasks of short-term forecasting (a day, a week or a month ahead), the initial conditions of loading units of power system are naturally known.At the same time, the values of the w g (T + 1) parameters are assumed to be zero [1].
Solving problems with initial conditions provides short-term forecasting of dynamic load modes of power system units.In general, the load regimes of the units at the beginning and at the end of the forecast period do not coincide.This means that the received forecast solutions cannot be propagated beyond the forecast interval by their periodic reproduction.Therefore, such prediction intervals are non-autonomous (figure 1).
Figure 1.Non-autonomous period T of power units load in the UC-model.Loads of power units in t+0 and t + T do not match.

MILP formulation of the UC problem with boundary conditions
Planning of long-term development of generating capacities of electric power systems is carried out on the forecasting time horizon, which is usually replaced by a representative set of short intervals of daily or weekly duration.Each short interval corresponds to a separate month or season of the selected forecast years.At such short intervals with hourly detail, the UC constraints of the optimization tasks of minimizing investment and operating costs for the entire planning horizon are formed.The UC constraints ensure the adequacy of optimization solutions in modern conditions of significant penetration of electricity production technologies from renewable energy sources and increased requirements for maneuverability and flexibility of electric power systems [2, 6-8, 10, 12].
A common drawback of the known MILP formulations of the UC problem defined on short autonomous intervals is the impossibility of providing their initial conditions.Therefore such conditions are either set arbitrarily, without justification, or are not set [2,7,8,10,12].To avoid the need for initial conditions, the domains of the UC constraints that require such conditions are reduced.In both approaches, we have a methodological problem and its negative impact on the adequacy of solutions to the problems of planning the long-term development of electric power systems.
We propose a solution to this problem by an MILP formulation of the UC problem as an MILP formulation with boundary conditions.For this purpose let us assume that the load schedules of the power system units are periodic, that is, they repeat with a time period T (figure 2).
Taking into account the constraints (2) and equations of the type w g (T + 1) = w g (1) we obtain, respectively p g (t) + r g (t) ≤ (P g − P g )u g (t) − Instead of constraints (3-4) with equations of the type p g (0) = p g (T ) we obtain respectively and Instead of constraints (5) with equations of the type u g (0) = u g (T ) we obtain Instead of constraints (6) with parameters (8c) we obtain Similarly, instead of constraints (7) with parameters (8d) we obtain Finally, instead of constraints (1m) with parameters (8e), we obtain MILP UC formulation (1) with changes (1d) to ( 9), (1e-1f) to (10)(11), (1i) to ( 12), (1j) to ( 13), (1k) to ( 14) and (1m) to (15) can be considered as a problem with boundary conditions.The obtained MILP UC formulation is correct under the conditions U T g ≤ T, (16b) DT g ≤ T, (16c) ∀s ∈ S, ∀g ∈ G Taking into account the values of the parameters T s g , U T g and DT g of generating units of thermal and nuclear power plants, it is advisable to choose weekly forecasting intervals to fulfill condition (16).

Computational experiments with MILP formulation of the UC-problem with boundary conditions
To perform computational experiments, the MILP formulations of the UC problem with initial and boundary conditions were implemented in IBM ILOG CPLEX Optimization Studio.
The weekly graphs of the load of the generating units, obtained using the MILP formulations of the UC problem with initial and boundary conditions, are presented in figure 3. A comparison of these graphs indicates a fundamental difference between them.In the case of the MILP UC formulation with initial conditions, the graph is not autonomous and sensitive to these arbitrarily given conditions.In the case of the MILP UC formulation with boundary conditions, the graph is autonomous and can be reproduced periodically outside the weekly forecasting interval.

Figure 2 .
Figure 2. Autonomous period T of power units load in the UC-model.Loads of power units in t+0 and t + T coincide.

Figure 3 .
Figure 3. Weekly load schedules of NuScale power units obtained using models with (a) initial conditions and (b) boundary conditions.
Minimum up time for generator g (h).RD g Ramp-down rate for generator g (MW/h).RU g Ramp-up rate for generator g (MW/h).SD g Shutdown rate for generator g (MW/h).SU Time offline after which the startup category s is available, (T 1 g = DT g , T g Minimum power output for generator g (MW).g Startup rate for generator g (MW/h).T C g Time down after which generator g goes cold, i.e., enters state S g .Sg g = T C g ).W (t) Aggregate renewable generation available at time t (MW).2.3.Variables p g (t) Power above minimum for generator g at time t (MW), ≥ 0. p W (t) Aggregate renewable generation used at time t (MW), ≥ 0. p l g (t) Power from piecewise interval l for generator g at time t (MW), ≥ 0. r g (t)