1. Introduction
Economic dispatch (ED) methods aim to schedule generating units and allocate the demand power among them to determine the best-generating scenarios. ED can benefit power utilities in various ways by systematically minimizing the cost of energy production consistent with the load demand. For this purpose, ED typically increases the usage of the most efficient generators, which can yield lower fuel costs and reduced carbon emissions. A complex mathematical model is solved through multiple computations to satisfy the demand while achieving the minimum generating costs of fuel-based generation stations. These computations are restricted by various constraints of power systems [
1,
2]. The typical constraints of the ED problem are the capacity of generators, the ramp-rate of generating units, and the power balance.
Conventionally, an approximate quadratic function is employed to mathematically formulate the ED problem in order to reduce the computational complexity. Nevertheless, in practice, the input–output curves of generators have nonlinear characteristic due to the multi-valve steam turbines (so-called valve-point effects). In addition, various faults in the machines prohibit some generators from operating in some zones (i.e., prohibited zones). The existence of the valve-point effects and prohibited zones make the solution space of the ED problem highly nonlinear, and thus increasing the complexity of the optimization process [
3].
The static economic dispatch (SED) provides an economical solution to the power of generation stations at a certain load level. Differently, dynamic economic dispatch (DED) optimizes power generation for multiple load levels over a period of time (e.g., one day) [
4]. Notably, DED is a more realistic procedure compared with SED because of considering the variation in the demand load and the variation of power generation over the studied period.
After formulating the DED problem mathematically, an optimization method is required to concurrently the fuel cost and carbon emission while handling all constraints. Although the traditional power systems have satisfied the demand loads for a long time, they will not be able to meet the current challenges alone, knowing that the fossil fuel is predicted to run out [
5]. The integration of renewable energy sources (RES) into power systems can be attributed to environmental, economic, and social benefits. Driven by these benefits, the future power systems are predicted to be fed by energy sources that are totally renewable. Therefore, modern power systems combine non-conventional resources, such as RES and energy storage systems in order to provide sustainable power systems capable of meeting the significant increase in demand [
6,
7,
8,
9,
10,
11]. For a realistic DED framework and sustainable power systems, there is an essential need to incorporate RES and energy storage systems in the optimization model. The declining trend of the costs of battery energy storage with enhanced efficiency, along with an increasing need to alleviate the intermittent RES generation increased the penetrations of such energy storage systems in the transmission levels. A benefits of the storage systems is to curtail some of the variability challenges related to the RES power, and so smooth the fluctuated generation while providing further charging/discharging control options. Indeed, the DED optimization process can allow the optimal scheduling of the power outputs of generating units and RES according to the load demands over predefined time intervals. Concretely, RES contributes to reducing fuel costs and providing sustainable power and clean energy. Examples of RES are solar energy, wind energy, and tidal energy [
12,
13,
14]. Indeed, solar energy is a necessary form of renewable energy. The earth receives a massive amount of energy from the sun that can be converted into clean electricity directly through photovoltaics (PVs) [
15,
16] or indirectly through concentrating solar power (CSP) [
17]. Although the capital cost of PV farms is high, they are a preferable choice for generating clean electricity [
18]. The integration of PV to power systems guarantees efficient power delivery and reduces the amount of CO
2 emissions, thus protecting the surrounding environment.
In the last years, several heuristic optimization algorithms have been proposed for solving engineering problems in general due to their high performance and simplicity. These heuristic algorithms are mimicking natural phenomena or the social behaviors of creatures. For instance, particle swarm optimization (PSO) [
19], genetic algorithm (GA) [
20], ant–lion optimization (ALO) [
21], ant colony optimization (ACO) [
22], and grey wolf optimization (GWO) [
23] are applied to solve the SED problem. The authors of [
24] evaluated the performance of moth–flame optimization (MFO), moth swarm algorithm (MSA), GWO, ALO, sine cosine algorithm (SCA), and multi-verse optimization (MVO) with applying mutation operators in solving the SED problem.
Regarding DED, many optimization algorithms have been applied to solve the DED problem, such as symbiotic organisms search (SOS) algorithm, which combines GA, PSO, and SOS in a tri-base population [
25]. In [
26], the GA algorithm is implemented to optimize the demand side management and the DED as a complementary stage. An accelerated approach is proposed in [
27] to solve the DED problem with high computational speed. A hybrid PSO algorithm called BBPSO is presented in [
28] for solving the DED problem. In addition, chaotic differential bee colony optimization algorithm (CDBCO) [
29], optimality condition decomposition (OCD) technique [
30], biogeography-based optimization [
31], differential evolution algorithm (DEA) [
32,
33], and hybrid flower pollination algorithm (HFPA) [
34] are implemented to solve the DED problem. Furthermore, hybrid genetic algorithm and bacterial foraging (HGABF) approach [
35], weighted probabilistic neural network and biogeography based optimization (WPNN–BBO) [
36], multidisciplinary collaborative optimization (MCO) [
37], quasi-oppositional group search optimization (QOGSO) [
38], and improved real coded genetic algorithm (IRCGA) [
39] are also applied to solve the DED problem. In [
40], the MILP-IPM approach is applied to solve the DED, and it combines the mixed-integer linear programming (MILP) with the interior point method (IPM).
Indeed, the DED optimization problem is still a challenging issue because of the nonlinear and non-convex characteristics of DED objective functions. The DED optimization process also involves massive calculations to find the optimal values of tens of variables and parameters (high dimensionality) while satisfying the constraints of the power system. The nonlinear, non-convex, and non-differentiable characteristics of the DED, as well as the constraints, may squeeze the solution space. In addition, the intermittent nature of PV increases the difficulty of the DED optimization problem as it adds fluctuations to the power system. These aspects might push the search agents to deviate from the global optima and yield diverse local optima. To cope with these challenges, a robust optimizer is required to provide an accurate solution with low computational complexity.
To cope with the issues above, in this paper, we propose a comprehensive DED framework, deep learning-based forecasting models, and an improved optimizer. Specifically, these major three contributions can be summarized as follows:
Comprehensive DED framework: A comprehensive DED framework is formulated that includes fuel-based generators, PV, and storage devices in a sustainable power system, considering clear and cloudy profiles of PV.
Improved optimizer: We propose an improved salp–swarm optimizer that helps manage the global exploration of the DED algorithm and reach reasonable DED solutions. Specifically, we apply a mutation operator to the salp swarm optimizer to increase the exploitation of the search space for improved solutions. The proposed algorithm is validated with ten benchmark problems and then used to optimize the DED problem for a sustainable power system with PV within the studied period.
Deep learning-based forecasting models: We propose a DED handling strategy that involves the use of PV power and load forecasting models based on deep learning techniques.
The rest of this paper is organized as follows.
Section 2 presents the DED framework.
Section 3 explains the proposed DED solution algorithm and the deep learning-based forecasting models.
Section 4 provides the results.
Section 5 concludes the paper and gives some lines of future work.
2. Comprehensive DED Framework
The main objective of the DED problem is to minimize the costs of the fuel consumed by generator units. This optimization problem is heavily restricted by various practical considerations, e.g., generation constraints of units, ramp-rate constraints of generating units, and power mismatches constraints. Unlike the basic SED problems, DED aims to optimally dispatch the output power of all generators in a number of time instants. Furthermore, we have here considered the PV generation units and energy storage systems, complying with the global policy for establishing sustainable power systems.
Figure 1 presents a general structure of sustainable power systems in which renewable energy sources and energy storage systems are interconnected, besides the fuel-based generation stations, to feed various loads.
To address the DED problem, most approaches published in the literature can be itemized into (1) meta-heuristic based optimization methods and (2) conventional optimization-based methods. Examples of meta-heuristic based optimization methods are [
25,
26,
27,
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40]. Conventional optimization based methods involve: linear programming algorithm [
41], lambda iteration method [
42], and the interior algorithm [
43]. These later methods are usually computationally efficient; however, they are suitable mainly with convex functions [
44].
The optimal dispatch can be achieved through maximizing the total fuel-cost savings (TFS) of the fuel-based generator units, or in other words minimizing the total fuel-cost (TFC). During time intervals
, the corresponding demand power at these occasions must be optimally allocated among the generating units. The TFC is proportional to the number of thermal generators, and TFS can be formulated as follows:
in which
where
is the total fuel cost of all generator units to supply
without PV and energy storage systems, and
is the total fuel cost of all generator units to supply (
) with PV and energy storage systems.
,
,
, and
are, respectively, the total predicted load power, available active power generation of PV based on the predicted environmental conditions (i.e., solar radiation), active power curtailment of PV, and the charging/discharging power of the energy storage systems at time instant
. For generator
,
,
, and
are its cost coefficients, and
, and
are the coefficients of valve-point effects,
n is the number of thermal generators to be scheduled,
t represents the current time,
T is the total time of the studied period, and
is the thermal power generated from the
generator at time
t. The valve-point effect can be considered as a practical operation constraint of thermal generators. This effect introduces a ripple in the heat rate function, yielding a discontinuous nonlinear fuel cost function that has multiple local minima. In Equation (
2), a second order quadratic cost function is formulated with a rectified sinusoidal term for precise modeling of the cost function of generators considering the valve-point effect.
The DED is mathematically formulated as:
The equality and inequality constraints of the power system, PV, and energy storage systems are listed below:
Constraint (4) represents the balance between the active powers of all thermal generators, the PV system, and the energy storage systems with respect to the total load demand and active power losses in the power system at each time instant t. is the output power from the PV unit at time t expressed by Label (5), and are the charging and discharging power of the storage device at time t, respectively, is the total demand power at time t, and represents the transmission losses computed by Label (6) in which represents B-coefficients.
Constraints (7), (8), and (9) represent the upper and lower operational boundaries, the ramp-up limit, and the ramp-down limit of each thermal generator at each time instant t, respectively. and are the minimum and maximum limits of the output power from the thermal generator, respectively. and represent the ramp-up and ramp-down boundaries of the thermal generator, respectively.
Constraints (10) can set the maximum allowed curtailed power of PV according to the regulations of utilities. is a factor where its value ranges from 0 to 1. In this work, this latter factor is set to zero to prevent the active power curtailment. Regarding the energy storage device, it has two operational constraints represented by (11) and (12). and are the minimum and maximum charging rate limits of the storage energy devices, respectively. and are the minimum and maximum state of charge limits of the storage energy devices, respectively. Note that the decision variables in the DED optimization problem are: the output power from each thermal generator , curtailed active PV power , charging rate of the storage energy devices , and state of charge of the storage energy devices for all time instants during the dispatching period, which is the next day in this work.
Besides the consideration of valve point effects, the prohibited operating zones of generators can model their persistent physical operation boundaries (e.g., generator faults, excessive vibrations, turbine constraints). Therefore, the feasible zones for the
ith generator can be mathematically formulated as follows:
where
z is the number of the prohibited operating zones for the
generator.
and
represent the lower and upper (MW) power boundaries of
prohibited operating zone for the
generator, respectively.
Figure 2 illustrates the basic generator costs curve, the costs considering valve point effect, and the impacts of the prohibited operating zones on the two curves.
Figure 2 implies that, if a generator has
z prohibited zones, its operating region will be split into isolated feasible sub-regions, yielding multiple decision spaces for the DED problem. As noticed, by considering both valve-point effects and the prohibited operating zones of generators, the optimization model became more complex and non-convex, and so required robust optimization algorithms to be accurately solved.
The output power of a PV module depends on the ambient temperature and solar irradiation. It can be modeled as follows [
45,
46]:
where
is the cell temperature and
is the ambient temperature at time
t,
refers to the nominal operating cell temperature,
R is the solar irradiation,
is the short circuit current,
is the temperature coefficient of current,
is the temperature of the standard test conditions,
is the open-circuit voltage,
is the temperature coefficient of voltage,
is the fill factor,
is the number of cells in the module, and
and
are the voltage and current at the maximum power point, respectively. In this paper, one hour resolution of the data and the dispatch cycle have been considered. However, the proposed model is general, and can be applied to other resolutions.