Application of Dynamic Non-Linear Programming Technique to Non-Convex Short-Term Hydrothermal Scheduling Problem

Short-term hydro-thermal scheduling aims to obtain optimal generation scheduling of hydro and thermal units for a one-day or a one-week scheduling time horizon. The main goal of the problem is to minimize total operational cost considering a series of equality and inequality constraints. The problem is considered as a non-linear and complex problem involving the valve-point loading effect of conventional thermal units, the water transport delay between connected reservoirs, and transmission loss with a set of equality and inequality constraints such as power balance, water dynamic balance, water discharge, initial and end reservoir storage volume, reservoir volume limits and the operation limits of hydro and thermal plants. A solution methodology to the short-term hydro-thermal scheduling problem with continuous and non-smooth/non-convex cost function is introduced in this research applying dynamic non-linear programming. In this study, the proposed approach is applied to two test systems with different characteristics. The simulation results obtained in this paper are compared with those reported in recent research studies, which show the effectiveness of the presented technique in terms of total operational cost. In addition, the obtained results ensure the capability of the proposed optimization procedure for solving short-term hydro-thermal scheduling problem with transmission losses and valve-point effects.


Introduction
Power systems are faced with a series of challenging issues taking into account the advances and improvements within them.Remarkable research is being carried out in various areas such as the application and analysis of micro-grids (MGs) and distributed generations (DGs) in the optimal operation of power systems [1], transient stability analysis in power systems [2], dynamic operation and control of the systems [3], connection decisions of distribution transformers [4], and fault current analysis of power systems [5,6].The authors implemented a direct search method (DSM) in [1] for solving economic dispatch (ED) of a medium-voltage MG considering several kinds of DGs.A modified artificial bee colony (MABC) optimization technique is applied in [7] for obtaining the optimal solution of the ED problem, where a novel mutation strategy based on the differential evolution (DE) method is used for improving the capability of the method in providing the optimal solution.The valve-point loading effect of conventional thermal plants is considered in this study.The authors proposed a three-stage technique in [8] to solve the ED problem of distribution-substation-level MGs, where the main power grid and MGs are studied as two key parts of the system.In this reference, the ED of the main grid and local MGs are solved using sensitive factors and an improved direct search method in stages I and II, respectively, and the optimal reschedules from the original dispatch solutions are provided in stage III.The authors have addressed the ED problem considering voltage magnitudes and reactive power flows in [9], where linear programming method is utilized for solving the problem.In this study, thermal capacities of transmission lines and line power transmission, and exponential loads are studied using piecewise linear models.Power system expansion planning is studied in [10], where costs associated with the fuel and buying emission allowances, and benefits from selling emission allowances are considered.A piecewise linear objective function is proposed for calculating the sensitivity of operation cost with respect to limitations of emission.
Short-term hydro-thermal scheduling (STHTS) is defined as one of the most important and challenging issues in power systems operation.Thermal power plants operational costs are high; however, the initial costs of such generation units are lower.On the other hand, the operational costs of hydro power plants are insignificant; however, the construction costs of such plants are high [11,12].Accordingly, the combination of these two types of power plants can be considered as an appropriate choice considering economic viewpoints.The main goal of short-term scheduling of hydro-thermal system is determining the optimal power generation of the hydro and thermal plants.The optimal solution provides the minimum total operational cost of the thermal units, while satisfying load demand and a series of equality and inequality constraints of the hydraulic and thermal power system network.The STHTS problem is proposed as a complex non-linear, non-convex and non-smooth optimization problem considering the water transport delay between connected reservoirs, the valve-point loading effect related to the thermal units, transmission loss and many equality and inequality constraints [13,14].
Different optimization methods are employed to obtain optimal solution of generation planning of hydrothermal systems, including heuristic and classical methods.A modified dynamic neighborhood learning based particle swarm optimization (MDNLPSO) method is introduced in [15] to solve the STHTS problem.In this reference, the proposed approach is applied on two test systems with different characteristics.STHTS problem is solved in [16] by employing quadratic approximation based on differential evolution with valuable trade-off (QADEVT) that minimizes fuel cost and pollutant emission simultaneously.The predator prey optimization (PPO) procedure is used in [17] to obtain the optimal power production planning of hydro and thermal units.In [18], a hybrid method differential evolution with adaptive Cauchy mutation is utilized to obtain the optimal generation scheduling of hydro and thermal units, in which water transport delay between connected reservoirs and the effect of valve-point loading of thermal power plants is taken into account.Particle swarm optimization (PSO) is introduced in [19] to deal with STHTS problem with non-convex and non-smooth cost function.The real coded genetic algorithm (RCGA) is used for the solution of STHTS problem with a series of equality and inequality restrictions and non-smooth/non-convex cost function.The suggested algorithm in this reference is armed with a restriction-management approach which eliminates the requirement of penalty parameters.In [20], by using the Lagrangian Relaxation (LR) method, not only are the electrical and hydraulic constrains handled, but also the existing network constraints are considered by employing DC power flow.The lexicographic optimization and hybrid augmented-weighted ε-constraint method are applied in [21] to produce Pareto optimal solutions for STHTS problem.In this reference, mixed integer programming (MIP) is introduced to obtain the optimal power generation planning of hydrothermal system in a day-ahead joint energy and reserve market.In [22], an improved merit order (IMO) and augmented Lagrangian Hopfield network (ALHN) is proposed to solve short-term hydrothermal scheduling with pumped-storage hydro units.The proposed method in this reference considers thermal, hydro and pumped-storage unit commitment (UC).The STHTH problem is solved in [23] with the consideration of AC network constraints, which is implemented a combination of the Benders decomposition method and Bacterial Foraging oriented by Particle Swarm Optimization (BFPSO) method.The application of chaotic maps in a particular game problem called the Parrondo Paradox is studied in [24].The proposed approach was used in a three-game problem and a more general N-game problem in which non-linear optimization problem is considered to define the parameters for the studied game.
In this study, the STHTS problem is solved using dynamic non-linear programming (DNLP) using general algebraic modeling system (GAMS) software.The valve-point effect of conventional thermal plants, which increases the complexity of solving STHTS problem, is considered in the solution of the problem.In addition, the power transmission loss of the hydro-thermal system is studied in the proposed study.Different case studies are solved to evaluate the performance and ensure the effectiveness of the introduced method.The optimal solutions are compared with those reported in previous studies in terms of total operational cost, which demonstrates the capability of the proposed method to identify solutions having less operational cost.In addition, optimal solutions obtained in this paper ensures the capability of the proposed method to deal with valve-point loading effect of thermal units and system power transmission loss.
The rest of the paper is organized as follows: The mathematical formulation of the STHTS problem is provided in Section 2. Section 3 introduces the proposed solution method for STHTS problem.In Section 4, the proposed approach is implemented on two test systems and the obtained optimal solutions are compared with those reported in previous studies.Finally, the paper is concluded in Section 5.

Problem Formulation
The optimal scheduling of hydro-thermal plant includes a non-linear optimization problem involving objective function and a set of linear, non-linear and dynamic constraints.The objective function and equality and inequality constraints of the STHTS problem are explained in the following [25].

Objective Function
The main goal of short-term planning of hydro-thermal system is determining the optimal power generation of the hydro and thermal plants so as to minimize the total operation cost of the thermal units since the cost of hydro production is insignificant.It should be mentioned that various constraints on the hydraulic and thermal power system network should be considered in the solution of the problem.The objective function to be minimized can be represented as follows [26]: where C(P) is the total fuel cost.N S is indicator used for the number of thermal plants.Moreover, P t i is power generated by the ith thermal plant at time t. a i , b i , and c i are the cost coefficients of ith thermal plant.Considering multiple steams valves in conventional thermal power plants, it is essential to model the effect of valve-points on fuel cost.Valve-points effect can be modeled by a sinusoidal term, which will be added to the quadratic cost function [27].P min i is minimum power generation of thermal unit i.Moreover, e i and f i are valve-point coefficients of cost function of thermal unit i.

Power Balance Constraint
The total power generated by hydro and thermal plants should be equal to the sum of total load demand and transmission line losses.
where N h is the number of hydro units.P t j is the generation of hydro units in megawatts (MW).Moreover, P t D and P t L are load demand and total transmission loss in MW, respectively.P t L can be calculated using the Kron's loss formula known as B-matrix coefficients [28].Equation (3) calculates power transmission loss utilizing Kron's loss formula, which is defined as B-matrix coefficients method in this paper as follows: The coefficients are Kron's loss formulation used to calculate power transmission of the hydrothermal system.The power loss of the system taking into account N s hydro plants and N h thermal units can be calculated by using such formulation.B-matrix coefficients for calculating the power loss are shown by B ij , B io , and B 00 .In such formulation, B mn is element of matrix B with dimension of (N S + N h ) × (N S + N h ).In addition, B 0n is vector of the same length as P, and B 00 is considered as a constant.
The hydro power generation, P t j , is a function of water discharge and storage volume, which can be calculated as follows: where V t j is the storage volume of reservoir in m 3 , and C 1,j , C 2,j , C 3,j , C 4,j , C 5,j , and C 6,j represent hydro power generation coefficients.Moreover, Q t j is the water discharge amount in m 3 .

Limitations of Power Production
The generator capacity constraints are expressed as: where P min i and P max i are the respective lower and upper bounds of power generation of thermal units.In addition, the minimum and maximum amounts of power production of hydro units are indicated by P min j and P max j , respectively.

Water Dynamic Balance
The reservoir storage of hydro unit is related to previous inflow and spillage, and storage of reservoir discharge from upstream reservoirs, which can be formulated as: where I t j is the inflow rate of the reservoir, φ j is set of instant upstream hydro plants of the j th reservoir.Additionally, τ is time delay of immediate downstream plants.

Reservoir Storage Volume Limits
The operating volume of reservoir should be limited in interval between minimum and maximum values, which can be stated as: where V min j and V max j are the respective lower and upper bounds of operating volume of the reservoir of ith hydro unit.

Water Release Limits
The water release of hydro units should be limited to minimum and maximum values, which can be considered as: where Q min j and Q max j are the minimum and maximum release of the water reservoir of the ith hydro plant.

Initial and Final Reservoir Storage Volume
Initial and final volumes of reservoir storage should be taken into account in the formulation of STHTS problem as: where is the elementary volume of the reservoir and V end j is the final volume of the reservoir.

Solution Methodology
GAMS is defined as a practical tool to handle general optimization problems, which consists of a proprietary language compiler and a variety of integrated high-performance solvers.GAMS is specifically designed for large and complex problems, which allows creating and maintaining models for a variety of applications.GAMS is able to formulate models in many different types of problem classes, such as linear programming (LP), nonlinear programming (NLP), mixed-integer linear programming (MILP), mixed-integer nonlinear programming (MINLP) and dynamic nonlinear programming (DNLP).Nonlinear models created in GAMS area should be solved by using an NLP algorithm.This paper offers a novel approach based on the NLP method to obtain optimal planning of hydrothermal systems.Accordingly, the STHTS is modeled as a NLP in this study, and is solved by implementing OptQuest/NLP (OQNLP) solver.The STHTS problem is formulated as a nonlinear problem, which can be solved by GAMS software [29] using OQNLP solver [30].OQNLP is a multi-start heuristic technique, which calls an NLP solver from different starting points.All feasible solutions obtained by such solvers are kept, and the best solution is reported as the final optimal solution.Such a method is capable of finding global optimal solutions of smooth constrained NLPs.A scatter search implementation called OptQuest is employed by OQNLP to compute starting points [31].OQNLP is able to obtain global optimal solutions of smooth NLPs and MINLPs.A simplified pseudo-code is provided in Figure 1 for introducing the application of OQNLP to find the optimal solution of the optimization problems, which is divided into two levels.The first level generates candidate starting points and selects the best starting point among all of the points.Then, in the second level, new points are generated and evaluated in order to obtain the best solution in terms of generation cost.

Case Studies and Simulation Results
In this paper, the performance of the proposed solution is evaluated in several test systems.A Pentium IV PC with 2.8 GHz CPU and 4 GB RAM PC is used to solve the problem in GAMS.The scheduling horizon is chosen as 24 h of a day.

Test System 1
First test system consists of four hydro plants and an equivalent thermal plant.The hydraulic communication among hydro units of this system is demonstrated in Figure 2. Transmission losses are not considered in this test system.Cost coefficients of thermal plants are ai = 0.002, bi = 19.2, and ci = 5000.The lower and upper operation limits of this thermal plant are 500 and 2500 MW, respectively.Data of thermal unit and hydro plants are adopted from [25].Two different cases including convex and non-convex cost function are studied for this test system.In this case, optimal generation scheduling of test system 1 is solved without consideration of valve-point loading impact.The hourly water discharge of the hydro plants and hydro power production, which is calculated by employing Equation (7), are shown in Table 1.In addition, thermal power production for case 1 is provided in Table 1.According to Table 1, the sum of power generation by four hydro units and one thermal plant meets total demand of the system.Hourly hydro discharges of the optimal solution are demonstrated in Figure 3. Considering Figure 3, hydro plant 4 has the maximum discharge among four hydro units, which shows that the power generation of hydro plant 4 is more than the others.In addition, hourly hydro and thermal plant generations are illustrated in Figure 4.The thermal units participates in power demand supply more than the hydro plants according to Figure 4.Moreover, total load demand is satisfied by the power generation of four hydro units and the thermal plants, which is obvious in Figure 4.

Case Studies and Simulation Results
In this paper, the performance of the proposed solution is evaluated in several test systems.A Pentium IV PC with 2.8 GHz CPU and 4 GB RAM PC is used to solve the problem in GAMS.The scheduling horizon is chosen as 24 h of a day.

Test System 1
First test system consists of four hydro plants and an equivalent thermal plant.The hydraulic communication among hydro units of this system is demonstrated in Figure 2. Transmission losses are not considered in this test system.Cost coefficients of thermal plants are a i = 0.002, b i = 19.2, and c i = 5000.The lower and upper operation limits of this thermal plant are 500 and 2500 MW, respectively.Data of thermal unit and hydro plants are adopted from [25].Two different cases including convex and non-convex cost function are studied for this test system.In this case, optimal generation scheduling of test system 1 is solved without consideration of valve-point loading impact.The hourly water discharge of the hydro plants and hydro power production, which is calculated by employing Equation (7), are shown in Table 1.In addition, thermal power production for case 1 is provided in Table 1.According to Table 1, the sum of power generation by four hydro units and one thermal plant meets total demand of the system.Hourly hydro discharges of the optimal solution are demonstrated in Figure 3. Considering Figure 3, hydro plant 4 has the maximum discharge among four hydro units, which shows that the power generation of hydro plant 4 is more than the others.In addition, hourly hydro and thermal plant generations are illustrated in Figure 4.The thermal units participates in power demand supply more than the hydro plants according to Figure 4.Moreover, total load demand is satisfied by the power generation of four hydro units and the thermal plants, which is obvious in Figure 4.The obtained results are compared with those obtained by employing quantum-inspired evolutionary algorithm (QEA) [25], quantum-inspired evolutionary algorithm (WDA) [32], small population-based particle swarm optimization (SPSO) [33], real coded genetic algorithm (RCGA) [34], real-coded quantum-inspired evolutionary algorithm (RQEA) [25], DE [25], modified differential evolution (MDE) [33], differential real-coded quantum-inspired evolutionary algorithm (DRQEA) [25], hybrid chemical reaction optimization (HCRO)-DE [35], modified adaptive particle swarm optimization (MAPSO) [36], real-coded genetic algorithm and artificial fish swarm algorithm (RCGA-AFSA) [34], teaching learning-based optimization (TLBO) [37], smallpopulation-based particle swarm optimization (SPPSO) [33], self-organizing hierarchical particle swarm optimization technique with time-varying acceleration coefficients (SOHPSO_TVAC) [38], PSO [39], improved differential evolution (IDE) [40], fuzzy adaptive particle swarm optimization (FAPSO) [39], dynamic neighborhood learning based particle swarm optimization (DNLPSO) [15], and modified dynamic neighborhood learning based particle swarm optimization (MDNLPSO) [15], and is shown in Table 2.As it can be observed from this table, the best reported cost for this case is equal to $914,660, which is related to FAPSO [39], while total operational cost of the solution obtained by the proposed The obtained results are compared with those obtained by employing quantum-inspired evolutionary algorithm (QEA) [25], quantum-inspired evolutionary algorithm (WDA) [32], small population-based particle swarm optimization (SPSO) [33], real coded genetic algorithm (RCGA) [34], real-coded quantum-inspired evolutionary algorithm (RQEA) [25], DE [25], modified differential evolution (MDE) [33], differential real-coded quantum-inspired evolutionary algorithm (DRQEA) [25], hybrid chemical reaction optimization (HCRO)-DE [35], modified adaptive particle swarm optimization (MAPSO) [36], real-coded genetic algorithm and artificial fish swarm algorithm (RCGA-AFSA) [34], teaching learning-based optimization (TLBO) [37], smallpopulation-based particle swarm optimization (SPPSO) [33], self-organizing hierarchical particle swarm optimization technique with time-varying acceleration coefficients (SOHPSO_TVAC) [38], PSO [39], improved differential evolution (IDE) [40], fuzzy adaptive particle swarm optimization (FAPSO) [39], dynamic neighborhood learning based particle swarm optimization (DNLPSO) [15], and modified dynamic neighborhood learning based particle swarm optimization (MDNLPSO) [15], and is shown in Table 2.As it can be observed from this table, the best reported cost for this case is equal to $914,660, which is related to FAPSO [39], while total operational cost of the solution obtained by the proposed method is $884,733.965.Accordingly, the proposed method is capable to find better solution in comparison with previous methods in terms of total operational cost.

Test System 1 Case 2: Quadratic Cost Function with Valve-Point Loading
In this case, optimal power scheduling of test system 1 is obtained with consideration of valve-point loading effect.The parameters of valve-point loading impact of thermal unit are e i = 700 and f i = 0.085.The simulations are provided for case 2 with non-convex fuel cost.The optimal planning of discharge of four hydro units are reported in Table 3.In addition, power generation of hydro units, which is obtained by applying Equation (7), are provided in this table.In addition, power production of thermal power plants are presented in Table 3.It can be observed from Table 3 that the power demand during 24-h scheduling time is satisfied by total power generation of four hydro units and one thermal unit.

Test System 2
This test system consists of four cascaded hydro power plants and three thermal plants.Valve-point loading effect of thermal plants and transmission losses are considered in this test system.Data of hydro and thermal generation units are adopted from [42].Coefficients of transmission loss for this system are given as the following: This test system consists of four cascades hydro plants and three thermal plants considering valve-point loading effect for all thermal units.In this case, transmission loss is not considered.The optimal hydro discharges and hydro power generation of four hydro units are provided in Table 5.Moreover, power generations of three thermal plants are reported in this table.According to Table 5, the sum of power generation of four hydro units and three thermal plants meets the load demand during the scheduling time of the STHS problem.Proposed method provided the minimum fuel cost of $41,101.738,which is compared with simulated annealing (SA) [25], DE [11], chaotic artificial bee colony (CABC) [26], adaptive differential evolution (ADE) [23], RCGA [13], DE [10], SPPSO [12], RQEA [10], PSO [27], chaotic differential evolution (CDE) [23], clonal selection algorithm (CSA) [28], TLBO [29], TLBO [18], improved quantum-behaved particle swarm optimization (IQPSO) [30], quasi-oppositional teaching learning based optimization (QTLBO) [29], Improved differential evolution (IDE) [21], adaptive chaotic differential evolution (ACDE) [23], real coded chemical reaction based optimization (RCCRO) [22], differential real-coded quantum-inspired evolutionary algorithm (DRQEA) [10], and adaptive chaotic artificial bee colony algorithm (ACABC) [26], quasi-oppositional group search optimization (QOGSO), as shown in Table 6.Results show that proposed method is better than previous methods used in the test system 2, case 1.As it can be seen, the minimum obtained cost is $41,274.42which is related to ACABC [43] compare to $41,101.738obtained by proposed method.The valve-point effects and transmission losses are considered in this case, which make the problem more complex.The optimal result obtained by OQNLP is reported in Table 7.The hourly discharge of four hydro plants and the power generation of the hydro units are prepared in this table.In addition, power generation of three thermal plants are reported in Table 7.The power transmission loss of the hydrothermal system by applying Equation (13) during 24-h scheduling time interval is also reported in this table.In this case, considering Table 7, total generation of four hydro units and three thermal plants meets total load demand and power transmission loss of the system.

Conclusions
In this study, dynamic non-linear programming is introduced to obtain optimal scheduling of a hydrothermal system.The valve-point loading impact of conventional thermal units and system power transmission loss are considered in finding the optimal solution of the short-term hydro-thermal scheduling problem by studying two test systems.Optimal solutions are reported and analyzed, and are compared with those provided in recent papers.Results showed the capability of the proposed method to obtain better solutions in terms of total operational cost in comparison with other heuristic algorithms.Test system 1 includes four cascaded hydro units and one equivalent thermal plant, in which daily savings are $29,926.035and $13,468.026 in comparison with previously reported solutions for both cases 1 and 2, respectively.In addition, for test system 2, which contains four cascaded hydro units and three thermal plants, daily savings are $172.682and $242.9226 in comparison with reported solutions in previous studies for both cases 1 and 2, respectively.The optimal solutions show that the proposed method is an effective and high-performance technique to solve short-term hydro-thermal scheduling problem considering transmission losses and valve-point loading effects.The future research trends in the area of short-term hydro-thermal scheduling can be concentrated on consideration of limitations of AC network constraints.In addition, the unit commitment problem of hydrothermal systems, considering the start-up cost, minimum uptime, and minimum downtime of the generation units can be considered as another research topic in this area.Moreover, the unavailability of the generation units and consideration of renewable energy sources such as wind power are other exciting subjects to be investigated.Also, middle and long-term scheduling of hydro-thermal system, considering the installation and maintenance cost of hydro and thermal plants, may be introduced as interesting subject in the area of hydro-thermal systems.
Author Contributions: All authors have contributed equally work.

Conflicts of Interest:
The authors declare no conflict of interest.

18 Figure 2 .
Figure 2. Hydro subsystem used in the all test systems.

Figure 2 .
Figure 2. Hydro subsystem used in the all test systems.

Figure 3 .
Figure3.Hourly hydro discharges volumes of the optimal solution for test system 1, case1.

Figure 4 .
Figure 4. Hourly hydro and thermal plant generations for test system 1, case 1.

Figure 4 .
Figure 4. Hourly hydro and thermal plant generations for test system 1, case 1.

Table 1 .
Hourly plant discharges, power outputs and total thermal generation (test system 1, case 1).

Table 4 .
Comparisons of simulation results for test system1, case 2.

Table 5 .
Optimal discharges and power output for test system 2 case 1.

Table 6 .
Comparison of obtained optimal costs for test system 2 case 1.4.2.2.Test System 2 Case 2: Quadratic Cost Function with Valve-Point Loading

Table 7 .
Hourly plant discharges, power outputs and total thermal generation for test system 2 case 2.

j
The number of hydro units Constants a i , b i and c i Cost coefficients of ith thermal plant e i and f i Valve-point coefficients of cost function of thermal unit i P min Upper bounds of operating volume of reservoir of ith hydro unit Q min Minimum release of water reservoir of the ith hydro plant Q max

j
Maximum release of water reservoir of the ith hydro plantLoad demand at time t C 1,j , C 2,j , C 3,j , C 4,j , C 5,j , and C 6,j Hydro power generation coefficients