PROGRAMA DE PÓS-GRADUAÇÃO EM ENGENHARIA ELÉTRICA
Marcelo Zapella
Clustered Unit Commitment Applied to the Brazilian Hydrothermal Power System
FLORIANÓPOLIS 2019
Marcelo Zapella
Clustered Unit Commitment Applied to the Brazilian Hydrothermal Power System
Dissertação submetido(a) ao Programa de Pós Graduação em Engenharia Elétrica da
Universidade Federal de Santa Catarina para a obtenção do Grau de mestre em Engenharia Elétrica
Orientador: Prof. Dr. Erlon C. Finardi
Florianópolis 2019
Clustered Unit Commitment Applied to the Brazilian Hydrothermal Power System
O presente trabalho em nível de mestrado foi avaliado e aprovado por banca examinadora composta pelos seguintes membros:
Prof. Murilo Reolon Scuzziato, Dr. Instituto Federal de Santa Catarina
Prof. Fabrício Takigawa, Dr. Instituto Federal de Santa Catarina
Certificamos que esta é a versão original e final do trabalho de conclusão que foi julgado adequado para obtenção do título de mestre em Engenharia Elétrica.
____________________________ Prof. Richard Demo Souza, Dr.
Coordenador em Exercício
____________________________ Prof. Erlon Cristian Finardi, Dr.
Orientador Florianópolis, 22 de Novembro de 2019. Erlon Cristian Finardi:02036 474918 Assinado de forma digital por Erlon Cristian Finardi:02036474918 Dados: 2019.12.18 19:14:05 -03'00' Richard Demo
Souza:00426737989
Assinado de forma digital por Richard Demo
Souza:00426737989
This thesis is dedicated to those who seek for an optimum and efficient power system operation.
provide me guidance, constant support, encouragement and patience throughout the course of this project. I am highly obliged to him for his valuable suggestions and articles referenced, which helped me to present the scientific results in this thesis.
I am highly thankful to Msc. Gilseu Von Muhlen for his enthusiastic support in providing the thermal data input from DESSEM, which was a key part from all computational experiments with direct impact on the interpretation of the results.
I am also grateful to Dr. Murilo Reolon Scuzziato and Msc. Brunno Henrique Brito who kindly guided me through the unit commitment formulation from thermal and hydro power plants, respectively.
I would like to extend my sincere thanks to my work colleagues and friends, who have always motivated and supported me during the course of the master degree, knowing the challenges of working and studying for quite a few years until get to the end of this project.
Last but not least, I am extremely grateful to the support from my family and my roommate, who has always encouraged me and believed in me. A big thank you to my mother Denise, my girlfriend Mayra, my sister Michele, my brother-in-law Leandro and my roommate Lucas.
The goal is to hit the sweet spot of maximum value optimization, where foolish risk is balanced against excessive caution. (Steven J. Bowen, 2017)
área de planejamento da operação de sistemas de energia elétrica. Neste problema, o objetivo consiste em minimizar o custo operacional de geração baseando-se no estado de operação (ligado/desligado) e o nível de potência despachado de cada unidade geradora do sistema ao longo de um horizonte de tempo. Os novos requisitos do DESSEM (Modelo de Despacho Hidrotérmico de Curto Prazo) definem que a alocação da geração deve entregar um cronograma de despacho termelétrico semanal, com discretização semi-horária no primeiro dia. O modelo clássico de alocação conta com variáveis binárias indicando o estado ligado/desligado de cada máquina leva um tempo significativo para ser executado em sistemas com muitas usinas geradoras. Como o DESSEM será executado diariamente, o modelo clássico pode não ser a melhor opção, em virtude da dimensionalidade envolvida no caso Brasileiro. Assim, para amenizar o esforço computacional, esta dissertação sugere o agrupamento de unidades similares em um mesmo grupo, que é representado por uma variável inteira. Visto que a metodologia de agrupamento traz alguns erros inerentes que podem levar a uma solução inviável, esta dissertação também avalia uma solução mista em que ambas a metodologia de agrupamento e a clássica são utilizadas, uma seguida pela outra, usando o resultado do método de agrupamento como entrada no método clássico, fornecendo assim uma solução viável para o despacho da geração.
Palavras-chave: Alocação da Geração. DESSEM. Agrupamento. Planejamento
horário.
A introdução de um planejamento de curto prazo com discretização horária para a operação do sistema de energia elétrica de grande porte, como o DESSEM vem introduzindo no Brasil, apresenta alguns desafios. Aumentar a discretização resulta em um aumento do custo computacional, e visto que o planejamento operacional deve ser atualizado todos os dias para que seja colocado em prática, um modelo mais eficiente precisa ser considerado para executar a otimização em tempo suficiente para os engenheiros operacionais executarem a operação. Assim, esta dissertação propõe um método para agrupar unidades geradores, a fim de reduzir o número de combinações possíveis entre todas as unidades de potência disponíveis e, consequentemente, reduzir o esforço computacional. Além disso, uma nova estratégia mista também é proposta neste trabalho. Embora o foco desta dissertação esteja na Alocação de Geração Agrupadas aplicado ao Sistema Hidrotérmico Brasileiro, a metodologia e experimentos realizados têm um foco muito mais profundo nos sistemas térmicos, uma vez que este possui um maior número de restrições operacionais em comparação aos sistemas Hidrelétricos.
Objetivos
Como objetivo principal, novas modelagens de alocação de geração estão sendo propostas para fornecer o planejamento de operação do sistema de energia elétrica com uma discretização de uma hora, sempre com foco na minimização do custo da operação e no comprometimento de uma alocação de unidades geradoras viável para um horizonte de uma semana. Embora esperasse que o modelo DESSEM seja executado diariamente, um planejamento operacional com um horizonte de uma semana pode fornecer tendências para da alocação de unidades geradores ao longo da semana, o que é útil para coordenar a esperada operação de curto prazo e o cronograma de manutenção das máquinas, por exemplo. Para atingir esse objetivo, a alocação de unidades geradores agrupadas reúne máquinas semelhantes em um grupo, reduzindo grandemente o número de despachos possíveis quando comparado ao modelo clássico de alocação de unidades geradores, pois elimina decisões de despacho idênticas ou semelhantes. Além disso, o modelo com agrupamento de unidades reduz o número de variáveis e restrições presentes na formulação. Entretanto, como desvantagem, o modelo de agrupamento traz alguns erros inerentes que podem levar a uma solução de alocação de geração inviável. A troca entre precisão dos resultados e tempo computacional é avaliada nas experiências.
Metodologia
Considerando as diferentes características dos geradores no setor elétrico, o problema da Alocação da Geração consiste em minimizar os custos operacionais com base na demanda de energia em um horizonte temporal. A definição de estado mais tradicional para a Alocação da Geração é o modelo ligado / desligado, que geralmente é representado por uma variável binária de 2 estados. No modelo com agrupamentos, em vez de considerar o estado binário ligado / desligado de cada unidade geradora, sugere-se um número inteiro que conta o número de unidades que operam no cluster e, como consequência, o número de possíveis de despacho das unidades é amplamente reduzido. Por exemplo, considerando 5 unidades geradoras no modelo binário tradicional resultaria em 25 = 32 possibilidades de despacha-las, enquanto no
modelo de agrupamento haveria 6 despachos possíveis, de 0 a 5 unidades geradoras em operação. Uma outra abordagem avaliada nesse trabalho é a estratégia de solução mista, que consiste em usar os dois modelos, de agrupamento e em seguida o tradicional para definir o planejamento de operação do sistema de energia. Inicialmente, o modelo de agrupamento é usado para fornecer uma aproximação precisa de uma programação de despacho das unidades com uma operação de custo mínimo. Em seguida, um dos resultados do modelo de agrupamento é usado como ponto de partida no modelo tradicional, visando acelerar esse último e fornecer um planejamento de operação totalmente viável, o qual não é possível com o modelo de agrupamento apenas. As informações de transferência do modelo de agrupamento para o modelo tradicional podem ser diversas, como quantidade de unidades despachados em cada tempo ou custo operacional, resultando em diferentes abordagens possíveis ao usar a estratégia de solução mista. Os experimentos executados no trabalho consideram um sistema de barra única. A base de dados das unidades geradoras térmicas foram as mesmas utilizados pelo modelo DESSEM, e a carga utilizada foi com base em 5 semanas da demanda real do setor elétrico brasileiro.
Resultados e Discussão
Os resultados do modelo tradicional foram utilizados como referência e comparados aos resultados do modelo de agrupamento e estratégia de solução mista. O modelo de agrupamento mostrou um resultado muito satisfatório, com menos de 0,6% de erro de custo operacional e mais de 70% de redução do tempo médio de execução, considerando todos os grupos propostos no trabalho. Ao restringir a análise a um determinado grupo, o maior erro de custo das 5 semanas analisadas foi de 0,04% e a redução do tempo de execução foi de 87% em relação ao modelo tradicional. Entretanto, a alocação das unidades geradores resultante do modelo de agrupamento pode não ser viável, apesar de ser uma ótima aproximação de um caso viável com custo mínimo. Avaliando a estratégia de solução mista, se demonstrou alta precisão no planejamento de operação com custo operacional e alocação de unidades geradores muito semelhantes ou até com mesma precisão comparado ao modelo tradicional, entretanto o tempo de execução da estratégia mista muitas vezes superou o modelo tradicional, invalidando a proposta.
Considerações Finais
Apesar dos desafios ainda envolvidos, o modelo de agrupamento se demonstra ser bastante promissor visando reduzir o esforço computacional para buscar um planejamento de operação com discretização horária e custo mínimo. Uma abordagem recomendada para estudos futuros é usar o modelo tradicional para um dia (ou dois) inicial da com discretização horária, garantindo uma alocação da geração viável, enquanto os dias restantes da semana (ou mês) podem tirar proveito do modelo de agrupamento para ser executado mais rapidamente, ainda em uma discretização horária. Já na estratégia de solução mista, outras estratégias podem ser utilizadas, buscando uma formulação mista ou outros tipos de filtros que ajudem a uma convergência mais rápida quando executando o modelo tradicional, e assim garantindo uma alocação de unidades geradoras viável.
Palavras-chave: Alocação da Geração. DESSEM. Agrupamento. Planejamento
planning. It consists in minimizing the operational costs for power generation based on the operational status (on/off) and the power released from generating units in a given time horizon. The new requirements from DESSEM (Short Term Hydrothermal Dispatch Model) in Brazil demands the hydrothermal power unit commitment shall deliver a week-ahead schedule with hourly discretization. The classic individual unit commitment (IUC) with binary variables indicating the on/off operational state of each machine takes a significant amount of time to be executed in complex systems, with several generating units. As the DESSEM model will be executed in a daily basis, the classic IUC model may not be an option given the complexity from Brazilian power system. Thus, to save processing time, this thesis suggests the clustered unit commitment (CUC) to gather similar units into a cluster, using an integer to represent the units. Given the CUC methodology brings a few inherent errors that may lead to a non-feasible solution, this thesis also evaluates a mixed-solution in which both CUC and IUC models are used, one followed by the other, using the CUC result as an input in the IUC model, resulting in a feasible solution for the unit commitment.
Keywords: Unit commitment. DESSEM. Clusters. Hourly schedule.
Figure 2 – IUC vs CUC: lack of individual generation levels ... 41
Figure 3 – IUC vs CUC: lack of individual operating time records ... 42
Figure 4 – Mixed-solution workflow ... 47
Figure 5 – Single bus bar from the experiment ... 53
Figure 6 – IUC/CUC algorithm flow chart ... 53
Figure 7 – Mixed-solutions algorithm flow chart ... 54
Figure 8 – Maximum and minimum power per unit ... 55
Figure 9 – Maximum power output vs variable cost per unit ... 55
Figure 10 – Start-up cost vs variable cost per unit ... 56
Figure 11 - Relationship between RUD vs Delta P (Pmax-Pmin) ... 57
Figure 12 – Hourly loads from five weeks used in the computational experiments ... 59
Figure 13 – CUC cost differentiation compared to IUC ... 61
Figure 14 - Cost difference IUC x MIX Method 1,01*Cost- ... 65
Figure 15 - Cost Difference IUC x MIX Method ±1,1*unit_state ... 65
Figure 16 - Cost Difference IUC x MIX Method Cost+ ... 65
Figure 17 – CUC vs IUC: Number of operational units – Week 1 ... 69
Figure 18 – MIX (1.01*Cost-) vs IUC: Number of operational units – Week 1 ... 69
Figure 19 – MIX (±1.1*un_st) vs IUC: Number of operational units – Week 1 ... 70
Figure 20 – CUC vs IUC: Number of operational units – Week 5 ... 70
Figure 21 – MIX (1.01*Cost-) vs IUC: Number of operational units – Week 5 ... 71
Table 2 – Parameters of a cluster with the two identical from Table 1. ... 40
Table 3 – Demand profile for Example 1 ... 41
Table 4 – Demand profile for Example 2 ... 42
Table 5 – Parameters from two different units ... 44
Table 6 – Parameters from two different units clustered ... 44
Table 7 – Conditions used for RUD estimation... 56
Table 8 – IUC cost and execution time results ... 60
Table 9 – CUC cost results... 61
Table 10 – CUC cost differentiation compared to IUC ... 61
Table 11 – CUC execution time results ... 62
Table 12 – CUC execution time differentiation compared to IUC ... 62
Table 13 – Mixed-solution cost results ... 64
Table 14 – Mixed-solution cost differentiation compared to IUC ... 64
Table 15 – MIX execution time results ... 66
Table 16 – MIX execution time differentiation compared to IUC... 66
Table 17 – Data input from individual thermal power units ... 83
Table 18 – Data input from G1 clustered thermal power units... 93
Table 19 – Data input from G2 clustered thermal power units... 98
IUC – Individual Unit Commitment CUC – Clustered Unit Commitment HUC – Hydro Unit Commitment
CVU – Unit Variable Cost (in Portuguese) RUD – Ramp-up/down power constraint SUC – Start-up cost
MIP – Mixed-Integer Programming
MILP – Mixed-Integer Linear Programming
MINLP – Mixed-Integer Non-Linear Programming DP – Dynamic Programming
LR – Lagrangian Relaxation
1.2 LITERATURE ... 29
2 THERMAL UNIT COMMITMENT PROBLEM………..33
2.1 INTRODUCTION ... 33 2.2 FORMULATION FOR THE IUC MODEL ... 34 2.3 FORMULATION FOR THE CUC MODEL ... 37 2.4 DEVIATIONS BETWEEN IUC AND CUC ... 39
2.4.1 Deviations when clustering identical units………..39 2.4.2 Deviations when clustering different units...44 2.4.3 Cost error between IUC model and CUC model in large networks...45
2.5 CHALLENGES IN THE CUC MODEL ... 45
3 MIXED-SOLUTIONS STRATEGY………..47
3.1 INTRODUCTION ... 47 3.2 DIFFERENT APPROACHES ... 48
3.2.1 Number of operational units (flexible) ………..48 3.2.2 Operational cost plus………...………...49 3.2.3 Operational cost minus………..………50
3.3 OTHER STRATEGIES ... 50
4 COMPUTATIONAL EXPERIMENTS………...53
4.1 TEST SYSTEM DESCRIPTION ... 53
4.1.1 Thermal power plants input………..………54 4.1.2 Thermal clusters………..………57 4.1.3 Load input………..………...58
4.2 COMPUTATIONAL RESULTS ... 59
4.2.1 IUC results………..………...59 4.2.2 CUC results………..………...60
4.2.3 Mixed-solution results………..……….63
4.3 RESULT ANALYSIS ... 68
5 CONCLUSIONS AND SUGGESTIONS FOR FUTURE STUDIES...73 REFERENCES………..………...77 APPENDIX A – Hydro Units model: CUC challenges………....79 APPENDIX B – Thermal Power Units………...83
1 INTRODUCTION
The electric power industry has lots of optimization problems in different applications given the complexity and the many variables involved in the generation, transmission, and distribution of electricity. For many years, engineers and researches have worked on a number of different methods and approaches for tackling the electric power industry’s problems, resulting in constant improvement in the algorithms and mathematical models used to handle these problems.
The unit commitment (UC) is a well-known optimization problem in the power operation planning area. It consists of minimizing the operational costs for power generation based on the operational status (on/off) and the power released from generating units in a given planning horizon. The UC problem is related to all types of controllable generating plants, such as thermoelectric and hydroelectric. The diversity
of plant types, the technology design of each of them and the number of physical, environmental and operational constraints involved in the power system lead to a variety of mathematical models for the UC problem.
The ongoing evolution of the electricity market also impacts on the UC problem. According to the Brazilian regulatory framework, for instance, an Independent System Operator (ISO) is responsible for defining the scheduling of the entire system. Currently, the Brazilian power system does not have a mathematical model to define UC of power generation, furthermore the shortest power schedule defined by ISO, called DECOMP, uses an optimization model with two months ahead in which the first month has a weekly discretization. However, with the introduction of a new optimization model called DESSEM into the Brazilian operational system, the power schedule will be updated every day with an hour or half-hour discretization, including the UC of power generation. DESSEM is planned to be officially released by January 2020, and it focuses on the hydrothermal modeling.
Introducing a short-term power schedule with an hourly discretization in large power electric schemes, such as DESSEM being introduced in Brazil, has some challenges. Increasing the discretization means the computational cost increases as well, but since the schedule is updated every day, a more efficient model has to be considered in order to provide enough time for operational engineers to process the output coming from the method. Thus, this thesis proposes a method to cluster units
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in order to reduce the number of possible combinations among all the power units available and consequently reducing the computational effort.
In addition to the computational issue, to minimize the operational cost is also a challenge to be considered in short-term power schedule. Since the electric power industry is a high-value market, even a slight percentage decrease in the operational costs may result in great savings. For this reason, the UC problems had several improvements over the years, as result from technological advances such as computer processing and optimization algorithms.
In summary, this thesis conducts studies to deliver a much faster algorithm in order to save computational time, but still provide a result that minimizes the operational cost with a feasible unit commitment in short-term power schedule, with an hour discretization. Although the focus of this thesis is on the Clustered Unit Commitment applied to the Brazilian Hydrothermal Power System, the next sections have much deeper focus on the Thermal systems, since it has higher number of operational constraints compared to Hydro systems. Nevertheless, the applicability of clustered unit commitment in Hydro systems is quick evaluated in APPENDIX A – Hydro Units model: CUC challenges.
1.1 OBJECTIVES
As the main objective, new UC modelings are being proposed to deliver the power planning with an hour discretization, always focusing on operation cost minimization and feasible unit commitment for a week-ahead power schedule. Although the DESSEM model will be executed daily, a week-ahead schedule can provide a trending to the UC along the week, which is helpful to coordinate with the short-term problem and for the machine maintenance schedule, for instance.
To achieve this objective, the clustered unit commitment (CUC) gathers similar units into a cluster, strongly reducing the number of possible dispatches when compared to the classical individual unit commitment (IUC), as it eliminates identical or similar commitment decisions, as proposed by [1]. In addition, the CUC model reduces the number of variables and constraints present in the formulation. However, as downside the CUC model brings a few inherent errors that may lead to a non-feasible solution, losing the accuracy on UC and operation cost, as shown in the next
chapter. The trade-off between accuracy and computational time is evaluated in the experiments.
Furthermore, given the inherent errors in the CUC model, this thesis also evaluates a mixed-solution in which both CUC and IUC models are used, one followed by the other, using the CUC results as an input for the IUC. The main idea behind mixed-solution is to eliminate the possibility of having a non-feasible power schedule resulted from CUC. Thus, the CUC model determinates an approximated result minimizing the operational cost, and later the IUC model refines the solution to ensure a feasible power schedule.
Thermal power generators are being considered in this study, given they have several constraints related to the machine design with a high impact on the operation costs. In addition, only a single bus with generators and load is being considered in this study. The time horizon considered is one week with hourly discretization.
As result, this thesis will contribute to future studies on UC model for power schedule with hour or half-hour discretization, looking for a fast model convergence with a feasible result in hydrothermal power sytems, similar to DESSEM model in Brazil. In addition, this study could be expanded to consider other types of generation, such as renewable energy, or to consider other aspects from power systems such as transmission lines.
1.2 LITERATURE
Recently, several studies are being conducted to evaluate the UC performance for hourly schedules, where the main challenge is the high computational cost to get a cost minimization with feasible UC in an hourly discretization. The studies are focusing on reducing the high number of possible operational schedules present in the UC, and as consequence, the time execution to run an optimization algorithm should decrease. Among the approaches being studied in the literature, the CUC is a model getting popular as it replaces the binary operational status (on/off) from each generating unit by an integer value for a group of units clustered together. Studies such as [1] illustrate the CUC model provides a gain in execution time of 60-70 factor when compared to the traditional UC. As disadvantage, clustering units bring some inherent errors that must be considered, even when clustering only identical units. Thus, clustering
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identical units are not exactly the same as considering them individually, although it may be intuitive to think the opposite, and this will be shown in the following section.
To overcome the errors inherent from CUC model, [1] suggests the use of IUC model followed by CUC — an approach defined as “Mixed-Solutions” in this thesis. In this mixed-solution approach, [1] suggests the total number of operational units resulted from the CUC model has to be equal to the number of operations in the IUC model, which is not always true given the inherent errors in the CUC. Based on this paper, this thesis evaluates the applicability of CUC considering the DESSEM database for thermal unit input, and also different approaches for mixed-solutions models considering possible differences between IUC and CUC results.
Given the combinatorial aspects and relationship between constraints in the thermal unit formulation, mathematical programming methods are required to find the optimal power production schedule of a set of thermal units while meeting all constraints. Many mathematical programming methods have been proposed by researches, such as Lagrangian relaxation (LR), dynamic programming (DP) and mixed-integer programming (MIP) to mention a few. The following topics present more details of a few mathematical programming methods used in the literature.
• Dynamic Programming (DP): solve complex problems by breaking it into steps (stages). The searching process could be either in a forward direction or backward direction, meaning that the search for a solution can start at the very beginning of the system or at the very end of the system. In [2], the DP model developed optimizes the number of generating units in operation at each hour. The methodology proposed by the authors highlights the tradeoff between start-up/shut-down of generating units and hydropower efficiency including the head effect, considering three situations: optimizing startup/shutdown costs, optimizing generation loss and the third combining both objectives.
• Mixed-Integer Nonlinear Programming (MINLP): to solve the HUC problem using a mixed 0-1 nonlinear programming, [3] uses a strategy that includes a two-phase approach based on dual decomposition. Firstly, a Lagrangian Relaxation (LR) is applied to obtain a primal point, and subsequently its result is used as a starting point in an inexact Augmented Lagrangian (AL). The method was implemented in Labview and the hydropower modeling had cascaded and head-dependent reservoirs.
Variables considered are the head variations, generator efficiency, hydraulic losses and at the end, the authors model the generators start-up and shut-down using as criterion water penalties, different from [2] which modeled it as cost.
• Mixed Integer Linear Programming (MILP): suggesting a new methodology for optimizing the HUC for the Three Gorges Project, [4] uses the MILP to model each generator unit individually with a large number of unit of various types, as well as the nonlinear time-varying head effect in the reservoir behind the dam. The two nonlinearities with the power release, the net head, and unit performance curve, are represented accurately by means of binary variables and piecewise linear approximations. For instance, to simulate the nonlinear power generation function of a unit, the authors use a three-dimensional interpolation method with nine data points to define the safe operating zone. In summary, the optimizer solver for this study case had to be a MILP considering the model proposed takes individual generating units with heterogeneous points of operation zone each (binary variables). In addition, recent studies such [1] and [5] recommend MILP as solver to achieve an accurate result in large scale problems from which the solving time is critical.
To summarize the comparison between the mathematical programming methods listed above, the DP was already largely used by the literature to solve UC problems, but in large networks this method is enormously challenging due to dimensionality. Though some complex coupling constraints can be relaxed by LR methods, the non-convexity of UC problem makes a calculation of a true dual function very hard. In addition, from the practical point of view, [6] mentions LR has a disadvantage that requires laborious computational implementation, requiring qualified staff with knowledge of several fields of optimization. The availability of highly efficient software environment encourages to solve large-scale real problems with MILP approaches, as it ensures an acceptable trade-off between the solution accuracy and the execution time.In addition, all the literature reviewed from [1] to [7] uses MILP or another method of Mixed-Integer Programming (MIP) to represent the problem.
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As other references to reduce the number of combinatory possibilities in the IUC, in [8] the author recommends methodologies to reduce the symmetry of IUC model, by having hierarchical decisions to set a unit as operating or not in order to relieve the execution time. However, as the author does not consider every constraint of thermal/hydro units, the proposed hierarchical decisions cannot be used depending on the formulation. In addition, the proposed method would always favor the same unit to be under operation, which may reduce the lifetime from one specific unit while the others are still conserved. The author in [5] recommends using mixed modeling of CUC and IUC. While using the integer model to define how many units are operational in the cluster, it defines a binary array for each cluster in order to monitor the individual operational status (on/off) from the generating units. Other studies have been conducted for hydro UC problems as well, such as in [4], [2], [3] and [7].
2 THERMAL UNIT COMMITMENT PROBLEM
2.1 INTRODUCTION
Considering the different characteristics of the generators in a power system, the UC problem consists in minimizing the operational costs based on the power demand over a time horizon. The most traditional state definition for the UC is the on/off model, which is usually represented by a 2-state binary variable. Although the literature has defined a few expansions such as the 3-state or 4-state shown in [9], they only add more constraints to the problem such as minimum / maximum operational level or start-up / shut down condition.
In CUC, instead of considering the binary on/off state of each generating unit, the methodology suggests an integer which counts the number of units operating in the cluster, and as consequence, the number of possible UC schedules is widely reduced. For instance, considering 5 generating units in the IUC model results in 25 = 32
possibilities of scheduling them, while in the CUC model there would be 6 possible schedules, from 0 up to 5 generating units operating.
To illustrate the possible differences of power schedules resulting among IUC and CUC models, Figure 1 shows 5 identical generating units during 8 time-steps where the only requirement is to fulfill the required number of units on. By considering the UC individually, any possible combination of those 5 units available may be taken under operation to fulfill the requirement of operating units, while the cluster only increment/decrement the total number of operational units based on the requirement. This is the main advantage of CUC model, as it greatly reduces the number of possible schedules compared to the IUC model.
Note the greater the number of identical generators in the problem, the more effective is the CUC model compared to the IUC model, as the scheduling possibilities exponentially increase in the IUC model while in the CUC model the increase is linear.
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Figure 1 – Example of an IUC compared to CUC
Given its effectiveness, the CUC model is attractive for UC with a discretization every hour, half-hour or even 15 minutes. However, the CUC model brings some inherent errors even when clustering identical units only, mainly because when clustering generators the model loses the individual operational on/off condition, and the status of each generator within the cluster is unknown. Such inherent errors may lead to a different number of units on when comparting IUC and CUC, and this will be explored in the next sections. At this point, it can be concluded the CUC greatly reduces the of possibilities compared to the IUC, but such advantage has to be used wisely.
2.2 FORMULATION FOR THE IUC MODEL
This section shows the individual unit commitment (IUC) formulation used in the computational experiments. As the main goal of this thesis is to compare the different UC approaches for dealing with large installed base generating units discretized hourly, the formulation does not consider transmission lines, renewable technologies, reservoirs or energy storages, but only a single bus problem in which all the generating units and the load are connected to.
Based on thermal generating units, the cost modeling will consider both the start-up cost and the operational cost, while the shut-down cost is not considered. Although
the thermal operational cost is a non-linear function, it can be accurately represented by a piecewise-affine function or approximately represented by one single linear function. In this formulation, the operational cost is only represented by a single linear segment for the whole range of the power plant. The objective function includes both the generating units’ startup costs and the operational generating costs. The thermal constraints were based in [10].
𝑚𝑖𝑛 ∑(𝑆𝑈𝐶𝑖. 𝑢𝑖,𝑡𝑠𝑢+ 𝐶𝑉𝑈𝑖 . 𝑃𝑖,𝑡)
𝑖,𝑡
(1)
where:
𝑢𝑖,𝑡𝑠𝑢 is the binary start-up decision of unit i and time t; 𝑆𝑈𝐶𝑖 is the start-up cost ($) of unit i;
𝐶𝑉𝑈𝑖 is the power generating cost ($/MW) of unit i.
𝑃𝑖,𝑡 is the operating power (MW) of unit i and time t;
The objective function is then subjected to the following constraints, the equation (2) represents the demand-supply balance from the system.
∑ 𝑃𝑖,𝑡
𝑖
= 𝐷𝑡 ∀𝑡 (2)
where:
𝐷𝑡 is the power demand (MW) in time t.
During the computational experiments, a fictitious generating unit with a high operating cost is added to represent the load curtailment. The following equations represent the operational machine constraints. Starting with equation (3), it represents the logic states (on/off, start-up, shut-down) of each generating unit:
𝑢𝑖,𝑡𝑜𝑛− 𝑢𝑖,𝑡−1𝑜𝑛 = 𝑢𝑖,𝑡𝑠𝑢 − 𝑢𝑖,𝑡𝑠𝑑 ∀𝑖, 𝑡 (3)
where:
𝑢𝑖,𝑡𝑜𝑛 is the binary operational decision of unit i and time t; 𝑢𝑖,𝑡𝑠𝑑 is the binary shut-down decision of unit i and time t.
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The constraints (4)-(7) represent the power output of each unit, including the minimum/maximum power output constraint, as well as the condition that the generating unit must be at its minimum power output in order to start up or to shut down. The constraint depends on the minimum uptime (tup) of each unit.
𝑃𝑖,𝑡 = 𝑃𝑡𝑖,𝑡+ 𝑢𝑖,𝑡𝑜𝑛. 𝑃𝑖𝑚𝑖𝑛 ∀𝑖, 𝑡 (4)
where:
𝑃𝑡𝑖,𝑡 is the power output (MW) above 𝑃𝑖𝑚𝑖𝑛 value of unit i over time t
𝑃𝑖𝑚𝑖𝑛 is the minimum power output (MW) of unit i
For power plants with tup = 0:
𝑃𝑡𝑖,𝑡 ≤ (𝑃𝑖𝑚𝑎𝑥 − 𝑃𝑖𝑚𝑖𝑛) . (𝑢𝑖,𝑡𝑜𝑛− 𝑢𝑖,𝑡𝑠𝑢) ∀𝑖, 𝑡 (5) 𝑃𝑡𝑖,𝑡 ≤ (𝑃𝑖𝑚𝑎𝑥 − 𝑃𝑖𝑚𝑖𝑛) . (𝑢𝑖,𝑡𝑜𝑛− 𝑢𝑖,𝑡+1𝑠𝑑 ) ∀𝑖, 𝑡 (6)
For power plants with tup ≥ 1:
𝑃𝑡𝑖,𝑡 ≤ (𝑃𝑖𝑚𝑎𝑥 − 𝑃𝑖𝑚𝑖𝑛) . (𝑢𝑖,𝑡𝑜𝑛− 𝑢𝑖,𝑡𝑠𝑢 − 𝑢𝑖,𝑡+1𝑠𝑑 ) ∀𝑖, 𝑡 (7)
where:
𝑃𝑖𝑚𝑎𝑥 is the maximum power output (MW) of unit i
The constraints (8) and (9) represent the ramp-up and ramp-down limits of power generating units.
𝑃𝑡𝑖,𝑡− 𝑃𝑡𝑖,𝑡−1 ≤ 𝑅𝑈𝑖 ∀𝑖, 𝑡 (8)
−𝑃𝑡𝑖,𝑡+ 𝑃𝑡𝑖,𝑡−1 ≤ 𝑅𝐷𝑖 ∀𝑖, 𝑡 (9)
where:
𝑅𝑈𝑖 is the maximum ramp-up (MW) of each unit i 𝑅𝐷𝑖 is the maximum ramp-down (MW) of each unit i
The last two constraints are the minimum up-time and minimum down-time, as represented by (10) and (11) respectively. Note the time start counting after the unit performs a start-up or shut-down.
∑ 𝑢𝑖,𝑡𝑠𝑢 𝑡 max (1,𝑡−𝑡𝑖𝑢𝑝) ≤ 𝑢𝑖,𝑡𝑜𝑛 ∀𝑖, 𝑡 (10) ∑ 𝑢𝑖,𝑡𝑠𝑑 𝑡 max (1,𝑡−𝑡𝑖𝑑𝑜𝑤𝑛) ≤ 1 − 𝑢𝑖,𝑡𝑜𝑛 ∀𝑖, 𝑡 (11) where:
𝑡𝑖𝑢𝑝 is the minimum uptime of unit i after start-up 𝑡𝑖𝑑𝑜𝑤𝑛 is the minimum downtime of unit i after shut-down
2.3 FORMULATION FOR THE CUC MODEL
Mathematically, the CUC formulation is similar to the IUC model, requiring only a few modifications. The main difference between that two models is that the CUC model uses integer variables to represent the total amount of units under on/off condition, start-up condition, and shut-down condition, while the IUC model uses binary variables to represent each generating unit’s condition. The value of these integer variables may vary from zero and to the maximum number of units present within the cluster, as illustrated in (12). In addition, instead of using i to index each unit, the formulation now uses the c to represent each cluster.
𝑛𝑐,𝑡𝑜𝑛 , 𝑛𝑐,𝑡𝑠𝑢 , 𝑛𝑐,𝑡𝑠𝑑 ∈ {0, 1, … , 𝑁𝑐} ∀𝑐, 𝑡 (12) where:
𝑛𝑐,𝑡𝑜𝑛 is the integer operational condition of cluster c and time t; 𝑛𝑐,𝑡𝑠𝑢 is the integer start-up condition of cluster c and time t;
𝑛𝑐,𝑡𝑠𝑑 is the integer shut-down condition of cluster c and time t.
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By only switching the binary variable for the respective integer variable in most cases, the following formulation represents the CUC model. As an objective function, (13) looks to minimize the operational cost in the CUC model.
𝑚𝑖𝑛 ∑(𝑆𝑈𝐶𝑐. 𝑛𝑐,𝑡𝑠𝑢+ 𝐶𝑉𝑈𝑐 . 𝑃𝑐,𝑡)
𝑐,𝑡
(13)
In the CUC model, equation (13) is subject to the following constraints from (14) to (23):
∑ 𝑃𝑐,𝑡
𝑐
= 𝐷𝑡 ∀𝑡 (14)
The equation (15) represented the integer conditions (on/off, starting up, shutting down) of each cluster:
𝑛𝑐,𝑡𝑜𝑛− 𝑛𝑐,𝑡−1𝑜𝑛 = 𝑛𝑐,𝑡𝑠𝑢 − 𝑛𝑐,𝑡𝑠𝑑 ∀𝑐, 𝑡 (15)
The constraints (16)-(19) represent the power output of each cluster, including the minimum/maximum power output, as well as the condition that generating units should start at its minimum power output in order to start up or before shut down. The constraints depend on the up time (tup) of each cluster.
𝑃𝑐,𝑡 = 𝑃𝑡𝑐,𝑡+ 𝑛𝑐,𝑡𝑜𝑛. 𝑃
𝑐𝑚𝑖𝑛 ∀𝑐, 𝑡 (16)
For power plants with tup = 0:
𝑃𝑡𝑐,𝑡 ≤ (𝑃𝑐𝑚𝑎𝑥 − 𝑃
𝑐𝑚𝑖𝑛) . (𝑛𝑐,𝑡𝑜𝑛− 𝑛𝑐,𝑡𝑠𝑢) ∀𝑐, 𝑡 (17)
𝑃𝑡𝑐,𝑡 ≤ (𝑃𝑐𝑚𝑎𝑥 − 𝑃
𝑐𝑚𝑖𝑛) . (𝑛𝑐,𝑡𝑜𝑛− 𝑛𝑐,𝑡+1𝑠𝑑 ) ∀𝑐, 𝑡 (18)
For power plants with tup ≥ 1:
The constraints (20) and (21) represent the ramp-up and ramp-down limits of each cluster, depending on the number of operational units.
𝑃𝑡𝑐,𝑡− 𝑃𝑡𝑐,𝑡−1 ≤ 𝑅𝑈𝑐 . (𝑛𝑐,𝑡𝑜𝑛− 𝑛𝑐,𝑡𝑠𝑢) ∀𝑐, 𝑡 (20)
−𝑃𝑡𝑐,𝑡+ 𝑃𝑡𝑐,𝑡−1 ≤ 𝑅𝐷𝑐 . (𝑛𝑐,𝑡𝑜𝑛− 𝑛𝑐,𝑡𝑠𝑑) ∀𝑐, 𝑡 (21)
The last two constraints of the CUC model are the minimum up-time and minimum downtime of units in the cluster, as represented in (22) and (23). The constraints do not keep track of each individual unit in the cluster, but it defines how many units are limited by the minimum up/down-time.
∑ 𝑛𝑐,𝑡𝑠𝑢 𝑡 max (1,𝑡−𝑡𝑐𝑢𝑝) ≤ 𝑛𝑐,𝑡𝑜𝑛 ∀𝑐, 𝑡 (22) ∑ 𝑛𝑐,𝑡𝑠𝑑 𝑡 max (1,𝑡−𝑡𝑐𝑑𝑜𝑤𝑛) ≤ 𝑁𝑐 − 𝑛𝑐,𝑡𝑜𝑛 ∀𝑐, 𝑡 (23)
2.4 DEVIATIONS BETWEEN IUC AND CUC
The difference between the results from the IUC model and the CUC model can be justified by two main reasons. The first is straightforward, gathering different units in a cluster will lead to errors in the result of the problem. The second, as described by [1], even if only identical plants are gathered into clusters there may be errors inherent to the CUC formulation, as the CUC model does not keep records of operational conditions of individual units, as shown in the following sections.
2.4.1 Deviations when clustering identical units
According to [1], the error inherent to the CUC formulation arises from limiting start-up and shut-down ranges. As this study considers start-up and shut-down constraints, the error from the CUC formulation will affect the computational experiments shown in this thesis. In order to illustrate the deviations caused by the
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CUC formulation, the results from two simplified examples will be compared to the IUC model as a reference. As parameters for the examples, consider two identical units with the characteristics as shown in Table 1. When gathering the two identical units into a cluster, it results in the characteristics as shown in Table 2. Note the only difference is the number of units gathered in the group as the units clustered are identical. In both models the start-up cost considered for each unit is $1,000.00. The tables contain the maximum and minimum power output (Pmax and Pmin), the ramp-up/down power constraint (RUD), the minimum ramp-up/down-time (t_up / t_dw) and the unit variable cost (CVU).
Table 1 – Parameters of two identical units
Unit Pmax (MW) Pmin (MW) RUD (MW) t_up (step) t_dw (step) CVU ($/MW) 1 and 2 400 150 50 3 1 100.00
Table 2 – Parameters of a cluster with the two identical from Table 1. Name Number of units Pmax (MW) Pmin (MW) RUD (MW) t_up (step) t_dw (step) CVU ($/MW) Cluster 2 400 150 50 3 1 100.00
In addition to the generating units, a fictitious generating unit is added to the IUC model and CUC model to cover the load curtailment. The fictitious unit is flexible and expensive.
2.4.1.1 Example 1: CUC model lacks individual generation level records
The demand profile considered for this example has four-time steps as shown in Table 3, and the results from both IUC and CUC models are presented in Figure 2. The results show that the IUC model could not find a feasible solution, resulting in load curtailment in step 2, while the CUC solution is feasible without any load curtailment.
Table 3 – Demand profile for Example 1 Time Step Load (MW)
1 650
2 650
3 550
4 400
Figure 2 – IUC vs CUC: lack of individual generation levels
In fact, this example cannot have a feasible solution without load curtailment, thus the IUC solution is correct. Note that both generators must be operating in steps 1, 2 and 3 to fulfill the demand, and more importantly, from steps 3 to 4 one of them must be powered off. Based on the shut-down constraint, the generators could only be powered off in 150 MW, and for this reason, the IUC model suggests that Gen 2 generation decreases 50 MW/step achieving 150MW in step 3. The CUC model overestimates the shut-down capability as it does not keep records of the operational level of each generator. Thus, the CUC model suggests a 100 MW ramping down from step 2 to 3, and without any records from the operational level of each unit, it shuts down one of them from step 3 to 4. The operational cost from the CUC model was lower in this example, as the CUC formulation relaxes the IUC constraints. This result is coherent, as in this case the IUC optimization had to use the fictitious generator. If this were real case, a third generator would have to be used as a backup to achieve a feasible solution, which would still cost more than the optimum cost resulted in the CUC model. 0 100 200 300 400 500 600 700 1 2 3 4 Po w er ( M W ) IUC Result
Generator 1 Generator 2 Load Curtailment
0 100 200 300 400 500 600 700 1 2 3 4 Po w er ( M W ) CUC Result
Cluster Gen. Load Curtailment n=2
n=1 n=2
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2.4.1.2 Example 2: CUC model lacks individual operating time records
The second example illustrates how the CUC model may lead to unfeasible solutions by not keeping the individual operating time records, even when only identical units are clustered. The demand profile considered in this example has six-time steps as shown in Table 4, and the results from both IUC and CUC models are presented in Figure 3.
Table 4 – Demand profile for Example 2 Time Step Load (MW)
1 400 2 400 3 550 4 550 5 400 6 400
Figure 3 – IUC vs CUC: lack of individual operating time records
Just as example 1, example 2 also cannot have a feasible solution without load curtailment. Thus, the IUC result illustrates the reality, while the CUC result is not feasible given the problem constraints. Targeting the cost optimization, the IUC model
0 100 200 300 400 500 600 1 2 3 4 5 6 Po w er ( M W ) IUC Result
Generator 1 Generator 2 Load Curtailment
0 100 200 300 400 500 600 1 2 3 4 5 6 Po w er ( M W ) CUC Result
Cluster Gen. Load Curtailment
n=1 n=1
n=2 n=2
keeps both units on during all periods, and from step 2 to 3, the generators ramp-up 100 MW in total. In step 4, the generation is kept the same as step 3, as from step 4 to 5 the total generation will have to ramp down 100 MW. As a result, the system would have a load curtailment of 50 MW in steps 3 and 4.
A completely different schedule is proposed by the CUC model. In steps 1, 2, 5 and 6 there is only 1 generator operating at 400 MW, its maximum output power. The second generator starts up in step 3, contributing to 150 MW in steps 3 and 4. The issue in the CUC result occurs from step 4 to 5 when there is a decrease of 150 MW and unit commitment shuts down one generator. However, as one of the generators is operating at 400 MW it could not be turned off, and the other generator was recently started-up in step 2 meaning it can only be shut down after the minimum up-time constraint, which is 3 steps. Thus, the unit commitment state from CUC model is not feasible. Even though the CUC formulation considers the minimum up-time from the cluster, it can only identify the number of units that have not reached the minimum up-time yet, but it cannot identify the individual unit(s) that is(are) limited by this constraint. For instance, in the example above the CUC model had the constraint that one unit was not allowed to be shut down, based on the minimum uptime constraint, but the model could not identify which of the two units had this restriction.
When comparing the operational costs of example 2, again the CUC model results in a lower cost than the IUC, as the CUC formulation relaxes the problem constraints by not keeping a record of the individual operational levels.
2.4.1.3 Final considerations of errors when clustering identical units
As shown in examples 1 and 2, the CUC model may lead to a feasible result from a modeling perspective, but realistically they are unfeasible. This may happen even when clustering only identical units as the formulation does not keep operational records of each individual unit in the cluster, such as power output and up/downtime. The main cause of this inconsistency is a start-up and shut-down constraints from generators, and a UC problem not considering such constraints will unlikely have these issues happening. Of course, the issues may not occur depending on the power demand profile (if load is flat over time, for instance), but this is rarely manageable.
The last consideration from this error is that the cost result from a CUC model with only identical unit clustered should always be equal (in case none issues occur)
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or lower (in case of issues occur) when compared to the result of its equivalent IUC model, as the CUC formulation relaxes the problem. The longer the power demand profile, the more likely the issues illustrated in examples 1 and 2 will appear.
2.4.2 Deviations when clustering different units
As expect, gathering different units in a cluster may not result in a feasible unit commitment solution, and it is easy to illustrate. As a quick example, take the parameters from the two different units as shown in Table 5, and consider a one-step demand of 600 MW. The optimum result of a power schedule would be that Gen 1 operates at its maximum power providing the 600 MW necessary to fulfill the demand. Now consider Gen 1 and Gen 2 are clustered as shown in Table 6, the maximum output from each unit would be 500 MW (in order to maintain the 1000 MW as max output power) and the unit commitment would now require two units operating to provide the 600 MW power demand.
Table 5 – Parameters from two different units
Unit Pmax (MW) Pmin (MW) RUD (MW) t_up (step) t_dw (step) CVU ($/MW) Unit 1 600 300 50 3 1 50.00 Unit 2 400 200 50 3 1 150.00
Table 6 – Parameters from two different units clustered Name Number of units Pmax (MW) Pmin (MW) RUD (MW) t_up (step) t_dw (step) CVU ($/MW) Cluster 2 500 250 50 3 1 100.00
Comparing the operational costs, the individual unit commitment would result in a $30k, while the clustered unit commitment would cost $60k. In conclusion, although the CUC formulation relaxes the problem, clustering different units may result in a higher cost if compared to the IUC, although this is not always true.
2.4.3 Cost error between IUC model and CUC model in large networks
In large networks, both the deviations from clustering identical units and different units will be present. The cost error will then depend on a series of factors. First, if the start-up and shut-down constraint are considered in the formulation, and if so, the issues will likely occur but it still depends on the power demand profile, which is not manageable. Secondly, if different units were clustered, and what strategy was used to define the parameters of the cluster. Note that both the issues depend on heuristic factors, and the operational cost result from a CUC model may be either higher or lower than the IUC model, depending on the characteristics of the units gathered together.
2.5 CHALLENGES IN THE CUC MODEL
Apart from the possibility of resulting in an unfeasible unit commitment, one of the main challenges when using the CUC model is to select which individual units will be gathered in each cluster. A good approach is to cluster only identical units, implying that the result will be either equal or lower when compared to the optimum IUC solution. However, clustering only identical units is not an easy option in large networks, as there is a wide range of units with different characteristics, in different locations and segregated by several transmission lines.
Thus, the selection of different units to form a cluster becomes a heuristic methodology. The recommendation when selecting different units in a cluster is to consider a few characteristics of the generating unit, such as production cost, maximum generation, minimum on and off period of time, and others, so the results can be compared in a useful way. Based on the experiments made on this thesis, even though a pattern was created to cluster different units, the unit selection has a great impact on the CUC results and should be taken carefully.
3 MIXED-SOLUTIONS STRATEGY
3.1 INTRODUCTION
As illustrated in the previous section, both the IUC and CUC models bring benefits and detriments. The IUC, on one hand delivers a feasible power schedule solution but on the other hand has a large number of possible combinations which may lead to excessive time to execute the model and thus not achieving the requirements from DESSEM. In the CUC model, on one hand provides a fast convergence by reducing the number of possible combinations during the unit commitment, but on the other hand the power scheduling feasibility cannot be guaranteed.
The mixed-solution strategy consists of using both models, the CUC and IUC in order to define the power scheduling. At first, the CUC model is used to provide an accurate approximation of a power scheduling with a minimum cost operation. Secondly, one of the CUC results is used as a start point in the IUC model looking towards accelerating the latter and provide a guaranteed feasible UC as result of IUC. The information may be any condition resulted from the CUC power scheduling, such as unit commitment or operational cost, resulting in different possible approaches when using the mixed-solution strategy.
The main challenge in this strategy is finding the a CUC result that indeed provides a good starting point for IUC in order to reduce the processing time. The information taken from the CUC and added to the IUC formulation must significantly decrease IUC execution time, so the total execution time of mixed solution (CUC plus IUC execution timing) is better when compared to a standalone IUC model. If the mixed-solution strategy execution time is not better than the IUC model, it may be invalidated.
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In [1], the authors suggest to use the constraint shown in (24) in order to equally use the number of operational units in each timestamp from the CUC solution as a requirement to IUC model. However, this approach cannot be achieved, as the total number of operational units proposed by the CUC model may not be feasible, as demonstrated previously in the section 2.4 in which the deviations were presented, demonstrating the actual number of operational units in the CUC model may be lower compared to the IUC model.
∑ 𝑢𝑖,𝑡𝑜𝑛
𝑖 ∈ 𝑐
= 𝑛𝑐,𝑡𝑜𝑛 ∀𝑐, 𝑡 (24)
3.2 DIFFERENT APPROACHES
As an alternative proposed by [1], this thesis uses the following approaches of getting the CUC result as an input for the IUC model.
3.2.1 Number of operational units (flexible)
Represented by (25), this approach uses the total number of operational units at each hour resulted from the CUC model as data input for the IUC model. In this condition, there is a flexibility of ±10% in the total number of operational units at each hour of the IUC model compared to the unit commitment resulted from CUC.
In addition, instead of comparing the number of operational units in group by group at each hour as equation (24), the proposed approach uses the total number of operational units at each hour, allowing the IUC model to redistribute the CUC operational units in order to get a feasible unit commitment.
0.9 . ∑ 𝑢𝑐,𝑡𝑜𝑛 ∀ 𝑐 ≤ ∑ 𝑛𝑖,𝑡𝑜𝑛 ∀ 𝑖 ≤ 1.1 . ∑ 𝑢𝑐,𝑡𝑜𝑛 ∀ 𝑐 ∀ 𝑡 (25)
For instance, if the CUC result recommends a total of 10 operational units in a period of time, the IUC model will restrict the total possible operational units between and including 9 and 11.
The 10% criterion was based on validation tests, in which lower and higher values were tested. The 10% value fitted in most cases to provide a feasible solution after running the IUC model. However, there are no restrictions on suggesting different percentage levels in either direction, lower to keep the problem more restrictive (which may lead to unfeasible solution) or higher to keep the problem more flexible (which may lead to higher execution time).
The main advantage of using the number of operational units approach is that it implies a large number of possible solutions, giving a good trending on the convergence for the IUC model. However, since the CUC result may have a lower or greater number of operation units, this approach brings the challenge of defining a multiplier to relax the parameter and avoid the equality suggested by [1], which can lead to a non-feasible unit commitment solution.
3.2.2 Operational cost plus
The second proposed approach for the mixed-solutions strategy is called operational cost plus. It uses the operational cost resulted from the CUC as reference to the target cost in the IUC. As shown in equation (26), this approach suggests the IUC resulting cost to be either equal or higher than the CUC operational cost result.
𝑐𝑜𝑠𝑡 (𝐼𝑈𝐶) ≥ 𝑐𝑜𝑠𝑡 (𝐶𝑈𝐶) (26)
only if clustering identical units
As previously presented in section 2.4.1, only when clustering identical units, the CUC operational cost will be either equal or lower than the IUC cost. For this reason, the operational cost-plus approach cannot be used in mixed-solutions strategies when using groups with different units clustered, as the CUC operational cost result will be likely higher than the actual optimum IUC cost.
The main advantage of the operational cost-plus approach is that it does not require any multiplier values to make the condition valid between the CUC result and the IUC expected result. On the other hand, the main challenge from this approach is that the main objective from the mixed-solution strategy algorithm is to minimize the IUC operational cost, while this restriction suggests the operational cost result must be higher than a pre-defined value. Starting the problem with such requirement may lead
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to a non-convergence depending on the optimization algorithm used in the problem. Further results will be presented in the computational experiments.
3.2.3 Operational cost minus
Called as operational cost minus, the third and last approach proposed for mixed-solutions strategy is a derivation from the previous one. As shown in (27), it is suggested a multiplier (1.01 in this case) to ensure the CUC cost is higher than the expected IUC operational cost result. The multiplier is required since the operational cost resulted from a CUC model may be actually lower than the operational cost from an IUC model. Otherwise, the mixed-solutions strategy using operational cost minus approach would not be able to achieve the operational cost proposed by this condition.
𝑐𝑜𝑠𝑡 (𝐼𝑈𝐶) ≤ 1.01 . 𝑐𝑜𝑠𝑡 (𝐶𝑈𝐶) (27)
Based on a few validation tests, adding 1% on the CUC operational cost result provides a higher operational cost target than the one expected as result from the IUC model in most cases. This is the main challenge when using this approach, and it may be different in future studies. The correlation may be either more restrictive (lower than 1%) or more flexible (higher than 1%), depending on what is expected to achieve, faster convergence or a feasible result, respectively.
Since this approach adds a percentage value over the CUC operational cost result, it can be used in both situations with clusters considering only identical or clusters with different units.
An advantage of the operational cost minus approach is that the main objective in the mixed-solution strategy is to minimize the operational cost, and (27) defines an upper bound for it as an input for the problem.
3.3 OTHER STRATEGIES
All the above mixed-solutions strategies consider first running the CUC model, taking one or more result as input reference to later run the IUC algorithm in order to find a feasible solution. As any CUC result output could be of interest as input in the IUC algorithm from mixed-solutions, a wide variety of combinations could be used as
strategy. Similar to the strategy proposed by section 3.2.1, another possibility would be to consider the total number of operational units inside a given group as reference to the same groups of individual units in the IUC model. Other possibilities would be to consider different multiplier factors for strategies represented in section 3.2.1 and 3.2.3. Lastly, the mixed solution approach could reduce the number of possibilities at each stage based on the CUC result, rather than add a new constraint to the problem. For instance, instead of add the restrictions shown in 3.2.1, 3.2.2 and 3.2.3, the problem could be reduce at each stage to consider the possibilities between those values, with a certain flexibility.
As other strategies, there are studies as in [5] that proposes a CUC formulation that better represent the individuality from each generation unit inside the clusters. In summary, it could be called a mixed formulation for CUC modeling, which is different from this thesis that proposes mixed approaches by running both models one followed by another.
4 COMPUTATIONAL EXPERIMENTS
4.1 TEST SYSTEM DESCRIPTION
All the experiments consider a single bus, as illustrated in Figure 5, in which all the thermal power generating units (Gen) and the power load (L) are connected during week-ahead planning on an hourly basis.
Figure 5 – Single bus bar from the experiment
….
The IUC and CUC models consist of three main blocks, as illustrate in Figure 6. The first is the generator data input, followed by the power demand input, and lastly, the optimization methodology was chosen (IUC or CUC). Note the generation input data is correlated to which model will be used, as a binary generation input data can only use in the IUC, and the clustered generation input can only be used in the CUC. On the other hand, the power demand does not depend on the methodology used, but it was adjusted to make sure the generators can provide the maximum power demand without load curtailment. Nevertheless, a virtual generator representing the load curtailment was kept active with an extremely high operational cost of 1,000,000 $/MWh. The possible combinations in the IUC model are only the week chosen, while in the CUC model both the week and the cluster may vary.
Figure 6 – IUC/CUC algorithm flow chart
L Gen 1 Gen 2 Gen 3 Gen n
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The mixed-solution algorithm flow is slightly different from the IUC and CUC models. As shown in Figure 4, the mixed-solution approach first considers the CUC model to later use one of its results as input for the IUC model. Figure 7 illustrates the mixed-solution flow in more detail, and the only step not repeated is the power demand input, as it does not depend on the methodology chosen. As other steps have to be duplicated, the mixed-solution methodology has 5 main blocks compared against three from IUC / CUC models as standalone. The possible combinations in the mixed-solution approach are on the cluster generating input, week chosen and the type of result transferred from the CUC optimization to the IUC model.
Figure 7 – Mixed-solutions algorithm flow chart
4.1.1 Thermal power plants input
To represent the thermal power plants, the database from DESSEM model was considered in all computational experiments as illustrated in APPENDIX B – Thermal Power Units. Originally, the DESSEM database includes most of the thermal power generators in Brazil, including future plant deployment. For this study, the plants which will be placed into operation in 2019 or later were not considered. Thus, the experimental system created to simulate the unit commitments has a total generating capacity of 23.76 GW based on 355 individual units.
In addition to the maximum output power from each unit, the DESSEM database also defines the minimum output power, minimum up/downtime periods and the
variable cost per unit ($/MWh). The maximum and minimum power per unit is illustrated in Figure 8.
Figure 8 – Maximum and minimum power per unit
Figure 9 illustrates the relationship between the maximum power output and CVU. As conclusion, it can be observed in most cases the greater the power capacity, the lower the operational cost, although this is not essentially true for every unit.
Figure 9 – Maximum power output vs variable cost per unit 0 200 400 600 800 1000 1200 1400 1600 1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103 109 115 121 127 133 139 145 151 157 163 169 175 181 187 193 199 205 211 217 223 229 235 241 247 253 259 265 271 277 283 289 295 301 307 313 319 325 331 337 343 349 355 P ow er (M W ) Power Unit P_max P_min 0 200 400 600 800 1000 1200 1400 1600 0 200 400 600 800 1000 1200 1400 1600 1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103 109 115 121 127 133 139 145 151 157 163 169 175 181 187 193 199 205 211 217 223 229 235 241 247 253 259 265 271 277 283 289 295 301 307 313 319 325 331 337 343 349 355 CV U Co st P ow er (M W ) Power Unit P_max CVU ($/MWh)
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Both the start-up cost and the power ramp-up/down were estimated based on the original DESSEM database. Constraints not considered in the computational experiments are the shut-down cost and maintenance schedule.
The start-up cost (SUC) estimation is inversely proportional to the CVU, and one of the cheapest operational units was set as reference. Figure 10 illustrates SCU vs CVU.
Figure 10 – Start-up cost vs variable cost per unit
The power ramp-up/down (RUD) estimation is based on the power delta (Pmax – Pmin) multiplied by a factor based on the CVU, as Table 7 below. The approximation considered in this thesis is the lower the CVU, the greater the number of steps needed between Pmin and Pman. Although this is a good approximation, the number of steps also depends on the unit technology and this is not always truth. Figure 11 illustrates the relationship between RUD vs Pmax-Pmin.
Table 7 – Conditions used for RUD estimation
CVU condition Factor Steps from Pmin to Pmax
0 ≤ CVU < 100 0.125 8 100 ≤ CVU < 300 0.20 5 300 ≤ CVU < 600 0.25 4 CVU ≥ 600 0.5 2 0 200 400 600 800 1000 1200 1400 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000 1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103 109 115 121 127 133 139 145 151 157 163 169 175 181 187 193 199 205 211 217 223 229 235 241 247 253 259 265 271 277 283 289 295 301 307 313 319 325 331 337 343 349 355 CV U Co st S UC Co st Power Unit
