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Binary grey wolf optimizer models for profit based unit commitment of price-taking GENCO in electricity market
电力市场中基于利润的 GENCO 价格承诺的二元灰狼优化器模型

Srikanth Reddy K a, ∗ a,  ∗ ^("a, "^(**)){ }^{\text {a, }{ }^{*}}, Lokesh Kumar Panwar b b  ^("b "){ }^{\text {b }}, B.K. Panigrahi a a  ^("a "){ }^{\text {a }}, Rajesh Kumar c c ^(c){ }^{\mathrm{c}}, Ameena Alsumaiti d d ^(d){ }^{\mathrm{d}}
Srikanth Reddy K a, ∗ a,  ∗ ^("a, "^(**)){ }^{\text {a, }{ }^{*}} ， Lokesh Kumar Panwar b b  ^("b "){ }^{\text {b }} ， B.K. Panigrahi a a  ^("a "){ }^{\text {a }} ， Rajesh Kumar c c ^(c){ }^{\mathrm{c}} ， Ameena Alsumaiti d d ^(d){ }^{\mathrm{d}} a a ^(a){ }^{a} Department of Electrical Engineering, IIT Delhi, India
a a ^(a){ }^{a} 印度 IIT Delhi 电气工程系 b b  ^("b "){ }^{\text {b }} Center for Energy Studies, IIT Delhi, India
b b  ^("b "){ }^{\text {b }} 印度印度理工学院德里分校能源研究中心 c c  ^("c "){ }^{\text {c }} Department of Electrical Engineering, MNIT Jaipur, India
c c  ^("c "){ }^{\text {c }} 印度斋浦尔 MNIT 电气工程系 d d ^(d){ }^{\mathrm{d}} Department of Electrical and Computer Engineering, Khalifa University, Abu Dhabi, United Arab Emirates
d d ^(d){ }^{\mathrm{d}} 阿拉伯联合酋长国阿布扎比哈利法大学电气与计算机工程系

A R T ICLE INFO

Keywords:  关键字：

Profit base unit commitment (PBUC)
利润基础单位承诺 （PBUC）
Meta-heuristic optimization
元启发式优化
Electricity market  电力市场
Binary grey wolf optimizer (BGWO)
二进制灰狼优化器 （BGWO）
Constrained optimization  约束优化
Nature inspired optimization
自然启发的优化

Abstract  抽象

The restructuration of electric power sector has renovated the power system operational planning. In the deregulated market, electricity is treated as an entity unlike in the traditional vertically integrated market model where it is treated as a service provided by the generation companies (GENCOs). The task of GENCO is to perform self-scheduling of available units so as to achieve maximum profit in restructured power sector. Therefore, the problem of profit based unit commitment (PBUC) in deregulated markets can be formulated as a commitment and generation allocation through self-scheduling procedure. The commitment schedule optimization, i.e whether the status of a thermal unit is to be changed to on or off state, is a binary problem. Thus, the PBUC problem requires binary optimization for commitment and real valued optimization for generation allocation. In this paper, three binary grey wolf optimizer (BGWO) models are presented to solve the profit based self-scheduling problem of GENCO. The BGWO models proposed in this paper differ with respect to the transformation function used to map real valued wolf position into a binary variable. The binary mapping of commitment status is carried out using sigmoid and tangent hyperbolic transfer functions. Also, in the sigmoidal transfer function, two binary transformation procedures, namely crossover and conventional sigmoidal transfer function, are presented. The effectiveness of the proposed models in improving the solution quality of PBUC problem is examined using two test systems, a 3 unit and a 10 unit test system. In addition, two cases of GENCO market participation with and without reserve market participation are simulated. In the test case with reserve market participation, two commonly used reserve payment models are examined. Simulation results are presented and compared to existing approaches that have been used to solve the PBUC problem. The simulation results and statistical analysis demonstrate the improved solution quality (profit or fitness value) of the PBUC problem and statistical significance of the BGWO models with respect to solution quality obtained as compared to other established metaheuristic approaches.
电力部门的重组更新了电力系统的运营规划。在解除管制的市场中，电力被视为一个实体，这与传统的垂直整合市场模式不同，在传统的垂直整合市场模式中，电力被视为发电公司 （GENCO） 提供的服务。GENCO 的任务是对可用机组进行自我调度，以便在重组后的电力部门实现最大利润。因此，在放松管制的市场中，基于利润的单位承诺 （PBUC） 问题可以通过自我调度程序表述为承诺和发电分配。承诺计划优化，即热量单元的状态是更改为 on 还是 off 状态，是一个二进制问题。因此，PBUC 问题需要对承诺进行二进制优化，对生成分配进行实值优化。在本文中，提出了三种二进制灰狼优化器 （BGWO） 模型来解决 GENCO 基于利润的自调度问题。本文提出的 BGWO 模型在用于将实值 wolf 位置映射到二进制变量的变换函数方面有所不同。承诺状态的二进制映射是使用 sigmoid 和切线双曲传递函数执行的。此外，在 S 形传递函数中，提出了两个二进制变换过程，即交叉和常规 S 形传递函数。使用两个测试系统，一个 3 单元和一个 10 单元测试系统，检查了所提出的模型在提高 PBUC 问题求解质量方面的有效性。此外，还模拟了有和没有储备市场参与的两种 GENCO 市场参与情况。 在储备市场参与的测试案例中，研究了两种常用的储备金支付模型。给出了仿真结果，并与用于解决 PBUC 问题的现有方法进行了比较。仿真结果和统计分析表明，与其他已建立的元启发式方法相比，PBUC 问题的解决方案质量（利润或适应度值）以及 BGWO 模型在获得的解决方案质量方面的统计显着性有所提高。

1. Introduction  1. 引言

The electricity market structure around the globe has been transforming into deregulated nature from conventional vertically integrated model. As a result, the unit commitment and generation scheduling practice of the generation company (GENCO) changed from cost minimization to profit maximization [1]. In this structure, GENCO acts as a price taking entity and schedules its available generation assets for various services as applicable, considering the generator’s operating and
全球电力市场结构已经从传统的垂直整合模式转变为不受管制的性质。因此，发电公司 （GENCO） 的机组承诺和发电调度实践从成本最小化转变为利润最大化 [1]。在这种结构中，GENCO 充当价格接受实体，并考虑到发电机的运营和
cost characteristics [2]. Therefore, the forecasted load and reserve requirement of the power system network provides the participation limit for GENCO. Therefore, in this self-scheduling procedure do not impose any obligations related to load/reserve fulfillment as in case of traditional/conventional market mechanism [3,4]. Thereupon, concurrent provisions to participate in various markets offering different services for a single GENCO were identified in the pursuit of profit maximization. However, earlier models have partially considered reserve costs and revenues [5]. Later, more appropriate and accurate models for
成本特征 [2]。因此，电力系统网络的预测负载和储备要求为 GENCO 提供了参与限制。因此，在这种自调度程序中，不要像传统/传统市场机制那样施加任何与装载/储备履行相关的义务[3,4]。因此，为了追求利润最大化，确定了参与为单个 GENCO 提供不同服务的各种市场的并发规定。然而，早期的模型部分考虑了储备成本和收入 [5]。以后，更合适和准确的模型
market participation are developed and are adopted in this work [6].
市场参与得到发展，并在这项工作中得到采用 [6]。
The non-linear nature of the generation scheduling was intensified through the profit based unit commitment problem in comparison with traditional cost minimization problem. In PBUC, some of the constraints, like load and reserve satisfaction, are modeled as soft constraints, unlike cost minimization problem in which both the load constraint and reserve constraint are equality constraints. The constraints pertaining to the minimum up and down time and ramp rate remained same in PBUC as that of conventional Unit Commitment (UC) problem. Over the past decade, the problem of GENCO self-scheduling through PBUC attracted considerable interest. As a result, different optimization approaches are proposed to solve the PBUC problem. Earlier models incorporated reserve market participation of GENCO using forecasted energy and reserve price [7]. The Lagrangian multiplier adjustment approach is used to accompany the thermal unit scheduling [8]. However, the non-convex nature of the PBUC problem led to convergence related issues [16]. Later, the convergence issue is resolved using gradient based Lagrangian Relaxation (LR) approach. However, the gradient based solutions are vulnerable to local minima for non-convex problems like PBUC. The local minima problem is solved using computationally intelligent and heuristic approaches like Genetic Algorithm (GA) in which softer demand constraints are modeled [3]. Thereafter, hybrid techniques like Lagrangian Relaxation-Evolutionary Programming (LR-EP) are proposed to suppress the issues with LR by using evolutionary programming for the Lagrangian multiplier update [9]. Similarly, Lagrangian Relaxation-Genetic Algorithm (LR-GA) is also proposed in which GA is used to update Lagrangian multipliers [10]. The evolutionary approaches are hindered by the fact the convergence of the algorithm is highly dependent on various factors such as mutation rate, crossover rate, parent selection method etc. Consequently, one has to perform multiple experimentations to find optimal parameters otherwise which the problem can encounter premature convergence. Later, the optimal allocation of energy and reserve capacities is carried out using Tabu-Search (TS) [11]. While, the TS approach scores in terms of excepting from local minima by allowing non-improving solution, it also requires optimal tuning of various parameters similar to GA. Thereafter, an integer based linear programming approach is developed with superior performance as compared to LR [12]. However, the same endured the computational problems with increment in system dimension.
与传统的成本最小化问题相比，基于利润的单位承诺问题加强了发电调度的非线性性质。在 PBUC 中，一些约束（如负载和预留满足）被建模为软约束，这与成本最小化问题不同，其中负载约束和预留约束都是相等约束。PBUC 中与最小启动和关闭时间以及斜坡速率相关的约束与传统的单位承诺 （UC） 问题相同。在过去的十年中，通过 PBUC 进行 GENCO 自我调度的问题引起了相当大的兴趣。因此，提出了不同的优化方法来解决 PBUC 问题。早期模型使用预测的能源和储备价格纳入了 GENCO 的储备市场参与 [7]。拉格朗日乘数调整方法用于配合热单元调度 [8]。然而，PBUC 问题的非凸性导致了与收敛相关的问题 [16]。后来，使用基于梯度的拉格朗日松弛 （LR） 方法解决了收敛问题。然而，对于像 PBUC 这样的非凸问题，基于梯度的解容易受到局部最小值的影响。局部最小值问题是使用计算智能和启发式方法解决的，例如遗传算法 （GA），其中对较软的需求约束进行建模 [3]。此后，提出了拉格朗日弛豫-进化规划 （LR-EP） 等混合技术，通过使用拉格朗日乘子更新的进化规划来抑制 LR 的问题 [9]。同样，还提出了拉格朗日弛豫遗传算法 （LR-GA），其中 GA 用于更新拉格朗日乘子 [10]。 进化方法受到以下事实的阻碍：算法的收敛高度依赖于各种因素，例如突变率、交叉率、亲本选择方法等。因此，必须进行多次实验才能找到最佳参数，否则问题可能会遇到过早收敛。之后，使用 Tabu-Search （TS） [11] 进行能量和储备容量的最佳分配。虽然 TS 方法通过允许非改进解决方案来排除局部最小值，但它也需要类似于 GA 的各种参数的最佳调整。此后，开发了一种基于整数的线性规划方法，与 LR 相比具有更好的性能 [12]。然而，同样忍受了系统维度增加的计算问题。
Recent literature showed the application of various meta-heuristic approaches like Particle Swarm Optimization (PSO) [13], Muller Method (MM) [14], Nodal Ant Colony Optimization (NACO) [15], Parallel Nodal Ant Colony Optimization (PNACO) [16], Gravitational Search-Lagrangian Relaxation Based Artificial Neural Network (GSLRANN) [17], hybrid Lagrangian Relaxation- Invasive Weed Optimization (LR-IWO) [18], Tabu-Search-Evolutionary Particle Swarm Optimization (TS-EPSO) [19], Particle Swarm Optimization-Dynamic Programming (PSO-DP) [20], Variable Neighborhood TabuS Parallel Enhanced Particle Swarm Optimization with island model (VTS-PEPSO) [21], Lagrangian Relaxation-Particle Swarm Optimization (LR-PSO) [22], Parallel Artificial Bee Colony (PABC) [23], Multi Agent System (MAS) [24] based approach, Imperialist Competitive Algorithm (ICA) [25], Ant Colony Optimization (ACO) [26], Improved Pre Prepared Power Demand (IPPD) [27], Binary Fireworks Algorithm (BFWA) [37], etc., to solve the PBUC problem. The approaches have different pros and cons for application to high dimension problems such as UC and PBUC. For example, PSO based algorithms present a simple way of implementation with minimum parameter tuning. However, the vulnerability of falling into local minima hinders the application of PSO to problems with high dimension. The other class of algorithms called ant colony algorithms (NACO, PNACO, ACO) gains over other algorithms with inherent parallelism. However, ACO algorithms are prone to random decisions and uncertain convergence time. The physical phenomenon based algorithms such as Gravitational Search Algorithm (GSA) are also prone to fall into local optima and solution precision for computationally challenging problems. Therefore, the heuristic approaches are either
最近的文献显示了各种元启发式方法的应用，如粒子群优化 （PSO） [13]、穆勒法 （MM） [14]、节点蚁群优化 （NACO） [15]、平行节点蚁群优化 （PNACO） [16]、基于引力搜索-拉格朗日弛豫的人工神经网络 （GSLRANN） [17]、混合拉格朗日弛豫-侵入性杂草优化 （LR-IWO） [18]、禁忌搜索-进化粒子群优化 （TS-EPSO） [19]、粒子群优化-动态规划 （PSO-DP） [20]、带岛模型的可变邻域 TabuS 并行增强粒子群优化 （VTS-PEPSO） [21]、拉格朗日弛缓-粒子群优化 （LR-PSO） [22]、并行人工蜂群 （PABC） [23]、基于多智能体系统 （MAS） [24] 的方法、帝国主义竞争算法 （ICA） [25]、蚁群优化 （ACO） [26]、改进的预制电力需求 （IPPD） [27]、二进制烟花算法 （BFWA） [37]， 等，以解决 PBUC 问题。这些方法在应用于 UC 和 PBUC 等高维问题时有不同的优缺点。例如，基于 PSO 的算法提供了一种简单的实现方法，只需最少的参数调整。然而，落入局部最小值的脆弱性阻碍了 PSO 在高维问题上的应用。另一类称为蚁群算法的算法（NACO、PNACO、ACO）优于具有固有并行性的其他算法。然而，ACO 算法容易出现随机决策和不确定的收敛时间。基于物理现象的算法，如引力搜索算法 （GSA），也容易陷入计算挑战性问题的局部最优和求解精度。因此，启发式方法是
simple to apply with local optimum vulnerability or need optimal parameter tuning for improved or best possible performance.
易于应用，具有 local optimal 漏洞，或者需要优化参数以提高或实现最佳性能。
The motivation behind this work is the successful implementation of meta-heuristic framework Binary Grey Wolf Optimizer [28] in solving several computationally challenging optimization problems such as PBUC in electricity market to maximize the GENCO profit. Some of the applications of GWO are blackout risk prevention [29], optimal reactive power dispatch [30], solar maximum power point tracking (MPPT) [31], optimization of photonic crystal filter [32], etc. Recently, binary wolf optimizer has been developed and applied successfully to feature selection problem [33]. The application of GWO to the engineering problems is based on the advantages of the meta-heuristic approach GWO, such as its requirement for parameter tuning corresponding to a particular problem. This paper presents the application of various Binary Grey Wolf Optimization (BGWO) models to solve non-convex, computationally challenging, bounded and constrained PBUC problem of GENCO in power system.
这项工作背后的动机是成功实现元启发式框架 Binary Grey Wolf Optimizer [28]，以解决几个计算上具有挑战性的优化问题，例如电力市场的 PBUC，以实现 GENCO 利润最大化。GWO 的一些应用包括停电风险预防 [29]、最佳无功功率调度 [30]、太阳能最大功率点跟踪 （MPPT） [31]、光子晶体滤波器优化 [32] 等。最近，二元狼优化器已经被开发并成功地应用于特征选择问题 [33]。GWO 在工程问题上的应用基于元启发式方法 GWO 的优势，例如它对对应于特定问题的参数调整的要求。本文介绍了各种二元灰狼优化 （BGWO） 模型在电力系统中解决 GENCO 的非凸、计算挑战、有界和约束的 PBUC 问题。
Rest of the paper is organized as follows: Section 2 introduces the PBUC problem of GENCO. Section 3 outlines the theory of GWO and various BGWO models. The application and solution procedure of PBUC using BGWO is described in Section 4. The test system description, simulation conditions, parameter settings are explained in Section 5. The same section also presents the simulation results and discussion along with statistical significance of the proposed BGWO models. Finally section 6 concludes the paper with key findings of the work.
本文的其余部分组织如下：第 2 节介绍了 GENCO 的 PBUC 问题。第 3 节概述了 GWO 理论和各种 BGWO 模型。使用 BGWO 的 PBUC 的应用和解决程序在第 4 节中描述。测试系统描述、仿真条件、参数设置在第 5 节中解释。同一部分还介绍了仿真结果和讨论以及所提出的 BGWO 模型的统计显着性。最后，第 6 节总结了本文的主要发现。

2. Problem formulation  2. 问题制定

In the self-scheduling problem such as PBUC, GENCO acts as price taker and schedules the available units to incur maximum profit. The different cash flows like GENCO costs and revenues in various market services i.e., energy and ancillary (reserve) markets, are illustrated along with GENCO objective in the following sub-sections.
在 PBUC 等自调度问题中，GENCO 充当价格接受者并安排可用单位以产生最大利润。不同的现金流，如各种市场服务（即能源和辅助（储备）市场）的 GENCO 成本和收入，与以下小节的 GENCO 目标一起说明。
Nomenclature.  命名法。
A. Sets and Indices  A. 集和索引
t t tt Hour of the scheduling day
t t tt 计划日的小时数
i i ii Conventional (coal fired) Generator(s)
i i ii 传统（燃煤）发电机
H H HH Time horizon (24 h)
H H HH 时间范围 （24 小时）
G G GG Conventional generators set
G G GG 常规发电机组
B. Decision variables  B. 决策变量
P ( i , t ) P ( i , t ) P(i,t)P(i, t) Energy market bid of i th i th  i^("th ")i^{\text {th }} thermal unit for t th t th  t^("th ")t^{\text {th }} hour
P ( i , t ) P ( i , t ) P(i,t)P(i, t) t th t th  t^("th ")t^{\text {th }} 一小时热力单位的 i th i th  i^("th ")i^{\text {th }} 能源市场出价
R ( i , t ) R ( i , t ) R(i,t)R(i, t) Reserve market bid of i th i th  i^("th ")i^{\text {th }} thermal unit for t th t th  t^("th ")t^{\text {th }} hour
R ( i , t ) R ( i , t ) R(i,t)R(i, t) i th i th  i^("th ")i^{\text {th }} 热力单位 t th t th  t^("th ")t^{\text {th }} 的小时储备市场出价
A ( i , t ) A ( i , t ) A(i,t)A(i, t) ON/OFF state of i th i th  i^("th ")i^{\text {th }} thermal unit at t th t th  t^("th ")t^{\text {th }} hour
A ( i , t ) A ( i , t ) A(i,t)A(i, t) t th t th  t^("th ")t^{\text {th }} 每小时热力装置的 i th i th  i^("th ")i^{\text {th }} 开/关状态
C. Parameters  C. 参数
P l ( t ) P l ( t ) Pl(t)P l(t) System load at t th t th  t^("th ")t^{\text {th }} hour
P l ( t ) P l ( t ) Pl(t)P l(t) 每小时的系统 t th t th  t^("th ")t^{\text {th }} 负载
R R q ( t ) R R q ( t ) R^(Rq)(t)R^{R q}(t) Required system reserve for t th t th  t^("th ")t^{\text {th }} hour
R R q ( t ) R R q ( t ) R^(Rq)(t)R^{R q}(t) t th t th  t^("th ")t^{\text {th }} 小时所需的系统预留
P i min P i min P_(i)^(min)P_{i}^{\min } Lower generation limit for t th t th  t^("th ")t^{\text {th }} thermal unit
P i min P i min P_(i)^(min)P_{i}^{\min } 热量单元的 t th t th  t^("th ")t^{\text {th }} 发电下限
P i max P i max P_(i)^(max)P_{i}^{\max } Upper generation limit for t th t th  t^("th ")t^{\text {th }} thermal unit
P i max P i max P_(i)^(max)P_{i}^{\max } 热量单元的 t th t th  t^("th ")t^{\text {th }} 发电上限
φ ( i , t ) φ ( i , t ) varphi(i,t)\varphi(i, t) Startup cost for i th i th  i^("th ")i^{\text {th }} thermal unit at t th t th  t^("th ")t^{\text {th }} hour
φ ( i , t ) φ ( i , t ) varphi(i,t)\varphi(i, t) t th t th  t^("th ")t^{\text {th }} 每小时热单元的 i th i th  i^("th ")i^{\text {th }} 启动成本
H S C ( i ) H S C ( i ) H_(SC)(i)H_{S C}(i) Hot startup cost of i th i th  i^("th ")i^{\text {th }} thermal unit
H S C ( i ) H S C ( i ) H_(SC)(i)H_{S C}(i) 热机组的 i th i th  i^("th ")i^{\text {th }} 热启动成本
C S C ( i ) C S C ( i ) C_(SC)(i)C_{S C}(i) Cold startup cost of i th i th  i^("th ")i^{\text {th }} thermal unit
C S C ( i ) C S C ( i ) C_(SC)(i)C_{S C}(i) 热机组的 i th i th  i^("th ")i^{\text {th }} 冷启动成本
M D T ( i ) M D T ( i ) M_(DT)(i)M_{D T}(i) Minimum down time of i th i th  i^("th ")i^{\text {th }} thermal unit
M D T ( i ) M D T ( i ) M_(DT)(i)M_{D T}(i) 热力装置的最小 i th i th  i^("th ")i^{\text {th }} 停机时间
M U T ( i ) M U T ( i ) M_(UT)(i)M_{U T}(i) Minimum up time of i th i th  i^("th ")i^{\text {th }} thermal unit
M U T ( i ) M U T ( i ) M_(UT)(i)M_{U T}(i) 热力装置的 i th i th  i^("th ")i^{\text {th }} 最短运行时间
O F F h ( i ) O F F h ( i ) OFF_(h)(i)O F F_{h}(i) Time(hours) for which i t h i t h i^(th)i^{t h} thermal unit is OFF
O F F h ( i ) O F F h ( i ) OFF_(h)(i)O F F_{h}(i) 热量单元关闭的时间 i t h i t h i^(th)i^{t h} （小时）
O N h ( i ) O N h ( i ) ON_(h)(i)O N_{h}(i) Time(hours) for which i th i th  i^("th ")i^{\text {th }} thermal unit is ON
O N h ( i ) O N h ( i ) ON_(h)(i)O N_{h}(i) 热量单元开启的时间 i th i th  i^("th ")i^{\text {th }} （小时）
R U R ( i ) R U R ( i ) R_(UR)(i)R_{U R}(i) Ramp up rate of i th i th  i^("th ")i^{\text {th }} thermal unit
R U R ( i ) R U R ( i ) R_(UR)(i)R_{U R}(i) 热单元的 i th i th  i^("th ")i^{\text {th }} 斜坡速率
R D R ( i ) R D R ( i ) R_(DR)(i)R_{D R}(i) Ramp down rate of i th i th  i^("th ")i^{\text {th }} thermal unit
R D R ( i ) R D R ( i ) R_(DR)(i)R_{D R}(i) 热机组的 i th i th  i^("th ")i^{\text {th }} 斜坡下降率
C S h ( i ) C S h ( i ) CS_(h)(i)C S_{h}(i) Cold start hours of i th i th  i^("th ")i^{\text {th }} thermal unit
C S h ( i ) C S h ( i ) CS_(h)(i)C S_{h}(i) 热力装置的 i th i th  i^("th ")i^{\text {th }} 冷启动时间
E P ( t ) E P ( t ) EP(t)E P(t) Spot price of energy market at t th t th  t^("th ")t^{\text {th }} hour
E P ( t ) E P ( t ) EP(t)E P(t) 能源市场 t th t th  t^("th ")t^{\text {th }} 小时现货价格
R P ( t ) R P ( t ) RP(t)R P(t) Reserve market price at t t h t t h t^(th)t^{t h} hour
R P ( t ) R P ( t ) RP(t)R P(t) 每小时的储备市场价格 t t h t t h t^(th)t^{t h}
r r rr Probability of contingency occurrence
r r rr 意外事件发生的概率
N G N G NGN G Number of generators
N G N G NGN G 发电机数量
T T TT Total number of hours
T T TT 总小时数

2.1. Objective function  2.1. 目标函数

In contrast to the traditional or vertically integrated conventional power system operations [6], in deregulated markets, the objective of GENCO is to maximize profit by self-scheduling procedures. The objective of GENCO in a deregulated market will maximize the difference between total revenue (RV) and total cost (TC) as given by,
与传统或垂直整合的传统电力系统运营 [6] 相比，在放松管制的市场中，GENCO 的目标是通过自我调度程序实现利润最大化。在不受管制的市场中，GENCO 的目标是使总收入 （RV） 和总成本 （TC） 之间的差异最大化，如下所示：
maximise profit = max . ( R V − T C )  maximise profit  = max . ( R V − T C ) " maximise profit "=max.(RV-TC)\text { maximise profit }=\max .(R V-T C)
Where, RV represents the total revenue that GENCO receives from market participation. The total revenue received by GENCO varies with respect to the payment methods pertaining to various market operating and clearing mechanisms in practice. Two of the widely used payment methods are discussed in detail as follows. The payment strategies for reserve allocated by generator in reserve market depend on the amount of time during which, payment is made and type of payment. The payment for reserve capacity can be of two types, reserve capacity payments and energy payments. The reserve capacity payment is made to the generators irrespective of the reserve capacity’s deployment. The energy payment is made for the duration for which the actual generation from the reserve capacity is allocated.
其中，RV 表示 GENCO 从市场参与中获得的总收入。在实践中，GENCO 收到的总收入因与各种市场运营和清算机制有关的支付方式而异。下面详细讨论两种广泛使用的支付方式。发电商在储备市场分配的储备金支付策略取决于支付的时间量和支付类型。预留容量的付款可以有两种类型，预留容量付款和能源付款。无论储备容量的部署如何，都会向发电机支付储备容量。能源付款是在分配预留容量的实际发电量的持续时间内支付的。

2.1.1. Reserve payment method A A AA (payment for energy)
2.1.1. 储备金支付方式 A A AA （能源费）

In this payment method (method A), the allocated reserve is paid only for the actual time it is called upon to serve. Therefore, the reserve payment ( r R P ( t ) R ( i , t ) ) ( r R P ( t ) R ( i , t ) ) (rRP(t)R(i,t))(r R P(t) R(i, t)) depends on the probability ( r r rr ) with which it is called to serve. The following expression represents the total payment to GENCO participating in both energy and reserve market.
在这种付款方式 （方法 A） 中，分配的准备金仅针对其被调用的实际时间支付。因此，准备金付款 ( r R P ( t ) R ( i , t ) ) ( r R P ( t ) R ( i , t ) ) (rRP(t)R(i,t))(r R P(t) R(i, t)) 取决于调用发球的概率 （ r r rr ）。以下表达式表示向参与能源和储备市场的 GENCO 支付的总款项。
R V = ∑ t = 1 T ∑ i = 1 N G [ { ( P ( i , t ) ) E P ( t ) A ( i , t ) } ] + ∑ t = 1 T ∑ i = 1 N G r R P ( t ) R ( i , t ) A ( i , t ) R V = ∑ t = 1 T   ∑ i = 1 N G   [ { ( P ( i , t ) ) E P ( t ) A ( i , t ) } ] + ∑ t = 1 T   ∑ i = 1 N G   r R P ( t ) R ( i , t ) A ( i , t ) RV=sum_(t=1)^(T)sum_(i=1)^(NG)[{(P(i,t))EP(t)A(i,t)}]+sum_(t=1)^(T)sum_(i=1)^(NG)rRP(t)R(i,t)A(i,t)R V=\sum_{t=1}^{T} \sum_{i=1}^{N G}[\{(P(i, t)) E P(t) A(i, t)\}]+\sum_{t=1}^{T} \sum_{i=1}^{N G} r R P(t) R(i, t) A(i, t)
In general, for this method, the reserve price is much higher compared to that of spot energy market price. This is to ensure that allocated reserve of GENCO makes considerable revenues. The total profit function for this method is given by,
一般来说，对于这种方法，底价比现货能源市场价格要高得多。这是为了确保 GENCO 的分配储备能产生可观的收入。该方法的总利润函数由下式给出，
profit = { ∑ t = 1 T ∑ i = 1 N G [ { ( P ( i , t ) ) E P ( t ) A ( i , t ) } ] + ∑ t = 1 T ∑ i = 1 N G r R P ( t ) R ( i , t ) A ( i , t ) } − { ∑ t = 1 T ∑ i = 1 N G [ ( 1 − r ) { F c ( P ( i , t ) ) A ( i , t ) } + r { F c ( P ( i , t ) + R ( i , t ) ) A ( i , t ) } + { φ ( i , t ) A ( i , t ) } ] }  profit  = ∑ t = 1 T   ∑ i = 1 N G   [ { ( P ( i , t ) ) E P ( t ) A ( i , t ) } ] + ∑ t = 1 T   ∑ i = 1 N G   r R P ( t ) R ( i , t ) A ( i , t ) − ∑ t = 1 T   ∑ i = 1 N G   ( 1 − r ) F c ( P ( i , t ) ) A ( i , t ) + r F c ( P ( i , t ) + R ( i , t ) ) A ( i , t ) } + { φ ( i , t ) A ( i , t ) } ] } {:[" profit "={sum_(t=1)^(T)sum_(i=1)^(NG)[{(P(i,t))EP(t)A(i,t)}]+sum_(t=1)^(T)sum_(i=1)^(NG)rRP(t)R(i,t)A(i,t)}],[-{sum_(t=1)^(T)sum_(i=1)^(NG)[(1-r){F_(c)(P(i,t))A(i,t)}+r{F_(c)(P(i,t):}],[+R(i","t))A(i","t)}+{varphi(i","t)A(i","t)}]}]:}\begin{aligned} \text { profit }= & \left\{\sum_{t=1}^{T} \sum_{i=1}^{N G}[\{(P(i, t)) E P(t) A(i, t)\}]+\sum_{t=1}^{T} \sum_{i=1}^{N G} r R P(t) R(i, t) A(i, t)\right\} \\ & -\left\{\sum _ { t = 1 } ^ { T } \sum _ { i = 1 } ^ { N G } \left[(1-r)\left\{F_{c}(P(i, t)) A(i, t)\right\}+r\left\{F_{c}(P(i, t)\right.\right.\right. \\ & +R(i, t)) A(i, t)\}+\{\varphi(i, t) A(i, t)\}]\} \end{aligned}
The first term denotes the revenue incurred by GENCO by scheduling generators in the energy market and the second term denotes the total cost (TC) of energy and reserve market schedules which again consists of two parts, fuel cost and startup cost. The shutdown costs, being a very small fraction of the startup cost, are often neglected. In (3), F c F c F_(c)F_{c}, the fuel cost of i th i th  i^("th ")i^{\text {th }} generator for t th t th  t^("th ")t^{\text {th }} hour, can be given as follows.
第一个术语表示 GENCO 通过在能源市场调度发电机而产生的收入，第二个术语表示能源和储备市场时间表的总成本 （TC），它同样由燃料成本和启动成本两部分组成。关闭成本占启动成本的很小一部分，经常被忽视。在（3） F c F c F_(c)F_{c} 中，发电机每小时 t th t th  t^("th ")t^{\text {th }} 的 i th i th  i^("th ")i^{\text {th }} 燃料成本可以给出如下。
F c ( P ( i , t ) ) = a i P ( i , t ) 2 + b i P ( i , t ) + c i F c ( P ( i , t ) ) = a i P ( i , t ) 2 + b i P ( i , t ) + c i F_(c)(P(i,t))=a_(i)P(i,t)^(2)+b_(i)P(i,t)+c_(i)F_{c}(P(i, t))=a_{i} P(i, t)^{2}+b_{i} P(i, t)+c_{i}
In the above equation a i , b i , c i a i , b i , c i a_(i),b_(i),c_(i)a_{i}, b_{i}, c_{i} denote cost coefficients of i th i th  i^("th ")i^{\text {th }} generator. In equation (3), φ ( i , t ) φ ( i , t ) varphi(i,t)\varphi(i, t) is the startup cost, which can either be hot or cold startup cost, given by,
在上述方程 a i , b i , c i a i , b i , c i a_(i),b_(i),c_(i)a_{i}, b_{i}, c_{i} 中表示发电机的成本 i th i th  i^("th ")i^{\text {th }} 系数。在等式 （3） 中， φ ( i , t ) φ ( i , t ) varphi(i,t)\varphi(i, t) 是启动成本，可以是热启动成本或冷启动成本，由下式给出：
φ ( i , t ) = { H S C ( i ) , if M D T ( i ) ≤ OFF h ( i ) ≤ C S h ( i ) + M D T ( i ) C S C ( i ) , if C S h ( i ) + M D T ( i ) < O F F h ( i ) φ ( i , t ) = H S C ( i ) ,  if  M D T ( i ) ≤ OFF h ( i ) ≤ C S h ( i ) + M D T ( i ) C S C ( i ) ,  if  C S h ( i ) + M D T ( i ) < O F F h ( i ) varphi(i,t)={[H_(SC)(i)","," if "M_(DT)(i) <= OFF_(h)(i) <= CS_(h)(i)+M_(DT)(i)],[C_(SC)(i)","," if "CS_(h)(i)+M_(DT)(i) < OFF_(h)(i)]:}\varphi(i, t)=\left\{\begin{array}{cc}H_{S C}(i), & \text { if } M_{D T}(i) \leq \operatorname{OFF}_{h}(i) \leq C S_{h}(i)+M_{D T}(i) \\ C_{S C}(i), & \text { if } C S_{h}(i)+M_{D T}(i)<O F F_{h}(i)\end{array}\right.

2.1.2. Reserve payment method B (payment for energy and capacity)
2.1.2. 储备金支付方式 B（能源和容量支付）

In method B, the GENCO is entitled for two types of payments for the amount of the reserve capacity that is used, i.e., capacity payment and energy payment. Both of these payments are dependent on the reserve
在方法 B 中，GENCO 有权就所使用的储备容量金额获得两种类型的付款，即容量付款和能源付款。这两项付款都取决于储备金
deployment probability. The GENCO will receive spot market price for the duration it generates electricity. On the other hand, the reserve price is paid for the amount of time in which the reserve capacity is not called to serve. Therefore, the total revenue can be expressed as follows.
部署概率。GENCO 将在其发电期间获得现货市场价格。另一方面，预留价格是针对未调用预留容量进行服务的时间量支付的。因此，总收入可以表示如下。
R V = ∑ t = 1 T ∑ i = 1 N G [ { ( P ( i , t ) ) E P ( t ) A ( i , t ) } ] + ∑ t = 1 T ∑ i = 1 N G { r E P ( t ) R ( i , t ) + ( 1 − r ) R P ( t ) R ( i , t ) } A ( i , t ) R V = ∑ t = 1 T   ∑ i = 1 N G   [ { ( P ( i , t ) ) E P ( t ) A ( i , t ) } ] + ∑ t = 1 T   ∑ i = 1 N G   { r E P ( t ) R ( i , t ) + ( 1 − r ) R P ( t ) R ( i , t ) } A ( i , t ) {:[RV=sum_(t=1)^(T)sum_(i=1)^(NG)[{(P(i","t))EP(t)A(i","t)}]+sum_(t=1)^(T)sum_(i=1)^(NG){rEP(t)R(i","t)],[+(1-r)RP(t)R(i","t)}A(i","t)]:}\begin{aligned} R V= & \sum_{t=1}^{T} \sum_{i=1}^{N G}[\{(P(i, t)) E P(t) A(i, t)\}]+\sum_{t=1}^{T} \sum_{i=1}^{N G}\{r E P(t) R(i, t) \\ & +(1-r) R P(t) R(i, t)\} A(i, t) \end{aligned}
The first term of (6) represents the energy payments to GENCO, while the second term represents the reserve payments. In the reserve payment, the first term ( r E P ( t ) R ( i , t ) ) ( r E P ( t ) R ( i , t ) ) (rEP(t)R(i,t))(r E P(t) R(i, t)) represents the amount paid to GENCO reserve when called upon to serve, which is a function of the reserve deployment probability. For the rest of the time, GENCO receives capacity payment ( ( 1 − r ) R P ( t ) R ( i , t ) ) ( ( 1 − r ) R P ( t ) R ( i , t ) ) ((1-r)RP(t)R(i,t))((1-r) R P(t) R(i, t)) for the amount of the capacity scheduled for reserve. Thus, the total profit earned by GENCO in Method B is given as follows.
（6） 的第一项代表向 GENCO 支付的能源款项，而第二项代表储备金。在储备金付款中，第一项 ( r E P ( t ) R ( i , t ) ) ( r E P ( t ) R ( i , t ) ) (rEP(t)R(i,t))(r E P(t) R(i, t)) 表示在被召唤服务时支付给 GENCO 储备金的金额，这是储备金部署概率的函数。在其余时间内，GENCO 将收到计划预留的容量的容量付款 ( ( 1 − r ) R P ( t ) R ( i , t ) ) ( ( 1 − r ) R P ( t ) R ( i , t ) ) ((1-r)RP(t)R(i,t))((1-r) R P(t) R(i, t)) 。因此，GENCO 在方法 B 中赚取的总利润如下。
profit = { ∑ t = 1 T ∑ i = 1 N G [ { ( P ( i , t ) ) E P ( t ) A ( i , t ) } ] + ∑ t = 1 T ∑ i = 1 N G { ( 1 − r ) E P ( t ) + r R P ( t ) } R ( i , t ) A ( i , t ) } − { ∑ t = 1 T ∑ i = 1 N G [ ( 1 − r ) { F c ( P ( i , t ) ) A ( i , t ) } + r { F c ( P ( i , t ) + R ( i , t ) ) A ( i , t ) } + { φ ( i , t ) A ( i , t ) } ] }  profit  = ∑ t = 1 T   ∑ i = 1 N G   [ { ( P ( i , t ) ) E P ( t ) A ( i , t ) } ] + ∑ t = 1 T   ∑ i = 1 N G   { ( 1 − r ) E P ( t ) + r R P ( t ) } R ( i , t ) A ( i , t ) } − ∑ t = 1 T   ∑ i = 1 N G   ( 1 − r ) F c ( P ( i , t ) ) A ( i , t ) + r F c ( P ( i , t ) + R ( i , t ) ) A ( i , t ) + { φ ( i , t ) A ( i , t ) } {:[" profit "={sum_(t=1)^(T)sum_(i=1)^(NG)[{(P(i,t))EP(t)A(i,t)}]+sum_(t=1)^(T)sum_(i=1)^(NG){(1-r)EP(t):}],[+rRP(t)}R(i","t)A(i","t)}-{sum_(t=1)^(T)sum_(i=1)^(NG)[(1-r){F_(c)(P(i,t))A(i,t)}:}],[{:+r{F_(c)(P(i,t)+R(i,t))A(i,t)}+{varphi(i,t)A(i,t)}]}]:}\begin{aligned} \text { profit }= & \left\{\sum_{t=1}^{T} \sum_{i=1}^{N G}[\{(P(i, t)) E P(t) A(i, t)\}]+\sum_{t=1}^{T} \sum_{i=1}^{N G}\{(1-r) E P(t)\right. \\ & +r R P(t)\} R(i, t) A(i, t)\}-\left\{\sum _ { t = 1 } ^ { T } \sum _ { i = 1 } ^ { N G } \left[(1-r)\left\{F_{c}(P(i, t)) A(i, t)\right\}\right.\right. \\ & \left.\left.+r\left\{F_{c}(P(i, t)+R(i, t)) A(i, t)\right\}+\{\varphi(i, t) A(i, t)\}\right]\right\} \end{aligned}
Where, the second term of (7) represents the total cost of operation incurred by the GENCO for energy as well as reserve market participation [37].
其中，（7）的第二项代表 GENCO 在能源和储备市场参与方面产生的总运营成本 [37]。

2.2. Constraints  2.2. 约束

Similar to the UC problem, all the thermal unit constraints are applicable to PBUC problem as well. However, the system constraints of UC problem undergo small variations in the context of self-scheduling approach in PBUC problem. The details of system constraints and thermal unit’s operation constraints are explained in the following sections.
与 UC 问题类似，所有热单位约束也适用于 PBUC 问题。然而，在 PBUC 问题中，UC 问题的系统约束在 PBUC 问题的自调度方法的背景下发生了微小的变化。以下各节介绍了系统约束和 Thermal Unit 的运行约束的详细信息。

2.2.1. System constraints
2.2.1. 系统约束

2.2.1.1. Load constraint. The load constraint in profit based unit commitment is different from the traditional unit commitment and suggests that the generators may or may not satisfy the whole load of the network and may generate less than or equal to the forecasted load requirements of the system.
2.2.1.1. 负载约束。基于利润的单位承诺中的负载约束不同于传统的单位承诺，这表明发电机可能会也可能不会满足网络的整个负载，并且可能会产生小于或等于系统预测的负载要求。
∑ i = 1 N G P ( i , t ) ≤ P l ( t ) ∑ i = 1 N G   P ( i , t ) ≤ P l ( t ) sum_(i=1)^(NG)P(i,t) <= Pl(t)\sum_{i=1}^{N G} P(i, t) \leq P l(t)