Two-Stage Multi-Objective Unit Commitment ... - IEEE Xplore

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Two-Stage Multi-Objective Unit Commitment Optimization Under Hybrid Uncertainties Bo Wang, Shuming Wang, Member, IEEE, Xian-zhong Zhou, Member, IEEE, and Junzo Watada, Member, IEEE

Abstract—Unit commitment, as one of the most important control processes in power systems, has been studied extensively in the past decades. Usually, the goal of unit commitment is to reduce as much production cost as possible while guaranteeing the power supply operated with a high reliability. However, system operators encounter increasing difficulties to achieve an optimal scheduling due to the challenges in coping with uncertainties that exist in both supply and demand sides. This study develops a day-ahead two-stage multi-objective unit commitment model which optimizes both the supply reliability and the total cost with environmental concerns of thermal generation systems. To tackle the manifold uncertainties of unit commitment in a more comprehensive and realistic manner, stochastic and fuzzy set theories are utilized simultaneously, and a unified reliability measurement is then introduced to evaluate the system reliability under the uncertainties of both sudden unit outage and unforeseen load fluctuation. In addition, a cumulative probabilistic method is proposed to address the spinning reserve optimization during the scheduling. To solve this complicated model, a multi-objective particle swarm optimization algorithm is developed. Finally, a series of experiments were performed to demonstrate the effectiveness of this research; we also justify its feasibility on test systems with generation uncertainty.

SR

Spinning reserve.

ORR

Outage replacement rate.

VaR

Value-at-risk.

FVaR

Fuzzy value-at-risk.

MO-UC

Multi-objective unit commitment.

MO-PSO

Multi-objective particle swarm optimization.

MILP

Mixed integer linear programming.

CPM

Cumulative probability method.

COPT

Capacity outage probability table.

PL

Priority list.

DPL

Descending order of priority list.

Constants Index of each scheduling period. Index of each generation unit. Time horizon. Set of generation units.

Index Terms—Multi-objective, stochastic and fuzzy uncertainties, unit commitment, value-at-risk.

Startup cost of unit . Shutdown cost of unit .

NOTATIONS

Generation cost coefficients of unit . Emission cost coefficients of unit .

Acronyms

Uncertain load of period .

UC

Unit commitment.

SUC

Stochastic unit commitment.

Lower and upper bounds of Maximal output of unit .

RUC

Robust unit commitment.

Minimal output of unit .

SOs

System operators.

Ramp rate of unit .

EENS

Expected energy not served.

Failure rate of unit .

LENS

Largest energy not served.

Outage replacement rate of unit .

LOLP

Loss-of-load probability.

System lead time.

.

Set of outage units. Minimal on time of unit . Manuscript received December 25, 2014; revised April 02, 2015 and June 18, 2015; accepted July 23, 2015. This work was supported by the Fundamental Research Funds for the Central Universities (No. 011814380003, No. 2062014286). Paper no. TPWRS-01742-20114. B. Wang and X. Zhou are with the School of Management and Engineering, Nanjing University, Nanjing 210093, China (e-mail: [email protected]; [email protected]). S. Wang is with the Faculty of Engineering, National University of Singapore, 117576 Singapore (e-mail: [email protected]). J. Watada is with the Graduate School of Information, Production and Systems, Waseda University, Fukuoka, Japan (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPWRS.2015.2463725

Minimal off time of unit . Functions Generation cost function of unit at . Emission cost function of unit at . Worst case evaluation at . VaR-based reliability measurement. Variables On/off (1/0) status of unit at .

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Output of unit at . Startup action of unit at . Shutdown action of unit at . Confidence level on outage uncertainty. Confidence level on load uncertainty. Spinning reserve of unit at . Periods that unit has kept on. Periods that unit has kept off. I. INTRODUCTION

U

NIT commitment (UC) in power systems is a decisionmaking process of scheduling and dispatching power generation resources so that the supply system can be operated with low cost but high reliability. Today's energy shortage and environmental concerns have made the UC optimization increasingly significant [1]. However, there are also some challenges arising when we try to find a satisfying UC decision. From the literature, these difficulties are generally stemmed from uncertainties that exist in both supply and demand sides, e.g., the uncertainty due to sudden unit outage, the uncertainty on account of unforeseen load fluctuation and the uncertainty associated with key cost factors such as fuel prices. In order to handle the above uncertainties, various models have been proposed by researchers with stochastic and/or robust concerns [2]–[8]. The idea of stochastic UC (SUC) uses scenario-based uncertainty representation in formulation [9]. In [4], the expected energy not served (EENS) and the loss-of-load probability (LOLP) were used to evaluate the reliability of a UC scheduling under the uncertainty of unit outage. Then, an offline method for setting the spinning reserve (SR) is developed prior to the traditional UC to prevent the load shedding with economic concerns. Similarly in [5], the EENS was utilized as a probabilistic criterion of a UC plan as well. Besides the SR, interruptible load was also taken into account as an operating reserve facility. Although, these two studies applied different approaches to build the optimization models, the future load uncertainty was not well addressed. Recently, Trivedi [6] et al. developed a multi-objective UC model which optimizes both the system cost and supply reliability. In addition, the uncertainty due to load forecast error was also incorporated by using a seven-step approximation while treating the future loads as normal distribution variables. However, the authors ignore the spinning reserve optimization and the unit ramp rate limits. Although stochastic programming has been recommended as a tailor-make tool to handle UC under uncertainties, problems and barriers do exist. In some real applications with huge number of scenarios, system operators (SOs) are in doubt about the complexity and transparency of the SUC models since the efficacy and high computational requirements still need to be further addressed [9]. Compared with SUC, robust UC (RUC) aims to capture randomness by using uncertainty sets, rather than exact probability distributions. Then, a solution of regular RUC (single-stage) must hedge against all possible realizations

within the uncertainty sets. In other words, RUC provides security against the worst-case scenario, but produces very conservative UC scheduling, which will translate to a high operation cost. In order to mitigate this disadvantage and obtain an adequate tradeoff between conservativeness and reliability, recent studies mainly improve the conventional RUC from two aspects: One is the development of two-stage RUC models, the other is the design and operation on the uncertainty set. Compared with single-stage models, two-stage RUC is less conservative and more cost-effective as it essentially enables the SO a recourse opportunity [10]. Consequently, various two-stage RUC formulations have been developed and implemented over the last few years, to ensure reliable power generation and dispatch [10], [11], [12]. From the uncertainty set perspective, which has a direct impact on the conservativeness of the final solution, the larger (wide coverage) the set is, the more conservative the solution will be. Accordingly, the notion of a budget of uncertainty is used to help balance between cost and reliability by improving the definition and/or adding different kinds of constraints on the uncertainty set [9]. For example, when a single uncertainty set cannot precisely describe the random factor, one may utilize different uncertainty sets to jointly define it; when an SO wants to adjust the level of overall reliability, he may update the scope of the uncertainty set or assign bounds on the worst case performance as constraints. As a result, the standard two-stage RUC can be improved to capture different random situations and balance the cost and reliability to meet various requirements. In this manner, study [10] applied multiple sets to improve the uncertainty description in RUC, while each set is assigned with a weight coefficient. The total allowed number of generation output cases introduced in [13] can be modified to enhance either reliability or cost concerns. The maximum allowed number of equipment failures defined in [14] and [15] can be varied as well to adjust the budget of uncertainty in contingency models. Nevertheless, there are two major concerns on the RUC approach: 1) it is known that extreme scenarios play a critical role in RUC; 2) the uncertainty set cannot include sufficient probabilistic information on the available data or imprecise knowledge from the experts' estimates, i.e., the information utilization level is low. Therefore, due to the nature of RUC, for any deterministic uncertainty set (even the bounded one), the optimization process may still pay excessive attention on the extreme scenarios which typically happen much less frequently. Although the modeling process and the computation time for RUC models are more acceptable than that of SUC, the obtained solutions bear underlying conservativeness. Based on the above analysis, it is necessary to develop UC models with both stochastic and robust concerns. Recently, a unified stochastic and robust UC model was proposed in [8], which assigns weight to the components of the stochastic and robust parts in the objective function to achieve a low operation cost while ensuring the supply system robustness. In this study, we also try to build a mathematical model that takes the advantages of both SUC and RUC, however different from most existing studies, the following three approaches are introduced to achieve a novel decision-making process of UC.

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First, the uncertainties due to unit outage and load forecast error are addressed respectively by using stochastic theory and fuzzy set theory. The reason mainly exists in the difference between these two types of uncertainties: The unit outage uncertainty normally can be represented as a state variable with binary outputs, i.e., a unit is either available or unavailable for generation, while the load uncertainty actually is with infinite outputs or should be approximated by using various discrete scenarios. Theoretically, it is much less challenging to construct a probability distribution of the unit outage uncertainty than that of the power load. In addition, there have been some statistical records on the failure rate of many units, and the time dependent outage replacement rate (ORR) can be obtained approximately by using some existing approaches [16] as well. Therefore, stochastic theory can be a reasonable tool to describe the unit outage uncertainty. However, in view of the load forecast process, some problems arise when we still use stochastic method to represent the load uncertainty. On the one hand, the various inputs such as precipitation, humidity, wind speed and temperature which form the basis of a load forecast, are often assessed with, more or less, some level of ambiguity. For example, when talking about the wind speed, one may use expression such as light air, strong breeze or a fresh gale. Such ambiguity (or vagueness) is usually non-statistical due to the practical difficulty of data acquisition, the nature of imprecision of the measurement and the ambiguity of human perception, as a joint fact in many real-world applications. Especially, in some practical situations, the reliable and precise data may not be sufficient enough to form a credible exact probability distribution. On the other hand, a predicted load is always subject to some error which can not be entirely accurate during the existing load forecast process. Therefore, to enhance the quality of forecasts and decision-making, an experienced SO may intend to incorporate his knowledge, based on his understanding and observations on the supply system with the available data, to construct and adjust the future load. This process involves linguistic knowledge processing and cannot be easily addressed by using probability theory, as explained in next section. Recently, fuzzy set theory has been applied as a widely acceptable tool which can deal with the above sophisticated linguistic knowledge (non-statistical uncertainties that exist in the load forecasts and SOs' knowledge on the supply system) appropriately. It is noteworthy that, one of the significant characteristics of fuzzy set theory is its linguistic elasticity, which accounts for ambiguities in an uncertain environment without becoming restricted by detailed level information [17]. This feature nicely matches the hybrid uncertainty representation of power load: First, using some probability-possibility consistency principles, any probability distribution function can be transformed into an appropriate membership function of fuzzy sets [17], [18], [19], without information loss, i.e., fuzzy variables can involve data information as well; Second, due to the flexibility of membership function, fuzzy variables can be adjusted in a much easier manner to involve linguistic knowledge than random variables. Therefore, compared with the probability distribution in SUC, the membership function of fuzzy load could incorporate more comprehensive information, on the basis of available data and

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linguistic knowledge; on the other hand, in contrast to the uncertainty set in RUC, fuzzy sets include date information and human knowledge of future events, thus can mitigate the over conservativeness of the solution rooted from the extreme scenarios concentration. Considering the above merits, it could be reasonable to apply fuzzy set theory as a suitable tool to handle the load uncertainty. It is also noticed that, there are various factors that influence the prediction of the unit outage rate, and the SOs' experience could be involved as well to improve the forecast. However, different from the load uncertainty, the Bernoulli distribution of the unit outage rate can be adjusted easily to incorporated the human knowledge. In this sense, the state probability distribution can serve essentially as a subjective probability with linguistic knowledge which is functionally equivalent to the membership function of fuzzy sets. In sum of the above considerations, we use probability theory to model the unit outage rate and fuzzy set theory to describe the load uncertainty, then the UC in this study is established as an optimization problem under stochastic and fuzzy uncertainties. Second, a unified reliability measurement is developed based on fuzzy value-at-risk (FVaR) to evaluate the supply system reliability under the aforementioned hybrid uncertainties. Actually, VaR has been popularly-used in many engineering problems, including power systems [7], [20], [21]. The VaR of an investment is the likelihood of the greatest loss under a given confidence level [22], and in the UC optimization, the VaR can be applied to measure the largest EENS under a confidence level predefined by the SO. In this research, we integrate the ORR and the load uncertainty into the FVaR to obtain a comprehensive evaluation of system reliability. The detailed knowledge of this measurement is introduced in late portions. Third, most of the existing studies construct the UC optimization as single-objective problems which minimize the total cost while taking the reliability as a constraint. Generally speaking, this type of models are concise and easy to be solved. However, sometimes the SOs might be interested in the inherent contradictions between the cost and the reliability, then they are able to select a UC scheduling that satisfies them best after analyzing various feasible solutions. Based on this consideration, we develop a multi-objective UC (MO-UC) model which optimizes the cost and the reliability simultaneously. Particularly, the total cost is calculated with environmental concerns in terms of emission cost. The remainder of this paper is organized as follows: Section II introduces the related knowledge on fuzzy set theory including FVaR. In Section III, we first provide the objective functions of the cost and the reliability, then build the mathematical model of the MO-UC. Section IV develops a multi-objective particle swarm optimization (MO-PSO) algorithm to solve the proposed model. Then, in Section V, we provide test system-based experiments to exemplify the effectiveness of this research. Moreover, a case study on generation mix UC optimizations is also performed to justify the feasibility of the model and algorithm when handling generation uncertainty. Finally, Section VI summarizes our conclusions.

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II. PRELIMINARIES A. Fuzzy Set Theory As introduced, the load uncertainty in this research is addressed by fuzzy set theory, and we use FVaR as a base to evaluate the system reliability. Therefore, in the following part, we provide some necessary concepts and features of fuzzy set theory. Let be a fuzzy variable with membership function , and is a real number. The possibility and necessity of event are expressed as follows [23]: (1) (2) The credibility measure is formed on the basis of the above knowledge (3) VaR of an investment measures the largest loss under a predefined confidence level [22]. Recently, researchers have introduced the VaR in fuzzy environment [24], as follows: Suppose is a variable that represents the fuzzy loss of one investment, then the FVaR of under confidence level is defined as follows: (4) where . Equation (4) tells us that the greatest loss under confidence level is . Recently, the FVaR has been applied to build optimization models in several engineering topics such as facility location [24] and portfolio selection [25]. And this approach can also be applied in the UC optimization: suppose denotes the imprecise future load, then (.) could be formed as a robust reliability measurement function which evaluates the largest energy not served (LENS) under an SO predefined confidence level. In addition, an SO may adjust the to obtain a tradeoff between conservativeness and optimality. Detailed knowledge about this function is introduced in Section III. B. Advantages of Using Fuzzy Set Theory Fuzzy set theory introduced by Zadeh [26] has been applied in many engineering problems, however few in power systems especially in the UC optimization. Although the motivations of using fuzzy set theory have been presented in Section I, we provide a more detailed discussion here to show the merit of implementing membership function to describe and adjust the load forecast uncertainty. First and foremost, fuzzy variable with a membership function is conceptually easier than random variable to understand as the mathematical concepts behind fuzzy reasoning are not complicated.

Fig. 1. Advantage of describing future load as fuzzy variable.

Second, fuzzy set theory is designed to allow the decisionmakers to incorporate the linguistic knowledge in a very flexible way. For instance, an SO in our UC problem can construct and modify (update) the load forecast based on his experience and observations on the supply system. Third, the membership function has a simpler mathematical structure than the probability distribution. Therefore, compared with probability distribution, it is much more convenient to modify in computation once information updates. This can be justified by the following simple example: Suppose that, based on available data, we obtain a random variable and a fuzzy variable respectively, to describe the future load of a scheduling period, as shown in the left side of Fig. 1. These two variables are assumed to be in normal distribution, then the SOs' experience is incorporated to adjust them to and respectively, for example the revised results are listed in the right side of Fig. 1. Now, some difficulties may arise for the treatment of : In order to incorporate the SO's knowledge into , one needs to design another function transformation system to meet both requirements: 1) the knowledge is incorporated in a correct and expected way; 2) the resulting function with knowledge added is still a probability density function. Such system in general is not easy to accomplish, by noting that both 1) and 2) need time and recourse to justify, and that sometimes 1) and 2) may conflict with each other. However, in view of fuzzy variable , it can be used directly to the UC optimization due to the flexibility of membership function of fuzzy sets. Therefore, we can say that it is more convenient to obtain a linguistic knowledge-based fuzzy variable than that of a random variable. We also noticed that probability theory as the most popular way to address uncertainty, has been well-developed in the past several decades. Obviously, it also outperforms fuzzy set theory in situations when the probability distribution is easy to calibrate. However, based on the feature of load forecasts, this study adopts fuzzy set theory as a more suitable mathematical vehicle to model the load uncertainty. III. MODELING The proposed MO-UC optimizes the on/off state and output (also available spinning reserve) of each unit over the scheduling horizon, to obtain a series of Pareto-optimal solutions which minimize the cost and maximize the reliability subjects

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to several constraints. The mathematical model of the MO-UC is constructed as follows. A. Objective Functions 1) Total Cost With Environmental Concerns: The total cost of a supply system includes the transition cost, the generation cost and the emission cost over the scheduling horizon:

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caused by the th uncertainty under confidence level of period , and reveals the minimal generation capacity caused by all the factors in . Finally, we can obtain the supply reliability under the realization of all the uncertainties by using (8). If only consider the hybrid uncertainties of ORR and load fluctuation, (8) degenerates to the following: (9)

(5) where is calculated as and equals . means the generation cost of unit with output , which is generally expressed as the following quadratic function:

where can be formed as a second stage optimization problem given as follows:

(6) where the numerical values of , , and are determined by the attributes of unit . It is noted that, this equation is an approximation of the generation cost, where some key cost factors such as fuel prices are not considered. Actually, recent SOs are faced with the challenge of dealing with increasing and volatile fuel prices, which not only has an impact on the cost of a generation company, but also increases the uncertainties involved in the UC optimization. From the literature, some researchers have investigated the uncertain fuel prices-based UC problems. For example, reference [27] analyzed the future generation portfolio on the all-Ireland system with respect to the impact of carbon costs. In addition, an assessment is made of the exposure of the portfolios to fuel price volatility and how portfolios may wish to diversify to avoid this. The readers may refer to [27] for the detailed knowledge. Function in (5) measures the emission cost which is similar to the generation cost function [6]: (7) and vary to different unit , and where the values of , it is assumed that the startup cost consists of the emission cost incurred during startup. 2) FVaR-Based Unified Reliability Measurement: We first introduce a generalized reliability measurement which could address the UC optimization under various uncertainties, then a special case is discussed in detail when considering the unit outage and load uncertainties. Suppose is a collection of all the uncertainties (except the load uncertainty) exist in the UC optimization, described as either fuzzy or random variable, represents the uncertainty, then a generalized reliability measurement is given as follows: (8) where is a fuzzy variable represents the forecast load of period , is the confidence level on , is a predefined confidence level on the th uncertainty, and denotes the related variables to . Then, calculates the worst case (the minimal remain generation capacity)

(10) where is determined by the smaller value of the remain capacity and ramp rate constraint of unit at period , and is a collection of outage units under confidence level . To solve this second stage MILP problem, we develop the following cumulative probability method (CPM), which is an algorithm with explicit formulations and can lead to an exact (also unique) solution. [Step 1]. Compute the ORR of each unit in this supply system, denoted as . The ORR of unit during a predefined is [6] (11) [Step 2]. For each period of a UC decision, we first compute the probability that all committed units are available, i.e., no outage happen in , expressed as : (12) [Step 3]. Then, for period , we calculate the committed unit-dependent outage probabilities of any single ( ), double ( ), triple ( ), and more (denoted as : ) units. The maximal number of is determined by the value of and different systems that include different units. Say, sufficient scenarios are considered in this step for the future calculations. For example, considering the unit on/off state, the probability of -unit outage in period is calculated as follows: (13) The above equation shows that the value of is equal to 0 if any unit in is off-line, i.e., this accident does not exist in period . Then, the probability values as well as the related unit knowledge are recorded in an archive in a descending order. Suppose is the element in , then it contains the following information: (14)

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B. Constraints There are several constraints to be considered when optimizing the above objectives. 1) Flexible Power Demand Requirement: In conventional UC optimization, the total generation of period (usually one hour) must be equal to an exact forecast load (18)

Fig. 2. Calculation of FVaR.

[Step 4]. Set probability :

, we calculate the following cumulative

In this research, such load is characterized by imprecise variable, therefore we modify the above power demand balance to a flexible constraint as follows:

(15)

(19)

with : if , Compare and repeat this step; otherwise, record the final value of as . [Step 5]. Based on (10), we first obtain the of the committed units in period ; then for each scenario , the is obtained as follows:

The following constraint is also utilized to ensure the realization of (19): (20) 2) Ramp Rate Limits: The ramp rate constraints can be formed as the following inequation:

(16) Using the above CPM to calculate the , we obtain the remain generation capacity at the worst case under confidence level . From the literature, the capacity outage probability table (COPT) [28] has been widely used to measure the EENS and the LOLP. The COPT can be visualized as a table with rows (when units in the system) and 3 columns (outage amount; outage probability; remain capacity). However, the main difference between the COPT and the proposed CPM is: Our approach involves not only the outage amount and the probability values, but also the knowledge of specific units that cause the accident. The significance of this point is, the spinning reserves of all committed units can be addressed in the UC optimization as well, since these values are disclosed directly in the CPM. It is also noticed that the runtime cost of the proposed CPM is higher than that of the COPT, as the above steps are repeated in each scheduling period. Nevertheless, the execution time of the whole algorithm is still reasonable when we use some high-speed computer, which can be proved by experimental experience in Section V. Based on the above analysis, we achieve the remain capacity due to sudden unit outage. Then, this value is integrated in the following equation to obtain a unified reliability measurement under the hybrid uncertainties of unit outage and future load: (17) is a fuzzy variable evaluates the power where shortage of period . Then, the FVaR operator is employed to measure the LENS under confidence level of , as shown in Fig. 2. It is noticed that and could be two different values assigned by the SOs.

(21) 3) Spinning Reserve Limits: As mentioned before, the spinning reserve is equal to the smaller value of the remained capacity and the ramp rate constraint of unit at period

(22) 4) Generation Limits: (23) 5) Minimum On/Off Time: The unit should remain the on/off state for a fixed time before it can be changed:

(24) C. Mathematical Model of the MO-UC Based on the above knowledge, the mathematical model of the MO-UC is established as problem (25), which aims to find the best power dispatch (as well as the related spinning reserve) to minimize the total cost and maximize the system reliability simultaneously. This model can also be adjusted to handle the UC problems with specified environmental concerns. e.g., for systems with rigid control on and/or , we may set emission level as a constraint, then optimize the UC scheduling when ensuring the emission lower than a predefined amount; in a more relaxed environment, the emission level could be taken as another objective and optimized together with the generation cost and the supply reliability. Besides, the objective functions as well as the constraints (especially the time-dependent ramp

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rate) of the hourly day-ahead MO-UC could be further improved to solve the optimization with a finer granularity of the scheduling period (e.g. half-hourly).

iterations. From the literature, PSO algorithm has been used in the UC optimization, readers may refer to [1] and [31].

IV. SOLUTION ALGORITHM

The classical PSO is a single-objective problem which cannot be applied directly to solve the MO-UC. Then, based on our previous studies [1], [25], [32], we develop an MO-PSO algorithm which can be summarized as follows. [Step 1]. Particle initialization: Consider a -unit -period UC problem, we first initialize particle swarm . The particle contains two matrices: a binary matrix (BM) denotes the on/off state of each unit over the scheduling horizon and a real value matrix (RM) expresses the corresponding power dispatch, as shown in the following:

The solution method to the UC problem has been studied extensively over the past decades. Among these techniques, the priority list (PL), branch-and-bound and Lagrangian relaxation methods are based on exact procedures [1], [29]. Nevertheless, the MO-UC is a complicated nonlinear problem with two objectives which cannot be solved by methods with explicit formulations:

B. MO-PSO

(28)

The initial values of and are generated randomly, where equals to either 0 or 1, is constrained by the generation limits. [Step 2]. Priority list: Make the PL of all units based on the ascending order of the average fuel cost ($/MW) at their maximum outputs. The priority coefficient of unit is represented as (29)

(25) Recently, meta-heuristic algorithms have been applied to handle such problems. This type of solutions are iterative search techniques that can find not only the local optimal solutions but also approximate global optima. Therefore, in this study, we develop a MO-PSO algorithm to solve the MO-UC. A. PSO Algorithm PSO was initially proposed by Kennedy et al. [30]. Suppose particles search in a -dimensional space for iterations, then the position of particle is expressed as , and the velocity and position are updated according to the following equations:

[Step 3]. BM adjustment: The BM initialized above is revised to satisfy constrains such as power demand requirement and minimum on/off time. [Step 3.1]. The total capacity of all committed units at period is examined by the following approach: (30) If , we commit an off-line unit with the lowest until . [Step 3.2]. Then, the BM is modified by using the following heuristic method to remove any violation of the minimum on/off time. [Step 3.2.1]. Compute the current on/off periods of all units over the scheduling horizon: (31) (32)

(26) (27) where means the velocity of particle at dimension with an upper and lower bounds of and , is an inertia weight, and are learning rates of the swarm, is a random value in , is the personal best tells the best position of particle in the past iterations, and is the global best reflects the best position of all particles after

[Step 3.2.2]. Set [Step 3.2.3]. Set [Step 3.2.4]. If . [Step 3.2.5]. If

(From period 1). (From the first unit). , , and

, , , we set . , , [Step 3.2.6]. If , is any integer, we set

, we set , and , and

.

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[Step 3.2.7]. Use (31) and (32) to update the and . [Step 3.2.8]. If , , then go to Step 3.2.4. [Step 3.2.9]. If , , then go to Step 3.2.3. Otherwise, stop the algorithm. [Step 3.3]. The above steps solve the minimum on/off problem by committing various off-line units, which may result in a redundancy of generation capacity. Then, the following method is applied to mitigate this disadvantage. Based on the descending order of priority list (DPL), we check the committed units one by one to detect whether some units can be de-committed without violating the power demand requirement and minimum on/off time. [Step 3.3.1]. Set . [Step 3.3.2]. Make the DPL of the committed units. Suppose the first unit in DPL is . [Step 3.3.3]. Compute the excessive capacity as (33)

Similar method can be developed in this step to repair the violation of , in which the DPL instead of the PL is used. [Step 4.4]. If , , go to Step 4.3. Otherwise, stop the algorithm. [Step 5]. Ramp rate constraints: the of unit has effects on not only the but also the output variation during two continuous periods. Therefore, the RM should be adjusted to remove any deviation from the ramp rate constraints. [Step 5.1]. Compute the output variations of all units in RM: (37) [Step 5.2]. Set when ). [Step 5.3]. Set [Step 5.4]. If Otherwise, if

(the unit initial states are considered . , , go to Step 5.5. , we first compute (38)

[Step 3.3.4]. If , we delete from DPL. If but de-commit unit will go against the minimum on/off time, we delete from DPL. If and de-commit unit will not violate the minimum on/off time, we then de-commit , remove from DPL and update the on/off periods of all units. [Step 3.3.5]. If DPL is not empty, go to Step 3.3.3. [Step 3.3.6]. If , , go to Step 3.3.2. Otherwise, stop the algorithm and record the BM matrix. [Step 4]. RM adjustment: the RM has been generated randomly considering the generation limits, then this matrix should be revised according to the above BM to obtain a power dispatch that fulfills the flexible power demand balance. [Step 4.1]. First, we multiply the corresponding elements of BM and RM to obtain a comprehensive matrix (CM):

(34)

. [Step 4.2]. Set [Step 4.3]. Calculate , if , , the following go to Step 4.4. Otherwise, if approach is applied to revise the above matrix. [Step 4.3.1]. We first calculate (35) [Step 4.3.2]. Select the first element in PL, denoted as , remove from PL. Otherwise, calculate

. If

(36) [Step 4.3.3]. If , then Step 4.4. Otherwise, from PL and go to Step 4.3.2.

, go to , remove

and let . Then, according to the DPL, we dispatch to specified units subject to the ramp rate and generation constrains. Similar method can be developed as well to handle the problem that , where the PL instead of the DPL is used. [Step 5.5]. If , go to Step 5.4. Otherwise, go to Step 5.6. [Step 5.6]. If , , go to Step 5.3. Otherwise, stop the algorithm and record matrix BM. [Step 5.7]. Finally, we multiply the BM and RM again to obtain a new matrix CM. [Step 6]. Fitness values: The matrix CM obtained above can be taken for a possible UC decision. Then, we calculate the objective values of the total cost and the reliability according to Equations (5) and (9), where the CPM is used. [Step 7]. Iterations: The multi-objective mechanism introduced in [1] is employed to determine the Pbest and Gbest of each iteration, whereafter the velocity and position of each particle in are updated. It is noticed that, the particle involves both the BM and RM, which are matrices with obviously different values. Therefore, two different types of settings are applied to PSO parameters such as and . [Step 8]. Results: Record the Pareto-optimal solutions after executing the MO-PSO for a predefined iterations. V. EXPERIMENTAL ANALYSIS In this section, a test system of 26 thermal units [16] (at a single bus with no transmission network) is employed to exemplify the effectiveness of the proposed model and algorithm. The unit ramp rate is considered as 40% of its maximal output, while the emission cost coefficients are assumed to be equal to 5% of the related generation cost coefficients, and the unit failure rates are listed in Table I. The dispatch period is considered as 24 h, and for the sake of convenience in expressions and calculations, the future loads are described as either normal distributed fuzzy variable ( ) with 3% standard deviation or trapezoidal one with 10% deviation on the exact data

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TABLE I UNIT FAILURE RATES (PER HOUR)

TABLE II 24-H FORECASTED LOADS (MW) OF THE 26-UNIT TEST SYSTEM

Fig. 3. Performance of the MO-UC and the MO-PSO.

TABLE III PARAMETER SETTINGS

Fig. 4. Pareto-optimal solutions to different

values when

.

Fig. 5. Pareto-optimal solutions to different

values when

.

given in [16], as shown in Table II. Then, all of the following experiments were implemented with C code on a Dell i7-4790 3.6-GHz CPU personal computer, where the knowledge of parameter settings are provided in Table III. A. Performance of MO-UC and MO-PSO We first exemplify the effectiveness of the proposed model and algorithm by solving the above UC optimization. The final Pareto-optimal solutions are shown in Fig. 3, while the total runtime cost is 5427.6 s. And, negative values of FVaR means the remain capacity of the supply system is able to satisfy the demand even the accidents happen. In MO-UC, the system reliability is evaluated by the summation of the FVaR over the scheduling horizon, and a smaller FVaR value represents a higher level of reliability. Then, Fig. 3 shows that the system reliability is improved with the raising of the operation cost, which is consistent with the real-life applications: on the one hand, in order to mitigate the influences caused by the load fluctuation, an SO has to increase the total generation capacity of the supply system; on the other hand, the outputs of units especially those with high priority may not arrive at their maximum, to avoid great loss due to sudden unit outage. Both of these operations will increase the total cost of the supply system. Therefore, we can say that the proposed model reflects the inherent conflict between the cost and the reliability. Additionally, the solutions in red square, for example (704475.1, -9.4) or (704272.4, 34.4) shows that the risk under confidence levels of and

is approximately equal to 0. Therefore, the related UC scheduling could be acceptable for conservative SOs when making the final decisions. B. Effect of FVaR measurement In this portion, we discuss about the performance of the FVaR and its variations to different confidence levels. Figs. 4 and 5 provide the final Pareto-optimal solutions to several and values. In Fig. 4, it can be observed that the Pareto-optimal solutions of smaller values dominate that of a higher . The reason is that, in the same supply system, a higher confidence

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TABLE IV COMPARISONS WITH THE EXISTING METHODS: FVAR

level on sudden unit outage generally involves more serious accidents than that of a lower confidence level. For example, in this 26-unit system, a confidence level of 0.95 could consist of two events: all committed-units are available and only one unit outage, while accidents such as two unit outages should be addressed if the confidence level raises to 0.97. In addition, the outage of two units with the largest capacities also results in the big gap between the Pareto-optimal solutions of and . Similarly, Fig. 5 also illustrates that the Pareto-optimal solutions of a higher is dominated by that of smaller ones, as the worst case LENS increases when the SOs apply higher confidence levels on the load uncertainty. Therefore, it can be concluded that the FVaR measurement is sensitive to different confidence levels of the unit outage and load uncertainties. C. Comparisons with Conventional Methods One of the main contributions of this study is the development of FVaR, which measures the system reliability under hybrid uncertainties of sudden unit outage and unforeseen load fluctuation. Due to the essential differences between stochastic and fuzzy set theories, it might be difficult and not meaningful to compare the above experimental results with that of the conventional methods. However, the advantages of FVaR still can be indicated in the following Table IV. As shown in Table IV, the classical LOLP cannot guarantee the system fulfil the demand in some extreme conditions, while the robust optimization normally results in a very-conservative scheduling. However, the FVaR developed in this study involves the worst-case analysis and the confidence levels could be assigned by SOs according to their understanding on the system to obtain a proper decision. Although, the runtime cost of the FVaR is relative higher than that of RUC, the execution time of the MO-PSO is still reasonable for this case study, e.g., the average cost of 18 running is 5498.8 s. D. A Case Study on UC Optimization With Wind Penetration We also notice that, the increasing penetration of renewables such as wind power introduces salient difficulties to solve the generation mix UC problems, and some recent studies have incorporated generation uncertainty in modeling [33], [34]. Therefore, in this portion, we provide the following case study to exemplify the feasibility of our approach when solving those optimizations with wind penetration. The aforementioned 26 thermal units test system (SYS) is modified as follows to involve wind generation. Test system 1 (SYS1): Three thermal units of SYS are replaced by various wind turbines with the same total generation capacity, as shown in Table V, where its right side shows the

TABLE V MODIFICATIONS ON THE 26 THERMAL UNITS TEST SYSTEM

quantity and attribute of each type of wind turbine, and , , , are the rated output, cut-in speed, cut-out speed and rated speed of wind turbine . Test system 2 (SYS2): The wind turbines listed in Table V are added to SYS while no thermal unit is removed. For the sake of convenience in expression and calculation, the wind turbines are assumed to be in the same location, and wind speed uncertainty of each period is modeled as the same fuzzy Gaussian distribution . The other knowledge such as thermal unit data and future load remain unchanged. Then, the generation technology mentioned in [35] and [36] is employed to obtain an imprecise output of wind turbine at period , and the total generation of all turbines at period is computed as (39) Based on the above knowledge, a realization of (8) is introduced to handle the imprecise wind generation together with the load forecast and unit outage uncertainties:

(40) where the formulation of has been explained in (10), reflects a reliable output of all wind tur, which is formed by using the bines under confidence following -cut method [37]

(41) Then, the related portions of the MO-PSO are updated to solve the UC problems of SYS1 and SYS2. Especially, fuzzy simulation [38] is employed to realize the calculation of (39) and (41), the readers may refer to [25] and [39] for the detailed knowledge of this approach. Finally, we report the corresponding experimental results and discuss about the difference between UC optimization with and without wind penetration. Fig. 6 shows the Pareto-optimal solutions obtained by the original or revised MO-PSO after solving the optimization problem of each test system, while , , and . The other parameter values are set the same as Table III. From cost perspective, it is obviously that, the operation expense of a generation mix system (SYS1) is lower than that of

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Fig. 6. Pareto-optimal solutions of test systems with and without wind penetration.

the system consists only thermal unit (SYS), since both the fuel and emission costs can be avoided for renewables. The transverse dash line in Fig. 6 also shows that the highest costs of SYS and SYS2 are almost the same. The reason is that, a conservative commitment and output decision on thermal units of SYS can be implemented on SYS2 without difference, to achieve high reliable scheduling with the same cost. This phenomenon also reflects the consistency of the MO-PSO when solving optimization problems on different test systems. From reliability aspect, the main difference between SYS and the other two exists in the potential power shortage amount. As the vertical dash line in Fig. 6 indicates, the FVaR of SYS1 and SYS2 could be much larger than that of the SYS due to the additive uncertainty involved in wind power, i.e., generation uncertainty of renewables introduces remarkable operational risk in power systems. Nevertheless, it is also noteworthy that such risk could be relieved via taking appropriate commitment and output decisions on thermal units, as justified by the solutions in squares. According to the above analysis, we can conclude that generation uncertainty is significant for UC optimization with renewables, and the proposed model and algorithm can be utilized to solve the generation mix problems under manifold uncertainties. However, there is still much room for future discussions and improvements. For example, the issue of wind speed uncertainty description, the fact that wind turbines may be connected to different locations and the improvement of emission cost representation. Then, our future work will focus on the application of some skewed distributions to model wind speed uncertainty at different locations and further improve the emission cost calculation with respect to current carbon trading market. VI. CONCLUSIONS According to different feature of the uncertainties involved in the UC optimization, this study applied stochastic and fuzzy set theories to characterize these imprecise information respectively. Then, a VaR-based reliability measurement is established to provide the SOs with a unified evaluation on the supply reliability under the uncertainties of load fluctuation and unit outage. In addition, the FVaR can be improved easily to address some

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other uncertainties such as generation variability of renewable energy. Compared with SUC and RUC, the MO-UC built in this research involves the worst-case analysis, produces less-conservative solution and lets an SO rely on his experience and confidence on the supply system. To solve this two-stage multi-objective nonlinear optimization, we first developed a CPM method for the second stage problem, which calculates the worst-case capacity due to sudden unit outage under a predefined confidence level. Then, an MO-PSO is developed as a solution to the MO-UC. Experimental results show that the FVaR is sensitive to different confidence levels and the MO-UC reflects the conflict between operation cost and system reliability, which provide effective information for the SOs and help them make proper decisions. More importantly, we also justify the feasibility of our approach when handling UC optimizations with wind penetration. Our future work will focus on some other effective representations on generation uncertainty as well as emission cost, and further improve the model and algorithm to address the multi-node UC optimization. REFERENCES [1] B. Wang, Y. Li, and J. Watada, “Supply reliability and generation cost analysis due to load forecast uncertainty in unit commitment problems,” IEEE Trans. Power Syst., vol. 28, no. 3, pp. 2242–2252, Aug. 2013. [2] Y. Chen, P. B. Luh, C. Guan, Y. Zhao, L. D. Michel, M. A. Coolbeth, P. B. Friedland, and S. J. Rourke, “Short-term load forecasting: Similar day-based wavelet neural networks,” IEEE Trans. Power Syst., vol. 25, no. 1, pp. 322–330, Feb. 2010. [3] S. Takriti, J. R. Birge, and E. Long, “A stochastic model for the unit commitment problem,” IEEE Trans. Power Syst., vol. 11, no. 3, pp. 21497–1508, Aug. 1996. [4] M. A. Ortega-Vazquez and D. S. Kirschen, “Optimizing the spinning reserve requirements using a cost/benefit analysis,” IEEE Trans. Power Syst., vol. 22, no. 1, pp. 24–33, Feb. 2007. [5] F. Aminifar, M. Fotuhi-Firuzabad, and M. Shahidehpour, “Unit commitment with probabilistic spinning reserve and interruptible load considerations,” IEEE Trans. Power Syst., vol. 24, no. 1, pp. 388–397, Feb. 2009. [6] A. Trivedi, D. Srinivasan, D. Sharma, and C. Singh, “Evolutionary multi-objective day-ahead thermal generation scheduling in uncertain environment,” IEEE Trans. Power Syst., vol. 28, no. 2, pp. 1345–1354, May 2013. [7] Y. Huang, Q. P. Zheng, and J. Wang, “Two-stage stochastic unit commitment model including non-generation resources with conditional value-at-risk constraints,” Elect. Power Syst. Res., vol. 116, no. 2, pp. 427–438, 2014. [8] C. Zhao and Y. Guan, “Unified stochastic and robust unit commitment,” IEEE Trans. Power Syst., vol. 28, no. 3, pp. 3353–3361, Aug. 2013. [9] Q. P. Zheng, J. Wang, and A. L. Liu, “Stochastic optimization for unit commitment—a review,” IEEE Trans. Power Syst., to be published. [10] Y. An and B. Zeng, “Exploring the modeling capacity of two-stage robust optimization: Variants of robust unit commitment model,” IEEE Trans. Power Syst., vol. 30, no. 1, pp. 109–122, Jan. 2015. [11] C. Lee, C. Liu, S. Mehrotra, and M. Shahidehpour, “Modeling transmission line constraints in two-stage robust unit commitment problem,” IEEE Trans. Power Syst., vol. 29, no. 3, pp. 1221–1231, May 2014. [12] D. Bertsimas, E. Litvinov, X. Sun, J. Zhao, and T. Zheng, “Adaptive robust optimization for the security constrained unit commitment problem,” IEEE Trans. Power Syst., vol. 28, no. 1, pp. 52–63, Feb. 2013. [13] R. Jiang, J. Wang, and Y. Guan, “Robust unit commitment with wind power and pumped storage hydro,” IEEE Trans. Power Syst., vol. 27, no. 2, pp. 800–810, May 2012. [14] A. Street, F. Oliveira, and J. Arroyo, “Contingency-constrained unit commitment with N-k security criterion: A robust optimization approach,” IEEE Trans. Power Syst., vol. 26, no. 3, pp. 1581–1590, Aug. 2011.

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Bo Wang received the B.Sc. degree in software engineering from Southeast University, Nanjing, China, and the M.Sc. and Ph.D. degrees from the Graduate School of Information, Production and Systems, Waseda University, Japan. He was a Research Assistant of the Global COE Program, Waseda University, Ministry of Education, Culture, Sports, Science and Technology, Japan. He was a Special Research Fellow of the Japan Society for the Promotion of Science (JSPS). He is currently with the School of Management and Engineering, Nanjing University, Nanjing, China. His research interests include, power system planning, economics analysis, fuzzy set theory, and Meta-heuristic algorithms.

Shuming Wang (M’09) received the B.Sc. and M.Sc. degrees in applied mathematics from Hebei University, China, in 2004 and 2007, respectively, and the Ph.D. degree in engineering from Waseda University, Japan, in 2011. He was a Special Research Fellow of the Japan Society for the Promotion of Science (JSPS), Japan (2009–2011), and worked as a Researcher in Research Institute and Risk Management Division of China Galaxy Securities Co. LTD (HQ), Beijing, China (2011–2012). Since 2012, he has been working as a Research Fellow of National University of Singapore (NUS), Singapore. He is also an Adjunct Researcher of Waseda University, Japan. His research interests are optimization under uncertainty, imprecise probability theory, energy system planning, and risk assessment and management.

Xian-zhong Zhou (M’11) received the B.Sc. and M.Sc. degrees in system engineering and the Ph.D. degree in control theory and application from Nanjing University of Science and Technology, Nanjing, China. Currently, he is a professor of Control Science and Engineering at the School of Management and Engineering, Nanjing University, Nanjing, China. His professional interests include mission planning theory and technology of networked swarms system, novel decision-making system theory and human ware technology. He is an executive director of the Systems Engineering Society of China and the vice-president of the Systems Engineering Society of Jiangsu Province.

Junzo Watada (M’94) received the B.Sc. and M.Sc. degrees in electrical engineering from Osaka City University, Osaka, Japan, and the Ph.D. degree from Osaka Prefecture University, Osaka. Currently, he is a Professor of Management Engineering, Knowledge Engineering and Soft Computing at the Graduate School of Information, Production Systems, Waseda University, Fukuoka, Japan. His professional interests include soft computing, tracking system, knowledge engineering, and management engineering. Prof. Watada is a recipient of the Henri Coanda Medal Award from Invention in Romania in 2002. He is a Life Fellow of Japan Society for Fuzzy Theory and Intelligent Informatics (SOFT).