Reliability constrained unit commitment with combined hydro and thermal generation embedded using self-learning group search optimizer

Energy xxx (2015) 1e10

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Reliability constrained unit commitment with combined hydro and thermal generation embedded using self-learning group search optimizer J.H. Zheng a, J.J. Chen a, Q.H. Wu a, b, *, Z.X. Jing a a b

School of Electric Power Engineering, South China University of Technology, Guangzhou, 510640, China Department of Electrical Engineering and Electronics, The University of Liverpool, Liverpool, L69 3GJ, UK

a r t i c l e i n f o

a b s t r a c t

Article history: Received 18 January 2014 Received in revised form 11 December 2014 Accepted 12 December 2014 Available online xxx

This paper proposes a reliability constrained unit commitment problem with combined hydro and thermal generation embedded (RCHTUC), solved by a SLGSO (self-learning group search optimizer). The RCHTUC problem aims at minimizing the sum of fuel costs and start-up costs of thermal plants subject to various operation constraints. Furthermore, the problem takes into account the combination of hydro and thermal systems and the reliability constraints in hydrothermal power systems so as to respond to unforeseen outages and changes of load demands. In order to solve the RCHTUC problem, a SLGSO is developed from the GSO (group search optimizer), applying adaptive covariance matrix to design the
vy flights to increase the diversity of group. This paper optimum searching strategy and employing Le reports on the simulation results obtained by the proposed method. The results are compared with those obtained by other methods on different hydrothermal systems over the scheduling horizon. The simulation results demonstrate the efficiency of the SLGSO for tackling the RCHTUC problem. © 2014 Elsevier Ltd. All rights reserved.

Keywords: Hydrothermal unit commitment Spinning reserve Group search optimizer Adaptive covariance matrix
vy flights Le

1. Introduction In general, the short-term HTUC (hydrothermal unit commitment) in power systems refers to an optimization problem that determines the power generation of hydro and thermal plants over the scheduling horizon so as to minimize the total fuel costs and start-up costs of the thermal plants while satisfying various constraints of hydro and thermal plants. The optimal hydrothermal scheduling results in saving fuel costs and start-up costs by decreasing the dependency on thermal generating units during the peak load periods with hydro power generation and making use of the extra thermal power generation to pump the water back up into the reservoir over lower demand horizon. Therefore, the optimal HTUC plays a significant role in energy saving and reliable operation in hydrothermal power systems. Moreover, in order to comprehensively simulate the performance of power systems, the complex hydraulic connections of hydropower plant [1], the nonsmooth valve point effects of thermal plant [2] and the

* Corresponding author. School of Electric Power Engineering, South China University of Technology, Guangzhou, 510640, China. E-mail address: [email protected] (Q.H. Wu).

transmission loss of network [3] are taken into account in this paper. Consequently, the short-term hydrothermal generation scheduling problem is formulated as a nonlinear and non-smooth optimization problem that is difficult to be tackled. In the past few decades, various classical mathematical approaches have been successfully employed to solve the HTUC problem, such as dynamic programming [4], mixed integer linear programming [5], gradient search method [6], nonlinear programming with network flow [7], Lagrange relaxation [8], and decomposition and coordination technique [9]. These methods for the operation scheduling problem vary strongly from one to another depending on the mix status and the particular operating constraints of hydro and thermal plants. However, the fuel cost curves of thermal plants and the inputeoutput curves of hydro plants are usually represented as nonlinear and non-smooth ones with the effect of valve point loading and the mathematical approaches are prone to initial points and they might suffer from handling inequality constraints [10]. Hence, most of the conventional gradient-based methods encounter difficulties due to the non-smooth feasible region of the optimization problem and these methods are extremely easy to fall into local optimum. Although dynamic programming is able to handle the problem without strict

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requirements on convexity and various constraints, the computation complexity of this method exponentially grows with the increase of the system size. In recent years, various EAs (evolutionary algorithms) have aroused intense interest due to their versatility, flexibility, and robustness in seeking the global optimum solution. These algorithms seem to be efficient in solving HTUC problem since they do not place any restrictions on the shape of cost curves and other system non-differentiable in model representation. Several EAs like GA (genetic algorithm) [11], SA (simulated annealing) [12], EP (evolutionary programming) [13], PSO (particle swarm optimization) [14], DE (differential evolution) [15] and DRQEA (differential real-coded quantum-inspired evolutionary algorithm) [16] have been applied to solve the HTUC problem. Although these algorithms have shown great potential in solving the non-smooth optimization problems, they may get trapped in the local optimum when handling large-scale problems with heavy equality and inequality constraints because of their limited local or global search capabilities [17]. Recently, a GSO (group search optimizer) has been proposed based on the animal searching behavior and group living theory [18]. A set of experimental results on function optimization shows that GSO is competitive with some conventional biologicalinspired optimization algorithms, such as GA and PSO [19]. However, there is still an insufficiency in GSO regarding its producerscrounger search model, which is good at exploration but is poor at exploitation. Based on the producer-scrounger model in GSO algorithm, only the information obtained from the best member of the group is studied, and that gathered from the other elite members from each generation are ignored. Thus the performance of GSO would degrade if the optimization problem is complicated. In order to mitigate the imperfections of GSO for solving the RCHTUC problem, a SLGSO (self-learning group search optimizer) based on the study of GSO is proposed. The SLGSO algorithm consists of three type group members: producer, organizers, and rangers. In each generation, the member conferred the best fitness value is chosen as producer, and a number of members expected producer are randomly selected as organizers, then the rest of members are named rangers. The producer performs the crappie search behavior which is characterized by maximum pursuit angle, maximum pursuit distance, and maximum pursuit height [20]. The organizers employ the concepts based on adaptive covariance matrix [21,22] to design optimum searching strategy. Moreover,
vy flights, which are found to be more efficient than random Le walks for searching resource [23,24], are also employed by the rangers to increase the diversity of group. To verify the efficiency of the proposed algorithm, the SLGSO is tested on hydrothermal test systems, and the results obtained by the SLGSO are compared with those by other reported methods. The rest of this paper is organized as follows. The next section formulates the model of reliability constrained hydrothermal unit commitment. Section 3 presents the proposed SLGSO and its implementation to the reliability constrained hydrothermal unit commitment problem. In section 4, simulation studies are carried on applying different hydrothermal systems so as to verify the efficiencies of the proposed SLGSO and the presented reliability constrained UC model. The last section draws the conclusion of this paper. 2. Problem formulation The RCHTUC problem is formulated as an optimization problem. As the costs of hydro plants is insignificant compared with that of thermal plants [25], the objective function is composed of fuel costs

of thermal units for electric power generation, start-up costs and shutdown costs of individual thermal unit over the scheduling horizon. Note that the shutdown costs are always treated as a constant [26] which is omitted in this paper. The constraints include (a) system power balance, (b) system spinning reserve requirement, (c) unit ramping up/down limits of thermal plants, (d) unit minimum ON/OFF time limits of thermal plants, (e) unit generation limits of hydro and thermal plants, (f) water balance, (g) reservoir storage limits and (h) water discharge rate limits. The proposed objective function and corresponding constraints are formulated in detail as follows: 2.1. Objective function The objective function which consists of the fuel costs for electric power generation and the start-up costs of thermal units is denoted as follows:

f ðx; uÞ ¼

" NGt T X X t¼1

utGti

     t Fi PGt þ si utGti i

# (1)

i¼1

t Þ. There are two where the power generation cost function is Fi ðPGt i popular models used to represent the relationship between fuel costs and power output: one is smooth quadratic cost function and another is non-smooth fuel cost function. For each generator per hour, the smooth quadratic fuel cost function is expressed as follows [27]:

   2 t t Fi PGt i ¼ ai PGt þ bi PGt þ ci i i

(2)

However, when the steam admission valve of the thermal plant starts to open, the fuel costs increase sharply owing to the wire drawing effects [28]. This phenomenon is called as valve point effects which can be represented by the non-smooth fuel cost function precisely. Taking the valve point effects into consideration, a sinusoidal component is added to the fuel cost function which is modified by:

    2      t t t Fi PGt i ¼ ai PGt þb P þc þ  ei sin fi  Pmin;i PGt i i Gti i i (3) where ai, bi, ci, ei and fi are coefficients of fuel cost function.In addition, si ðutGti Þ is the start-up cost, which can be described by:

    si utGti ¼ Ki 1  ut1 Gti

(4)

where Ki is a two-value start-up coefficient. If the down-time of unit i is less than its cold-start time, Ki refers to hot start-up coefficient; Otherwise, Ki refers to cold start-up coefficient. 2.2. Constraints 2.2.1. System power balance Power generations of hydro plants and thermal plants must meet system load demands including transmission losses, otherwise the system cannot operate at a balanced condition which is described as follows: NGt X i¼1

t PGt ut þ i Gti

NGh X

t t PGh ¼ PD þ PLt j

(5)

j¼1

Please cite this article in press as: Zheng JH, et al., Reliability constrained unit commitment with combined hydro and thermal generation embedded using self-learning group search optimizer, Energy (2015), http://dx.doi.org/10.1016/j.energy.2014.12.036

J.H. Zheng et al. / Energy xxx (2015) 1e10 t where the hydropower generation PGh can be represented by a j quadratic function of water discharge rate and reservoir storage volume:

 2  2 t PGh ¼ c1j $ vtj þ c2j $ qtj þ c3j $vtj $qtj þ c4j $vtj þ c5j $qtj þ c6j j

(6) where c1j, c2j, c3j, c4j, c5j and c6j are coefficients of hydro output function. Besides, the power loss of power system is taken into consideration in this paper. Based on the Kron's loss formula [29], the transmission loss can be calculated by using the B loss coefficients matrix, which is known as a practical method to calculate incremental loss.

PLt ¼

NG X NG X

t PG B Pt þ i ij Gj

i¼1 j¼1

NG X

3

Furthermore, spinning reserve is synchronized with the system, which is available to serve loads at a short notice. By maintaining sufficient spinning reserve across all generating units, system generation is much more effective in dealing with frequency deviations, fast system load pick-up, unit-forced outages, and other system disturbances [32]. In power systems, a certain amount of spinning reserve is required from the viewpoint of the reliability of system operation. Thus, the satisfaction of spinning reserve requirement is of great importance for both real time dispatching and daily scheduling in power system operations. NGt X

t rit  rD

(17)

i¼1 t B0i PG þ B00 i

(7)

i¼1

where the hourly spinning reserve supply is given by:

  n t ; rit ¼ min utGti Pmax;Gti  PGt i 2.2.2. Unit generating output limits The output capacity of both the hydro and thermal units must be restricted between the minimum and maximum limits, which define the bounds of the corresponding variables.

 o utGti ti $RUGti

(18)

Furthermore, the ramp-up/down rate and the ON/OFF time of thermal units are limited by their physical characteristics. They must satisfy the following constraints:

In the proposed model, the amount of spinning reserve requirement is determined by system load demands and system security demands at different times. The system load demands are determined by peak load demands on a hourly base. Hourly peak load demands, which are denoted by ldt are taken into consideration. Moreover, the system reliability demands are associated with the probability of contingencies Pc and the Expected Energy Not Supplied EENSt. Pc is estimated based on network elements FOR (forced outage rate). In this paper, only the FOR of generators is taken into consideration. FOR is probability of element outage and calculated based on the statistical data of the element [31]:

  t t1 t1 PGt  PGt  ut1 Gti RUGti þ 1  uGti Pmax;Gti i i

(10)

FOR ¼

  t1 t  PGt  utGti RUGti þ 1  utGti Pmax;Gti PGt i i

(11)

where MTTR is the mean time taken to repair and MTTF is the mean time taken to failure of the generators in the tested system. Then, the probability of each contingency can be determined by:

t utGti Pmin;Gti  PGt  utGti Pmax;Gti i

(8)

t Pmin;Ghj  PGh  Pmax;Ghj j

(9)

tk t ut1 Gti þ uGti  uGti  0; tk t ut1 Gti  uGti þ uGti  1;

ck : 1  k  ðt  1Þ  Ton ck : 1  k  ðt  1Þ  Toff

(12) (13)

2.2.3. Hydraulic constraints The reservoir storage of a hydro plant is determined by the inflow and spillage, reservoir storage at previous period and discharges from upstream reservoir [30]. They must meet the hydraulic continuity equations as follows [16]:

vtj

¼

vt1 j

þ

Ijt



qtj



stj

þ

NGh  X

qm;ttmj þ sm;ttmj



(14)

Pc ¼

MTTR MTTR þ MTTF

NG FOR Y ð1  FORi Þ 1  FOR i¼1

(19)

(20)

Moreover, the value of EENS, is the energy loss expectation based on network elements' FOR, the load shedding when a contingency event occurs and the EENS coefficient set by the system operators. Combining all these factors, EENS is calculated as follows:

EENSt ¼

Nc X

Pci $ldlosti

(21)

i¼1

m¼1

Moreover, the water discharge rate and reservoir storage volume must be limited between the minimum and maximum operating condition, which is given by:

where ldlosti represents load lost when contingency i occurs and Nc is the number of contingencies. Hence, the required spinning reserve capacity is:

qmin;j  qtj  qmax;j

(15)

t rD ¼ ldt þ EENSt

vmin;j  vtj  vmax;j

(16)

Consequently, the constraint of spinning reserve requirement in (17) can be modified as follows: NGt X

2.2.4. Spinning reserve requirement Supplied by free capacity of generating units, spinning reserve are able to be activated on the demand of system operators [31].

rit  ldt þ EENSt

(22)

(23)

i¼1

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3. Optimization of the RCHTUC

mðgþ1Þ ¼

3.1. The self-learning group search optimizer

m X

ðgþ1Þ

ui xi:p

(25)

i¼1

As well known, it is necessary to award the population-based optimization algorithms with exploration and exploitation properties. The exploration refers to the ability to investigate the various unknown regions in the solution space to discover the global optimum while exploitation refers to the ability to apply the knowledge of the previous good solutions to find better solutions [33]. In practice, exploration and exploitation are contradictory to each other. Under the producer-scrounger model, only the information from the best member of the group is studied, and the other elite members from each generation are ignored. In order to well balance the exploration and exploitation, a SLGSO (self-learning group search optimizer) is proposed in this paper, utilizing a producerorganizer modal to improve exploitation ability.

where m X

ui ¼ 1;

u1  u2  /  um > 0

i¼1

p denotes the population size and m denotes the number of the members selected from the elite group. The recombination weights are generated by

u0 ui ¼ Pm i j¼1

(26)

u0 j ðgþ1Þ

0

3.1.1. Producing mechanism The producer’s search behavior in SLGSO is the same as the producer’s in GSO [18], which will first scan at zero degree and then scan laterally by randomly sampling three points in the scanning field [20]: one point at zero degree, one point in the right hand side hypercube and one point in the left hand side hypercube. The best point with the best resource (fitness value) is chosen as producer. If the best point has a better resource than its current position, then the producer will fly to this point. Otherwise the producer will stay in its current position and turn its head to a new randomly generated angle. And if the producer cannot find a better area after a generations, it will turn its head back to zero degree. 3.1.2. Organizers' behaviors During each generation searching, a number of group members are selected as organizers. The organizers keep searching for opportunities based on the organized information from entire group. By learning for the group information, organizers determine the evolution path and step-size, and then offspring are accordingly generated. In order to make organizers be adaptive learning for group members, the adaptive covariance matrix [21] has been employed. The organizers mainly perform following three steps: a) Firstly, the group members are partitioned into an elite group and an inferior group based on their fitness values, then the elite group members are used to generate mean vector by exponential weighting using (25); b) The covariance matrix, which is used to determine the evolution path and step-size, is updated by mean vector using (27); c) The offspring of organizers are obtained by the evolution path and step-size using (24).Here g(g ¼ 0,1,2,/) denotes the generation number and the offspring of the kth organizer can be modeled as follows [22]:

ðgþ1Þ xk

m

ðgÞ

ðgÞ

þs

N



 0; C ðgÞ ;

k ¼ 1; /; l

(24)

where “” denotes equality in distribution. s(g) 2 ℝþ is the stepsize. And N (0,C(g)) is a multivariate normal distribution with mean 0 and covariance matrix C 2 ℝnn. l is the organizer size. By learning for the group members, mean value m(gþ1) of the search distribution is a weighted average of m(m  p) selected points ðgþ1Þ ðgþ1Þ from the sample x1 ; /; xp :

where ui ¼ lnðp=2 þ 0:5Þ  ln i; ði ¼ 1; /; mÞ: And xi:p denotes ðgþ1Þ ðgþ1Þ the ith best individual out of x1 ; /; xp which are ranked ðgþ1Þ ðgþ1Þ ðgþ1Þ follow, f ðx1:p Þ  f ðx2:p Þ  /  f ðxp:p Þ; and f is the objective function to be minimized. The update formula of covariance matrix C is computed by

  ðgþ1Þ ðgþ1ÞT C ðgþ1Þ ¼ 1  c1  cm C ðgÞ þ c1 pc pc þ cm

m X

ðgþ1Þ ðgþ1ÞT yi:p

ui yi:p

(27)

i¼1

where ðgþ1Þ

pc

ðgþ1Þ

yi:p

ðgÞ

¼ ð1  cc Þpc þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mðgþ1Þ  mðgÞ cc ð2  cc Þmeff sðgÞ

.  ðgþ1Þ sðgÞ ¼ xi:p  mðgÞ

ðgÞ

ð0Þ

pc 2ℝn , is the evolution path, and pc and cm are shown as follows:

¼ 0: The values of cc,c1,

. 4 þ meff n . cc ¼ n þ 4 þ 2meff n

c1 ¼

(28)

2

(29)

ðn þ 1:3Þ2 þ meff

. ! meff  2 þ 1 meff . cm ¼ min 1  c1 ; am ðn þ 2Þ2 þ am meff 2

(30)

P where meff ¼ 1= mi¼1 u2i and am ¼ 2. The update equation of step-size s is given by

sðgþ1Þ

cs ¼ sðgÞ  exp ds

psðgþ1Þ ¼ ð1  cs ÞpðgÞ s þ

   ðgþ1Þ  ps  EjjN ð0; IÞjj

!! 1

(31)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 mðgþ1Þ  mðgÞ cs ð2  cs Þmeff C sðgÞ (32)

ðgÞ

ð0Þ

where ps 2ℝn , is the conjugate evolution path, and ps ¼ 0:

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J.H. Zheng et al. / Energy xxx (2015) 1e10

3.1.3. Rangers' walks In GSO algorithm, the rangers perform random walks. However,
vy flights [23] is more efficient than random biologists found that Le
vy walks [23,34] in random search. Therefore, in this paper, Le flights are introduced as rangers' search behavior in this paper. Firstly, the rangers choose a random stepsize value:

 1=b   u ðgÞ ðgÞ stepsizei ¼ 0:01$ i $ xi  xp vi

(33)

where u ¼ f,randn(n), v ¼ randn(n), b ¼ 1.50, n is the number of variables. The randn(n) function generates a uniform integer between [1,n]. f is computed by:

 f¼

Gð1 þ bÞ$sinðp$b=2Þ Gðð1 þ bÞ=2Þ$b$2ðb1Þ=2

1=b (34)

where the G denotes gamma function. Then, rangers will move to the new point based on the above stepsize: ðgþ1Þ

xi

ðgÞ

¼ xi

þ stepsizei $randnðnÞ

(35)


vy flights Consequently, the rangers' walks are employing the Le instead of random walks, and the simulation studies in the next section verify its efficiency in searching for a superior solution.

5

Step 2.1 In the second step, the status and outputs of each thermal unit are determined. The committed thermal units in each hour over the scheduling horizon is determined by means of a priority list. The UC scheduling calculated by the priority list is based on the priority index of GCR (generation cost rate) which is defined as follows:

GCRi ¼

  dFit Pmax;Gti Pmax;Gti

(36)

where dFit ðPit Þ is the first derive of the fuel cost function of unit i at time t. The priority list of units will be formulated based on the priority index GCRi, in which a unit with lowest GCRi will have the highest priority. Step 2.2 In this step, units are primarily committed satisfying power balance and spinning reserve requirement. Based on the priority list computed by the priority index of GCR, set the states of the unit with higher priority utGti ¼ 1 until the power balance and spinning reserve requirement are satisfied over 24-h period. Step 2.3 In General, the primary unit scheduling may not satisfy the unit minimum ON time limits and unit minimum OFF time limits. Therefore, the primary unit scheduling should be modified to avoid the unit minimum ON/OFF time violations. The ON/ OFF time of unit i up to the hour t can be calculated as follows:

3.2. Implementation of SLGSO for the RCHTUC According to the description stated in the previous section, the procedure for the SLGSO is summarized in Table 1. As a consequence, the procedure of the proposed method to solve the RCHTUC problem is summarized as follows: Step 1. Firstly, randomly choose the initial water discharge rate qti in the search space. Apply the proposed SLGSO to figure out the optimal output of all the hydro units in power systems. Note that if the water discharge rate qti < 0, it means that, the corresponding hydro plant is in the pumping mode; Otherwise, the corresponding hydro plant is in the generating mode. Afterwards, calculate the total power demands of thermal units based on (5). Table 1 Procedure for the SLGSO. Set g:¼0; Randomly initialize position xi and head angles 4i of all members; Calculate the fitness values of initial members: f(xi); WHILE (the termination conditions are not met) FOR (each members i in the group) Choose producer: Find the member which conferred the best fitness value as producer xp; Perform producing: The producer randomly samples three points; Perform organizing: Randomly select 70% from the rest members to perform organizing: 1) By learning for the elite group members, mean vector m is generated using (25); 2) Based on mean vector m, covariance matrix C is updated using (27), to determine evolution path and update step-size using (31); 3) By obtained mean vector m, covariance matrix C and step-size s, the new organizers are generated using (24); Perturbation operation: For the rest members, they will be dispersed from their current position to perform ranging:
vy flights using (33); 1) Choose a random stepsize value based on Le 2) Move to the new point using (35); Calculate fitness: Calculate the fitness value of current member: f(xi); END FOR END WHILE OUTPUT the best fitness value f(xbest) and the corresponding member xbest.

( t Ti;on ¼

( t Ti;off

¼

t1 þ 1; Ti;on 0;

uti ¼ 1 uti ¼ 0

(37)

t1 Ti;off þ 1; 0;

uti ¼ 0 uti ¼ 1

(38)

Step 2.4 After the modification of the unit scheduling in Step 2.3, the constraints of over time horizon are satisfied. Nevertheless, there may be not enough generation for the load demands or redundant generation leading to extra generation cost because of the rescheduling. Thus it is necessary to check out the violations and modify it by starting up more units with higher priorities or shutting down the unnecessary units. Step 3. Applying the steps presented above, the status of each thermal units are scheduled over 24-h period. Subsequently, the proposed SLGSO is carried out to optimize the outputs of all the thermal plants over the scheduling horizon aiming at minimizing the total costs in (1) while satisfying the boundary constraints of the thermal plants. 4. Simulation studies In this section, in order to verify the suitability and effectiveness of the proposed method, the SLGSO is tested on two typical hydrothermal systems, respectively. One test hydrothermal system is with 4 hydro units and 1 equivalent thermal unit (Test System 1) while the other is with 4 hydro units and 40 thermal units (Test System 2). These test systems consist of two sub systems: hydro sub-system and thermal sub-system [35]. In the hydro sub-system, there are multi chain cascade hydro plants on one stream that can represent most of the complex hydro network in reality. The natural inflow, river transport delay between successive reservoirs and

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J.H. Zheng et al. / Energy xxx (2015) 1e10

variable water head are also taken into consideration [16]. In the thermal sub-system, the characteristics of all the thermal plants that may consist of several units can be represented by an equivalent virtual thermal unit. Note that the parameters used in this case are chosen as the same as those used in the other references in order to make comparisons with other algorithms fair enough. The simulation studies are carried out based on MATLAB 7.11.0 and executed on a PC with Intel core 2 Duo CPU (3.10 GHz) and 3.17 GB RAM. The parameters of SLGSO set as follows: p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi are 40 ¼ ðp=4; /; p=4Þ; a ¼ roundð ðn þ 1ÞÞ; qmax ¼ p/a2,amax ¼ qmax/ 2, the percentage of rangers is 30%, and population size is set to 100. In SLGSO, it contains these parameters and these parameters perform the same performance as being used in GSO. Thus, the tunable parameters of SLGSO are the population size and the percentage of the rangers. The values for population size are selected from the set {30, 50, 100, 150, 200, 250, 300}, respectively. The percentage of the rangers increases from 0 to 100% in steps of 10%. For all the parameters setting above mentioned, performance of the SLGSO is evaluated 30 times. Hence the parameters of the SLGSO are set based on the test in order to find the best options for the proposed problem in our paper. 4.1. Case 1: smooth fuel cost function of the thermal power production in Test System 1 In this case, there four cascaded hydropower plants and an equivalent thermal plant in the tested system which is obtained from Ref. [16]. The equivalent thermal plant with and without valve point effects are taken into consideration over the scheduling period of 24 h. Applying the proposed SLGSO, the HTUC problem on the tested system is solved, and the obtained results are reported as follows: Table 2 lists the optimal hydro plant water discharge rates, hydro power outputs, thermal power outputs and total generation over 24-h period calculated by the proposed method. Moreover, the hourly hydro reservoir volumes of the optimal scheduling results on the tested system are plotted in Fig. 1. It is obvious that the

Fig. 1. Reservoir storage volume of the hydro plants in Case 1.

simulation results calculated by the proposed SLGSO satisfy all the constraints. In order to verify the efficiency of the proposed SLGSO, the obtained best fuel cost, worst fuel cost and average fuel cost are reported in Table 3. In this case, the proposed SLGSO algorithm is implemented to optimize the fuel costs without considering value point effects for 30 times independent runs with different random initial population. The obtained best fuel cost, worst fuel cost and average fuel cost are reported in Table 3, compared with those of APSO [36], BCGA [11], CEP [13], DE [37], DRQEA [16], EGA [38], EPSO [38], FEP [13], GA [35], IFEP [13], IPSO [39], LWPSO [14], MAPSO [36], MDE [40], PSO [38], RCGA [11], SPPSO [41], and TLBO [28]. Although the best solution of SLGSO is not guaranteed to be the global solution, the proposed SLGSO has shown the superiority to the existing methods. The optimal fuel costs of the proposed algorithm in 30 independent simulations are $920423.15 as seen in Table 3. It can be seen from the table that the proposed algorithm yields the best cost among all the techniques considered. Moreover, it can also be found from Table 3 that the proposed algorithm yields the best statistical properties such as the worst and the average values in 30 independent simulations among the considered algorithms. Meanwhile, the comparisons of the convergence process among SLGSO, GSO and SPSO [42] are provided in Fig. 2. It is quite apparent from the figure that the proposed SLGSO algorithm can

Table 2 Hydro plant discharge rates, hydro power outputs, thermal power outputs and total generation in Case 1. Hour

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Hydro plant discharges (104 m3/h)

Hydro power output (MW)

Plant 1

Plant 2

Plant 3

Plant 4

Plant 1

Plant 2

Plant 3

Plant 4

7.3410 8.1018 8.8320 8.8391 7.8905 8.5709 8.8535 9.7684 7.8130 7.9455 8.5812 7.5431 7.0951 8.8003 9.3531 6.7398 8.6273 8.5164 8.6335 6.9367 8.4334 7.3927 7.9093 6.5517

6.5323 6.6828 8.7680 6.3037 7.8472 8.4942 6.4295 8.5735 7.5973 7.0394 7.3872 9.2737 9.3164 7.7020 9.9740 7.0220 10.2210 7.0829 9.5894 8.6116 8.1858 11.3991 9.4675 12.4996

19.1977 22.1811 21.3096 20.4189 19.1209 17.7416 16.4084 19.3119 15.4991 15.7920 17.1319 15.5594 17.2329 18.5973 18.2451 13.6518 14.9375 15.5446 15.4444 15.1298 16.3285 17.2647 15.5263 13.9668

6.7722 6.3560 7.5161 7.0965 7.5865 6.6189 10.7628 11.0247 13.9439 13.2001 16.4028 15.9948 14.7139 17.1713 15.8045 17.4381 16.8741 18.8416 17.3333 18.3822 18.0619 18.9926 19.4539 18.9133

71.5146 76.7800 80.9089 80.3784 74.1248 77.7239 79.0083 83.3563 73.3222 75.0835 79.9609 73.9431 71.5548 83.5949 87.1546 69.9977 83.4835 82.6960 83.1665 71.2106 81.4725 74.4933 78.3059 68.5887

53.5633 55.3484 68.2232 54.3947 64.5772 67.5093 54.2832 66.7516 61.2461 58.8317 61.9862 72.1665 71.5926 63.2233 75.0608 58.9020 74.8101 56.7913 69.4239 63.9319 62.0576 75.8174 66.3598 75.1989

44.7260 25.5023 26.3187 29.3350 34.9675 41.5608 45.6515 35.6548 48.2804 48.1680 44.9905 49.3432 46.1873 41.5253 43.1687 54.3737 54.2406 54.1914 54.6650 56.2593 54.6312 52.8344 56.9234 58.7969

138.5014 129.3968 138.3122 125.9270 143.6598 145.6138 205.3532 216.5680 251.1454 247.9240 276.8567 276.5987 266.0437 285.4663 275.6587 286.9521 282.9879 296.9212 287.1487 290.0151 284.6562 287.1554 285.5838 278.1875

Thermal output (MW)

Total output (MW)

1061.6948 1102.9726 1046.2370 999.9650 972.6706 1077.5923 1265.7037 1597.6693 1806.0060 1889.9929 1766.2057 1837.9485 1774.6216 1726.1903 1648.9573 1599.7745 1634.4779 1649.4002 1745.5958 1798.5831 1757.1825 1629.6994 1362.8271 1109.2279

1370 1390 1360 1290 1290 1410 1650 2000 2240 2320 2230 2310 2230 2200 2130 2070 2130 2140 2240 2280 2240 2120 1850 1590

Please cite this article in press as: Zheng JH, et al., Reliability constrained unit commitment with combined hydro and thermal generation embedded using self-learning group search optimizer, Energy (2015), http://dx.doi.org/10.1016/j.energy.2014.12.036

J.H. Zheng et al. / Energy xxx (2015) 1e10 Table 3 Comparison of simulation results in Case 1. Bold values in Table 3 signify superior solutions for the corresponding problems. Algorithms

Best cost ($)

Worst cost ($)

Average cost ($)

APSO BCGA CEP DE DRQEA EGA EPSO FEP GA IFEP IPSO LWPSO MAPSO MDE PSO RCGA SPPSO TLBO GSO SLGSO

926,151.54 926,922.71 930,166.25 923,991.08 922,526.73 934,727.00 922,904.00 930,267.92 932,734.00 930,129.82 922,553.49 925,383.80 922,421.66 922,555.44 928,878.00 925,940.03 922,336.31 922,373.39 925,173.35 920,423.15

e 929,451.09 930,927.01 e 925,871.51 937,339.00 923,527.00 931,396.81 939,734.00 930,881.92 e 927,240.10 923,508.00 e 938,012.00 926,538.81 923,083.48 922,873.81 928,239.73 922,027.88

e 927,815.35 930,373.23 e 923,419.37 936,058.00 924,808.00 930,897.44 936,969.00 930,290.13 e 926,352.80 922,544.00 e 933,085.00 926,120.26 922,668.45 922,462.24 927,024.68 921,202.44

avoid premature convergence effectively and that this algorithm has faster convergence speed compared with the reformulated algorithms for this case. Furthermore, in order to demonstrate the robustness of the proposed method, two different load demands which are generated by adding 20% and þ20% to the origin load demands of the tested system, are applied to test the SLGSO. The optimal solutions obtained by GSO and SLGSO are summarized in Table 4, compared with results from Ref. [16]. From the table, it can be seen clearly that the proposed method can get superior solutions than other methods, which implies the better robustness to adapt for different system of the SLGSO.

4.2. Case 2: non-smooth fuel cost function of the thermal power production with value point effects in Test System 1 In this case, the equivalent thermal plant with valve point effects is taken into consideration in the same hydrothermal system as Case 1. The comparative performance of 30 independent simulations between the SLGSO and the other algorithms are given in Table 5. The optimal costs of the SLGSO are found to be $923417.49 for this case, which is smaller than the other results obtained by other algorithms: DE [37], DRQEA [16], IPSO [39], MDE [40], MHDE [43], QEA [16] and RQEA [16]. The comparisons of the convergence process among SLGSO, GSO and SPSO are provided in Fig. 3. According to the figure, it is quite apparent that the proposed SLGSO algorithm can avoid premature convergence effectively compared with GSO and SPSO. The optimal hourly hydro discharges and

Fig. 2. Cost convergence characteristic among SLGSO, GSO and SPSO in Case 1.

7

Table 4 Comparisons of simulation results for different load demand. Bold values in Table 4 signify superior solutions for the corresponding problems. Minimum fuel cost ($) Method

80% load demands

120% load demands

DE DRQEA QEA GSO SLGSO

718,149.50 697,168.49 743,651.31 698,516.63 695,103.47

1,198,022.47 1,167,310.21 1,267,351.08 1,167,202.43 1,163,342.68

Table 5 Comparison of simulation results in Case 2. Bold value in Table 5 signifies superior solutions for the corresponding problems. Algorithms

Best cost ($)

Algorithms

Best cost ($)

DE DRQEA IPSO MDE MHDE

928,662.84 925,485.21 925,978.84 925,960.56 925,547.31

QEA RQEA GSO SLGSO

930,647.96 926,068.33 934,444.35 923,417.49

thermal power generation obtained by SLGSO are provided in Table 6, and the hourly volume of reservoirs of hydro plants is shown in Fig. 4. 4.3. Case 3: reliability test in Test System 2 As stated in previous sections, the PS unit can be applied to adjust spinning reserve by means of peak clipping and valley filling over the scheduling horizon. In this study, the hydro plants can operate in the pumping mode, the idle mode or the generating mode, which means that the water discharge rates can positive values, negative values or zeros. The parameters of hydro subsystem are chosen as the same as those used in the above cases, and the parameters of thermal sub-system for unit commitment are chosen as the same as those given in Ref. [26]. In addition, the data applied to calculate the spinning reserve requirement are referred to [44]. Applying the proposed implementation stated in Section 3.2, the results are reported by making a comparison between the test system with and without the hydro units. Fig. 5 depicts the hydro power output over 24-h period obtained by the proposed method. As shown in the figure, the hydro plants are operating in the pumping mode during the period of lower load demands; Otherwise, during the period of higher load demands, the hydro plants are operating in the generating mode. The total thermal power output over 24-h period in test systems with PS units and without PS units are plotted in Fig. 6. As shown clearly in the figure, over the scheduling horizon, the curve of the total thermal power output in the system with PS units seems to be

Fig. 3. Cost convergence characteristic among SLGSO, GSO and SPSO in Case 2.

Please cite this article in press as: Zheng JH, et al., Reliability constrained unit commitment with combined hydro and thermal generation embedded using self-learning group search optimizer, Energy (2015), http://dx.doi.org/10.1016/j.energy.2014.12.036

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J.H. Zheng et al. / Energy xxx (2015) 1e10

Table 6 Hydro plant discharge rates, hydro power outputs, thermal power outputs and total generation in Case 2. Hour

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Hydro plant discharges (104 m3/h)

Hydro power output (MW)

Plant 1

Plant 2

Plant 3

Plant 4

Plant 1

Plant 2

Plant 3

Plant 4

8.2300 8.6961 6.4946 8.3104 8.9104 6.2026 9.5509 8.2646 7.7969 8.1016 9.3626 9.8010 11.0650 7.4139 9.6994 7.7599 8.1176 8.4605 7.4245 8.1634 6.8717 7.3672 9.2576 6.8251

7.6879 7.8565 7.1812 8.2560 8.2207 7.2190 7.3551 8.1385 7.1025 8.1605 7.2089 7.5086 10.5291 8.4567 8.8509 7.6908 8.4028 9.2714 8.3621 7.3361 9.6792 9.5157 12.3867 9.6231

20.9120 20.4103 20.3419 16.8764 18.7706 16.5260 15.4288 20.5717 18.0487 20.3589 14.9593 17.0366 16.9143 19.3616 16.8732 18.6495 13.9014 15.3088 16.1913 14.1460 15.2481 16.2309 15.8321 15.9409

10.4844 9.3860 9.2761 7.4698 9.8670 7.8840 9.6018 9.1248 13.6876 14.2281 14.0505 14.0235 14.5203 14.8550 16.6721 16.6983 17.4606 18.1209 18.9173 19.0376 18.8906 17.6466 18.7431 17.7417

77.0725 79.9356 65.5727 77.7090 80.2774 62.5140 83.3577 76.5475 74.2709 77.0172 85.0998 87.4220 93.0312 74.0427 88.4565 77.0914 79.6716 81.8257 74.7148 79.4226 70.2968 74.0779 86.1872 70.4309

60.3212 61.4280 58.3294 65.3372 64.9999 58.7512 58.8110 62.8642 57.0555 64.0598 59.2895 61.4752 76.0296 65.9422 68.2178 61.7061 65.0574 67.6586 61.8859 56.2142 68.7713 67.6253 76.0829 63.9426

36.7625 34.8170 32.0248 45.5396 38.8421 46.9646 50.7708 32.3444 42.6381 31.7940 49.6209 45.1915 47.0747 39.0438 48.8904 43.8570 55.7405 55.0810 53.7289 57.3659 57.0421 56.0142 56.5667 56.6405

178.3901 159.1785 149.6617 120.9339 159.9527 150.3089 180.8956 182.4857 235.4762 242.6524 242.4006 248.3951 256.2370 264.2926 278.0187 278.5487 283.8069 289.7629 293.0327 293.4005 287.2523 276.5253 281.0857 270.6688

more flat than that of the output in the system without PS unit. The result implies that the PS units are fully utilized to clip the peak load and fill the valley, the start-up costs of the thermal units will accordingly be reduced (see Table 7). Fig. 7 plots the curves of spinning reserve capacities over the scheduling 24-h period obtained by the proposed model in test systems with PS units and without PS units with a comparison with capacities obtained by the traditional criteria, the capacity of the maximum online unit ðmaxðutGi Pmax;i ÞÞ, utilized in some real power systems.

Thermal output (MW)

Total output (MW)

1017.4537 1054.6408 1054.4114 980.4802 945.9278 1091.4613 1276.1648 1645.7582 1830.5593 1904.4766 1793.5891 1867.5161 1757.6275 1756.6786 1646.4167 1608.7967 1645.7236 1645.6711 1756.6377 1793.5968 1756.6375 1645.7573 1350.0775 1128.3172

1370 1390 1360 1290 1290 1410 1650 2000 2240 2320 2230 2310 2230 2200 2130 2070 2130 2140 2240 2280 2240 2120 1850 1590

Table 7 summarized the start-up costs, fuel costs and total costs calculated by the proposed method in the tested systems with and without PS units. According to the results shown in the table, the PS units can not only reduce the start-up costs but also reduce the fuel costs in the hydrothermal system, which indicates the significant role of the PS units in power systems. 5. Conclusions This paper has proposed a reliability constrained unit commitment with combined hydro and thermal generation embedded, solved by a self-learning group search optimizer. Simulation studies have been carried out using the proposed method to solved the RCHTUC problem.

Fig. 4. Reservoir storage volume of the hydro plants in Case 2.

Fig. 6. Thermal power output over 24-h period obtained by the proposed method.

Table 7 Comparison of costs in test systems. Bold values in Table 7 signify superior solutions for the corresponding problems.

Fig. 5. Hydro power output over 24-h period obtained by the proposed method.

System

Start-up costs ($)

Fuel costs ($)

Total costs ($)

without PS units with PS units

22,500.0 17,920.0

2,226,220.866 2,148,515.551

2,248,720.866 2,166,435.551

Please cite this article in press as: Zheng JH, et al., Reliability constrained unit commitment with combined hydro and thermal generation embedded using self-learning group search optimizer, Energy (2015), http://dx.doi.org/10.1016/j.energy.2014.12.036

J.H. Zheng et al. / Energy xxx (2015) 1e10

Fig. 7. Spinning reserve scheduling for 24-h using the proposed model and the traditional model.

The combination scheduling of hydro and thermal generation and the reliability constraints in hydrothermal power systems have been taken into consideration, which is modeled as a nonlinear and non-smooth optimization problem. Meanwhile, the spinning reserve requirement is optimized in the RCHTUC problem aiming at responding to unforeseen outages and load demand changes in power systems. Based on GSO, the proposed SLGSO not only employs an adaptive covariance matrix to design optimum searching strategy but
vy flights to increase the diversity of group also employs Le searching, which results in better exploration and exploitation properties than that of GSO. To validate the efficiency of the propose method of solving the RCHTUC problem, simulation studies have conducted on three different cases. The minimum costs of case 1 obtained by the SLGSO are $ 920,423.15, which can save $ 1913.16 compared with the best reported results. In case 2, the minimum costs computed by the proposed method is $ 923,417.49, saving $ 2067.72 compared with the best reported results. The simulation results have indicated that compared with other methods published in literature, the SLGSO is able to find a superior solution for the RCHTUC problem with a fast convergence in finding the global optimal solution, which accordingly results in significant energy saving in hydrothermal power systems.

Acknowledgement The project is funded by the State Key Program of National Natural Science of China (Grant No. 51437006) and Guangdong Innovative Research Team Program (No. 201001N0104744201).

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Nomenclature Bij: B loss coefficients matrix EENSt: expect energy not supplied at hour t, MWh FOR: forced outage rate Iit : inflow of hydro unit i at hour t, m3/h ldlosti: load lost when contingency i occurs, MW ldt: hourly peak load demands at hour t, MW MTTF: mean time taken to failure, hour MTTR: mean time taken to repair, hour Nc: number of contingencies NG: number of units NGh: number of hydro units NGt: number of thermal units Pc: probability of each contingency Pmax,i: maximum output limit of unit i, MW Pmin,i: minimum output limit of unit i, MW PDt : load demands at hour t, MW t : the hydro power generation of unit i at hour t, MW PGh i

t : the thermal power generation of unit i at hour t, MW PGt i PLt : network transmission power loss at hour t, MW qmax,j: maximum water discharge rate limit of hydro unit j, m3/h qmin,j: minimum water discharge rate limit of hydro unit j, m3/h qti : water discharge rate of hydro unit i at hour t, m3/h t : control variable for spinning reserve requirements at hour t, MW rD rit : control variable for spinning reserve of unit i at hour t, MW RUi: ramp-up rate of unit i, MW/h sti : water spillage of hydro unit i at hour t, m3/h T: number of scheduling hour Toff: maximum down-time limits of unit i, hour Ton: maximum up-time limits of unit i, hour ti: time available for unit i to ramp up its power generation utGti : the ON/OFF status of unit i at hour t vmax,j: maximum reservoir volume limit of hydro unit j, m3 vmin,j: minimum reservoir volume limit of hydro unit j, m3 vti : reservoir volume of hydro unit i at hour t, m3

Abbreviations APSO: adaptive particle swarm optimization BCGA: bit-coding genetic algorithm CEP: classical evolutionary programming DE: differential evolutionary DRQEA: differential real-coded quantum-inspired evolutionary algorithm EGA: enhanced genetic algorithm EPSO: enhanced particle swarm optimization FEP: fast evolutionary programming GA: genetic algorithm IFEP: improved fast evolutionary programming IPSO: improved particle swarm optimization LWPSO: local vision of particle swarm optimization with inertia weight MDE: modified differential evolutionary MHDE: modified hybrid differential evolutionary MPSO: modified adaptive particle swarm optimization PSO: particle swarm optimization QEA: quantum-inspired evolutionary algorithm RQEA: real-coded quantum-inspired evolutionary algorithm SPPSO: small population-based particle swarm optimization SPSO: standard particle swarm optimization TLBO: teaching learning based optimization

Please cite this article in press as: Zheng JH, et al., Reliability constrained unit commitment with combined hydro and thermal generation embedded using self-learning group search optimizer, Energy (2015), http://dx.doi.org/10.1016/j.energy.2014.12.036