A Novel Quantum-Inspired Approximate Dynamic Programming Algorithm for Unit Commitment Problems Hua Qin1, a

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1 3rd Iteratioal Coferece o Materials Egieerig Maufacturig Techology ad Cotrol (ICMEMTC 2016) A ovel Quatum-Ispired Approximate Dyamic Programmig Algorithm for Uit Commitmet Problems Hua Qi1 a 1 Guagxi Key Laboratory of Power System Optimizatio ad Eergy Techology Guagxi Uiversity aig Chia a qihua263@163.com Keywords: Approximate Dyamic Programmig; Quatum Computig; Uit Commitmet Abstract. A ovel quatum-ispired approximate dyamic programmig algorithm (ADP) is proposed for solvig uit commitmet (UC) problems. The quatum computig theory is applied to tackle some ew issues risig from ADP. I details the uit states i UC problems are expressed by the quatum superpositio. The the collapsig priciple of quatum measuremet is applied to solve the Bellma equatio of ADP speedily. Based o the quatum rotatio gate the pre-decisio states of the ADP are geerated by quatum amplitude amplificatio techology. I the proposal algorithm the quatum computatio balaces betwee state space exploratio ad exploitatio automatically. Test cases of UC are performed to verify the feasibility of the proposal approximate algorithm for the rage of 10 to 100 uits with 24-hour with ramp rate costraits. The experimetal results show that the quatum-ispired ADP algorithm ca fid the better sub-optimal solutios of large scale UC problems withi a reasoable time. 1. Itroductio Uit commitmet (UC) is a importat task i the power system operatio. A good schedule of geeratig uits may save the electric utilities may productio costs. The UC problem ca be formulated as a large-scale oliear mixed-iteger combiatorial optimizatio problem but it is difficult to obtai the global optimal solutio. Though a lot of methods have bee used to solve the UC problem such as the priority list (PL) [1] dyamic programmig (DP) [2][3] mixed-iteger programmig (MIP) methods [4][5] meta-heuristics [6]-[8] quatum-ispired meta-heuristic methods [8]-[10] etc. they still have some flaws. For example DP method for solvig UC is easy to ecouter the curse of dimesioality problem [2][3]. Fortuately this problem ca be solved by the approximate dyamic programmig (ADP) method efficietly. Moreover ADP ca gai the optimal or better sub-optimal solutios of may egieerig problems. To implemet ADP geerally the DP algorithm is costructed over a limited set of states. For istace the priority list is used to geerate the limited set of uit commitmet problem s states [2] the the ADP is implemeted o the limited set to solve the UC problem. ADP is implemeted usig quatum amplitude amplificatio i [11]. Recetly the method of quatum amplitude amplificatio based o Grover iteratio was applied to the actio selectio of the Bellma equatio i dyamic programmig ad solve the curse of dimesioality [12]. I [13] this method is also used to solve the Bellma equatio of reiforcemet learig ad makes a good tradeoff betwee exploratio ad exploitatio. However the quatum amplitude amplificatio based o the quatum rotatio gate is much easier to implemet ad combie with combiatorial optimizatio problems compared with the oe based o Grover iteratio [14]. The most importat work of ADP i recet years is the forward iteratio ADP algorithm theory proposed by Prof. Powell of Priceto Uiversity i the year 2013[15]. The forward ADP algorithm ca overcome the curse of dimesioality. However there are still some ope issues. The mai oe is how to make a balace betwee exploratio ad exploitatio which affects the quality of solutio of ADP sigificatly. I the case for solvig UC problem the geeral exploratio strategies such as Boltzma machie [15] epsilo-greedy [15] kowledge gradiet ad Bayesia method [15] R-max [15] etc. could ot explore the state space of UC problem effectively [3][15]. Therefore it The authors - Published by Atlatis Press 1257

2 is ecessary to seek some ew explorig strategies for the UC problems specially. I this paper a ovel quatum ispired ADP algorithm (QI-ADP) for solvig large scale UC problems is proposed by usig the quatum computig theory to tack some issues i ADP. The uit states i UC problems are expressed by the quatum superpositio. The the collapsig priciple of quatum measuremet is applied to solve the Bellma equatio of ADP speedily. The pre-decisio states of the ADP are geerated by quatum amplitude amplificatio techology. Test cases of UC problems which rage from 10 to 100 uits with 24-hour are performed to verify the feasibility of the proposed algorithm. The experimetal results show that the quatum-ispired ADP algorithm ca fid the better sub-optimal solutios of large scale UC problems withi a reasoable time. It s feasible that usig quatum computig theory tackle some ew issues risig from ADP. 2. The Model of UC Problem The UC problem is to determets a uit commitmet schedule to miimize the total operatig cost of all geeratig uits subject to a umber of system ad uit costraits. The objective fuctio of UC problem is give as the followig T mi F u [ f ( P ) (1 u ) C ] (1) c it i it it 1 it t1 i1 where u it presets the state of uit i at hour t (0 whe the uit is off ad 1 otherwise). The fuel cost fuctio of uit i at hour t is expressed as 2 (2) f ( P ) a ( P ) bp c i i t i i it i i t i where P it is the power geeratio of uit i at hour t. The a i b i ad c i are the cost coefficiets of uit i. C it is the startup cost ad ca be described as t Ci hot Ti off Ti Ti off Ti cold Cit (3) cold t Ci Ti Ti off Ti cold where C ihot is the hot startup cost of uit i ad C icold is the cold startup cost of uit i. T it is the cotiuous olie time (+) or offlie time (-) of uit i. T ioff is the miimum dow time of uit i. T icold is the cold start time of uit i. The power balace costrait is give by uit Pit PDt 0 (4) i1 where P Dt is the system demad at hour t. The system spiig reserve is expressed by uit Pimax PDt SRt (5) i1 where P imax is the maximum power output of uit i. S Rt is the required spiig reserve at hour t. The active power from uit i is limited as uit Pimi Pit uit Pimax (6) where P imi ad Pi max are the miimum ad maximum power output of uit i. The miimum up/dow time costrais are give by (7) ( uit 1 uit )( Tit 1 Tio ) 0 ( uit uit 1 )( Tit 1 Tioff ) 0 where T io is the miimum up time of uit i. T ioff is the miimum off time of uit i. The ramp rate costraits are idicated as below P P u P ( u u ) P (1 u ) P (8) it 1 it 1 it 1 iup it it 1 istart it imax P P u P ( u u ) P (1 u ) P (9) i t1 i t i t i dow i t1 i t i shut i t1 imax 1258

3 where P iup P istart P idow ad P ishut are the limit of uit i for ramp up ramp dow startup ramp ad shutdow. The UC model i this paper is costituted by the above equatios raged from (1) to (9). 3. The QI-ADP Algorithm for UC Problem 3.1 State of UC problem ad quatum state The state of UC problem is give by (10). The value of u it is 0 or 1. Therefore (10) has 2 T differet combiatio states which are from T tot. I this paper the quatum state is employed for (10) to express the states of UC problem for the sake of usig quatum theory sice u it ca be cosidered as a bit i the traditioal iformatio sciece. u11 u12 L u 1 u22 u22 u2 M L (10) M M M M ut1 ut2 L ut I the quatum iformatio sciece a iformatio uit (a quatum bit qubit) is represeted by a superpositio state as = 0> + 1 (11) where 0> ad 1> correspod to traditioal logi states 0 ad 1. α ad β are complex coefficiets ad satisfy But (11) is hard to be expressed ad stored i a traditioal computer. Based o the characteristics of α ad β i (11) a vector as (12) ca be expressed ad stored as followed: (12) The (10) is expressed by the quatum state as L 1 j L L 1 j L 1 Q j L L 2 Q 2 Q L 2 j L 2 (13) M Q M T T1 T2 L T j L T T1 T2 L T j L T where Q t expresses all the uit commitmet states of uits at hour t. (13) has 2 T differet base states which are from T to T. Thus the states of (10) ca be quatized by (13) The QI-ADP Algorithm for UC Problem Accordig to the ADP algorithm framework i [15] the quatum-ispired ADP algorithm for UC problem is described as follows. Step0: Iitializig. (a) Iitialize the quatum amplitude matrix (13). (b) Iitialize the value fuctio t V 0 0 t T. (c) Iitialize the iteratio couter =1 ad maximum umber of iteratios as. 0 (d) Iput the model parameters of UC problem. (e) Set pre-decisio state S 0 with the iitial states of uits. Step1: Get the pre-decisio state matrix P by quatum observig (13). 1259

4 Step2: do for t=012.t (a) Solve the Bellma equatio i (14) by quatum observig to obtai vˆt ad t 1 S. a t t t M a t vˆ mi C ( ) 1( ( )) t t S at V S S at ata (14) (b) If t>0 update t 1 V usig t1 ( t1 ) (1 ) t1 ( t1 ) t a 1 1 a 1ˆ V S V S v (15) (c) Get the ext pre-decisio state t 1 S from the t+1 lie of matrix P. Step3: process the solutio of th iteratio. (a) Costruct the curret solutio of th iteratio ad store it i matrix M. (b) Costruct the curret best solutio of th iteratio ad store it i matrix B. (c) Update (13) usig the values of objectio fuctio f(m) ad f(b). (d) =+1. If 0 go to step1. Step4: Retur the best solutio B of UC problem. The followig will discuss some mai issues i the above QI-ADP algorithm. I step1 the form of the pre-decisio state matrix P is the same as (10). The quatum observig method to costruct P is that for a pair of (α it β it ) i (13) u it is 1 if α it 2 < β it 2 otherwise u it is 0. After observig each pair (α it β it ) i (13) the matrix P is costructed. As P represets a schedule of UC problem it must satisfy UC costraits. If P violates the UC costraits a heuristic method [8]-[10] is used to adjust them. I step3 the solutio of th iteratio is give by 0 S a u11 u12 L u 1 1 Sa u21 u22 u2 M L (16) M M M M M T 1 Sa ut 1 ut2 L ut M is a schedule of UC problem. If M violates the UC costraits the heuristic method [8]-[10] is used to adjust them. Let B is the best solutio of -1 iteratio with the same formulatio as (10). I matrix B the meaig of b it is same as u it i (16). If f(m)< f(b) the curret best solutio is updated with B=M. I step3 the (13) is updated by the quatum rotatio gate. For a pair of (α it β it ) i (13) the updatig method is show as % it cos( it ) si( it ) it % (17) it si( it ) cos( it ) it where it is the quatum rotatio agle correspodig to (α it β it ). it ca be determied through the followig adaptive equatio (/ 0 ) it 0e (18) where 0 is the give iitial rotatio agle.τis the costat coefficiet. 4. Test results The proposed algorithm is simulated o a PC with Itel Core2 Quad CPU (2.5GHz) 2GB RAM. The algorithm is implemeted with Matlab o Widows platform. The data of test cases for UC problems are take from [5]. Six test cases are from 10 up to 100 uits with 24 hours with ramp rate costraits. Let P iup =P idow =0.2P imax which meas the ramp up ad ramp dow rate of each uit are take to be 20% of its maximum power output. Let P istart =P ishut =2P imi which meas that the startup ad shutdow ramp rate of each uit are its double miimum power output. The Outer-Ier Approximatio approach (OIA) outer approximate (OA) ad tighter several-step outer approximatio (TOA) are very successful determiistic mix-iteger mathematical programmig 1260

5 methods ad they ca obtai high quality solutios for UC problems [5]. The results show i Tab.1. Tab.1. The compariso of results o six test cases Uits OA[7] TOA[7] OIA[7] QI-ADP From Tab.1 Comparig results of QI-ADP to the oes of OA TOA ad OIA the average differeces are 0.87% which are very small. The results idicate that QI-ADP ca obtai the high-quality sub-optimal solutios of test cases with ramp rate costraits. Ruig time(s) QI-ADP OIA QBPSO umber of uits Fig.1. The compariso of ruig time of several methods Fig.1 illustrates the compariso of ruig time of QI-ADP OIA ad Quatum-Ispired Biary PSO(QBPSO)[10] which idetify that the characteristics of ruig time of QI-ADP ad OIA are early liear while the characteristic of QBPSO is oliear. That s to say QI-ADP ca solve the UC problems efficietly ad obtai the high-quality solutios. As a result it s feasible usig quatum computig theory to tackle some ew issues risig from ADP itself. 5. Coclusio A ovel quatum-ispired ADP algorithm is proposed i this paper. The quatum computatio theory is used to hadle the issues of ADP. The quatum amplitude amplificatio techology is applied to explore the state space of UC problem ad automatically makes the balace betwee exploratio ad exploitatio. The Bellma equatio is solved efficietly by the quatum observig method. The pre-decisio states of ADP are geerated by the quatum computatio theory. The results of simulated experimets show that QI-ADP ca solve large scale cases of UC problems ad ca obtai a better sub-optimal solutio withi a reasoable time. Ackowledgemet This work is sposored by the atioal Basic Research Program of Chia (973 Program) (Project o.2013cb228205) the atioal Sciece Foudatio of Chia (Project o ) ad the Humaities & Social Sciece Research Project of the Miistry of Educatio of Chia (Project o.11yjazh080). Refereces [1] Tomoobu Sejyu Kai Shimabukuro Katsumi Uezato Toshihisa Fuabashi. A Fast Techique for Uit Commitmet Problem by Exteded Priority List [J] IEEE Trasactios o Power Systems (2) [2] Syder W. L. Powell H. D. Jr Raybur J. C. Dyamic Programmig Approach to Uit Commitmet [J] IEEE Trasactios o Power Systems (2)

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