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A System of Systems Engineering Approach for Unit Commitment in Multi-Area Power Markets Amin Kargarian, Student Member, IEEE, Yong Fu, Senior Member, IEEE, Ping Liu, Student Member, IEEE, and Chunheng Wang, Student Member, IEEE
Abstract—In power systems, the main grid might be a group of several interconnected areas. The areas can be self-governing with their own polices and rules. According to the concept of system of systems (SoS) engineering, this paper presents a decentralized decision-making framework to determine an economical hourly generation schedule for a multi-area power system. Each self-governing area is modeled as an independent system, and the entire power system is modeled as a SoS. The proposed decentralized unit commitment algorithm takes into account the privacy of each independent system, and only a limited data information such as power exchange between the areas, needs to be exchanged between the systems. An iterative decentralized optimization model is presented to find the optimal operating point of all independent systems in the SoS-based power system architecture. The numerical results show the effectiveness of the proposed SoS framework and solution methodology. Index Terms— Multi-area power system, system of systems, decentralized decision-making framework, unit commitment.
C
I. INTRODUCTION
OLLABORATION between the areas in a multi-area power system improves the entire power market efficiency, and system security and reliability [1]-[3]. The areas with low-cost generation sources can supply electric power for loads located in the areas with high-cost generating units [4]. Moreover, during any disturbance or variation in one area, the other areas can control the entire system and for example keep the system away from instability point, and alleviate transmission congestion [5], [6]. However, the areas are connected together to build the entire power system, each of which might be a self-governing system with its own operation and control regulations. Thus, the multi-area power system might be defined as a system including different independent systems. When all systems collaborate together to improve security and reliability of the whole power system, each independent system intends to increase its own benefit [7], [8]. Hence, the operation and control of a multi-area power system can be described based on the concept of system of systems (SoS) engineering. A SoS is identified as a group of components that are separately considered as systems and are both administratively and operationally self-governing. Operational autonomy means The authors are with the Department of Electrical and Computer Engineering, Mississippi State University, Mississippi State, MS 39762 USA (e-mail:
[email protected],
[email protected],
[email protected],
[email protected]).
978-1-4799-6415-4/14/$31.00 ©2014 IEEE
that if the SoS is disassembled into its individual systems, these systems have to be capable of suitably operating independently to perform valid functions for their own right and continue to work to fulfill the customer purposes [9]-[12]. Fig.1 depicts a typical SoS including three independent systems. The systems m, n, and q have common parts and each of which has its own independent design requirements. Although there are many similarities between systems engineering and system of systems engineering, but they are different field of study. The traditional systems engineering intends to find the optimal operation point of and individual system. And the SoS tries to find the optimal operating point of the networks including interacting systems that work together to satisfy various objectives when guarantee constraints of the systems [7], [11]. System of systems
System m
System n
System q
Fig.1. A typical system of systems.
The unit commitment (UC) problem is an important issue in power systems operation, and in [13]-[16] the centralized optimization algorithm was applied to solve the UC problem in the multi-area power systems. However, in such a SoS-based multi-area power system, the dispatching and operational independence of each area should be respected. Using centralized optimization algorithms which need all the information of the autonomous systems might not be the appropriate way to find the optimal UC of such SoS-based multi-area power system since generators, loads and network information of each autonomous system, are usually considered commercially sensitive. On the other hand, determining the entire multi-area power system operation including huge numbers of design and control variables in one centralized optimization model might be challenging [17]. In this paper, a decentralized decision-making framework is presented in order to determine the UC solution of a multi-area SoS-based power system. In this SoS structure, each selfgoverning area is modeled as an independent system with its own rules and policies. The proposed decentralized decision
2
making framework considers the privacy of each independent system, and it needs to receive only limited number information, power exchange between the areas, from each independent system in order to coordinate the areas and find an economical hourly generation schedule for the entire power system. Therefore, the independent systems do not need to share all of their information of generators, loads and network with other systems. An iterative decentralized model is proposed to find the optimal operating point of all independent systems in the SoS-based power system architecture. The IEEE 48-bus and 118-bus test systems, are studies and the results are discussed to show the effectiveness of the proposed SoS framework and solution methodology.
operating point of the other systems. Therefore, there is a set of shared variables between the UC problem of the system m (UCm) and other systems represented by z . Because of these shared variables, the UC problems of the independent systems cannot be solved separately. In fact, the power exchange between two areas is the shared variable coupling the systems together. In order to decompose the UC problems and make them independently solvable, pseudo generations and loads are introduced for each independent system to model the shared variables and formulate the self-governing objective functions and constraints. Assume that in Fig.2, the power is transferred from system y ∈ {n, q} toward area m at operating hour t. From the
II. DECENTRALIZED SOS-BASED DECISION-MAKING FOR UC
perspective of area m, the line flow is modeled as a pseudo generator supplying to system m ( PGmyt ); and from the
Fig.2 shows a power system consisting of three individual areas connected through the tie-lines (T). Each area has its own local network, generation sources (G) and loads (L), and also the areas might have their own independent system operator. Therefore, each area is modeled as an autonomous system (area m is system m, area n is system n, and area q is system q), and the entire multi-area power system is a SoS. Area n
Area m [Lm]
[Ln] Tmn
Network m
[Gm]
Tmq
[Lq]
Network n
[Gn]
perspective of area y, the line flow is modeled as a pseudo load supplied by system y ( PDmyt ) as depicted in Fig.3. Therefore, at hour t, the shared variable is the pseudo generation for the system m and the pseudo load for system y. It should be noted that the pseudo generation might be negative which means the power is delivered to system y by system m, and the pseudo load of system y may also be negative. To satisfy the physical feasibility of operating points of the systems, the consistency constraint expressed by (2) is introduced. ct = PGmyt − PDmyt = 0 y ∈ {n, q} (2) Constraint (2) should be regarded in the optimization problems of both systems m and y. Using the penalty function, the consistency constraint can be relaxed. Then, the UC problem of system m (UCm) expressed by (1) is rewritten as follows.
Tnq
Network q
Min Area q [Gq]
s.t.
Fig.2. A three-area power system.
NT NGm
∑ ∑ Fi ( Pit ) I it + SUDit
(1)
t =1 i =1
s.t.
gm ( ~ xm , z) ≤ 0 h (~ x , z) = 0 m
NT
t =1 i =1
t =1
∑ ∑ Fi ( Pit ) I it + SUDit + ∑ π (cmyt )
m
m
myt
myt
NT NGm
NT
Min ∑ ∑ Fi ( Pit ) Iit + SUDit + ∑ (αmyt ( PGmyt − PD*myt ) t =1 i =1
+
t =1
βmyt o ( PGmyt − PD*myt )
2 2
), ∀y PDmn
Fi (.) is generation cost curve of unit i, Pit is the power
variables which are exclusively for independent system m. Since the areas are connected together through the tie-lines, the optimization problem of one of them may change the
(3)
In this paper, a second-order function called Augmented Lagrangian is used to model the penalty function π [18]. The objective function of the UCm is formulated by (4).
where SUDit is startup and shutdown cost of unit i at time t, generated by unit i at time t, I is binary variable representing ON/OFF states of the units, NT is the time horizon, NGm is number of generating units in the independent system m, and gm and hm are sets of equality and inequality constraints of the UC problem such as power balance, minimum on/off time xm is the set of local design limits, and ramping limits. In (1), ~
∀y
gm ( ~ x m , PGmyt , PDmyt ) ≤ 0 h (~ x , PG , PD ) = 0 m
Assume that the UC problem for the independent system m is formulated as follows.
Min
NT NGm
UCm
PGmn
PGmq
UCn
PGnq
PDmq
PDnq
UCq
Fig.3. Pseudo generation and load model for the three-area power system.
(4)
3
The penalty function consists of two terms, linear and quadratic. α myt and β myt are multipliers associated with linear and quadratic terms, respectively, and they will be updated during the iterative solving process. Note that in the penalty function, the pseudo generation PGmyt needs to be * is received determined, but the value of pseudo load PDmyt
Step 6: Set k=k+1 and s=0, and update the values of penalty multipliers α tk and β tk using (20) and (21), and go to Step 2.
α t( k +1) = α t( k ) + 2( β t( k ) ) 2 ( PGt − PDt )
β t( k +1) = λβ t( k ) (21) where the coefficient λ is necessary to be equal or larger than
one in order to get the converged optimal results.
from system y. Similarly, the objective function of UCn can be formulated by (5).
k=1 s=0
NT NGn
Min ∑
(20)
∑ Fi ( Pit ) I it + SUDit
t =1 i =1
Set initial value for
NT
2
t =1 NT
2
* + ∑ (αmnt ( PG*mnt − PDmnt ) + βmnt o ( PGmnt − PDmnt ) )
αtk
(5)
βtk
PDt*
PGt*
2
+ ∑ (αnqt ( PGnqt − PD*nqt ) + βnqt o ( PGnqt − PD*nqt ) )
s=s+1
2
t =1
And setting y ∈ {m, n}, the UCq is formulated as follows. NT NGq
Min ∑
∑ Fi ( Pit ) I it + SUDit
∀y
t =1 i =1
NT
+∑
t =1
(α yqt ( PG*yqt
− PD yqt ) +
(6) β yqt o ( PG*yqt
2
− PD yqt ) )
Update the value of
αtk
Solve UC for independent system s and update
βtk
PDt*
2
k=k+1 s=0
III. SOLUTION PROCEDURE An iterative procedure is addressed in this section in order to solve the proposed decentralized optimization algorithm and to determine the optimal UC results for each independent system and accordingly for the SoS-based multi-area power system. Fig.4 shows the flowchart of the solving process, and also its steps are summarized as follows.
s ≥ NIS
No
system s using related PGt* and PDt* . Step 3: Use the results obtained in Step 2, and update the value of those pseudo generations and loads which are among the design variables in Step 2. Step 4: If s is larger than number of independent systems (NIS) return to Step 2, otherwise go to Step 5. Step 5: Check the stopping criteria (7) and (8). If they are not satisfied, go to Step 6; otherwise, the converged optimal result is obtained and the solution procedure stops.
PGt − PDt ≤ ε1 ∀t f s (x
) − f s (x
f s (x
(k )
)
Stop
Fig.4. Flowchart of the solving process.
Step 2: Set s=s+1. Solve the UC problem for independent
(k −1)
converged?
Yes
pseudo generations PGt* and pseudo loads PDt* , and penalty multipliers α tk and β tk .
No
Yes
Step 1: Set s=0, the iteration index k=1, and initial values for
(k)
PGt*
)
(7)
≤ ε2
(8)
where f s is the objective function of the independent system S.
IV. NUMERICAL RESULTS This paper presents two case studies including the IEEE 48bus and IEEE 118-bus test systems in order to illustrate the performance of the proposed decentralized SoS-based UC algorithm. All cases utilize ILOG CPLEX 12.4 on a 3.4GHZ personal computer. A. 48-Bus System The entire 48-bus system has 20 generating units, and 34 demand sides. The power system includes 2 areas, as shown in Fig.5, each of which is an independent system with their own rules and policies. Independent system one (S1) has 10 units and 17 loads; and system two (S2) consists of 10 units and 17 demand sides. According to the characteristics of each area, a local UC is formulated for each independent system. There is one shared variable between the UC problems of these two systems which is modeled as a pseudo generator in UC problem 1 and a pseudo load in UC problem 2. The initial
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value for
PGt* = PDt* = 0,
αt0 = βt0 = 1 (t=1:24); and
convergence thresholds ε1, and ε2 are set to 0.1, 0.001, respectively. The proposed SoS-based UC is applied to find the optimal operating point of these two independent systems. The algorithm converges after 4 iterations. ON/OFF states of the generating units of each independent system are shown in Table I. And the generation dispatch over the time horizon is depicted in Fig.6. Generators 2, 7, and 8 of S1, and units 4, 5, and 6 of S2, which are very expensive units, are scheduled to be OFF; and generators 3, 6, and 9 of S1, and units 3 and 7 of S2 are expensive units and are OFF in the off-peak hours and are only committed when the load is high. The operating cost and total calculation time of the independent systems are summarized in Table II. Note that the calculation time shown in this table is the summation of the solver time in all iterations. Since the generating sources of S1 are cheaper that the units of S2, the operating cost of S1 is less than S2. The total operating cost of the SoS-based multi-area power system is $721996, and the total calculation time of the proposed decentralized algorithm is 2.8 s. G7
G7
G9
G8
G9
G8
(b) Fig.6. Generation dispatch for a) independent system 1 and b) independent system 2.
G10
G10 G6
G6
G4
G4 G5
G5
G1
G3
G2
G1
G3
G2
Fig.5. IEEE 48-bus test system. TABLE I ON/OFF STATES OF THE UNITS FOR EACH INDEPENDENT SYSTEM Ind. Syst.
S1
S2
Units 1 2 3 4-5 6 7-8 9 10 1-2 3 4-6 7 8 9 10
Hours (1-24) 1 0 0 1 0 0 0 1 1 1 0 0 1 0 1
1 0 0 1 0 0 0 1 1 0 0 0 1 1 1
1 0 0 1 0 0 0 1 1 0 0 0 1 1 1
1 0 0 1 0 0 0 1 1 0 0 0 1 1 1
1 0 0 1 0 0 0 1 1 0 0 0 1 1 1
1 0 0 1 0 0 1 1 1 1 0 0 1 1 1
1 0 1 1 0 0 1 1 1 1 0 0 1 1 1
1 0 1 1 0 0 1 1 1 1 0 0 1 1 1
1 0 1 1 0 0 1 1 1 1 0 1 1 1 1
1 0 1 1 0 0 1 1 1 1 0 1 1 1 1
1 0 1 1 1 0 1 1 1 1 0 1 1 1 1
1 0 1 1 1 0 1 1 1 1 0 1 1 1 1
1 0 1 1 1 0 1 1 1 1 0 1 1 1 1
1 0 1 1 1 0 1 1 1 1 0 1 1 1 1
1 0 1 1 1 0 1 1 1 1 0 1 1 1 1
(a)
1 0 1 1 0 0 1 1 1 1 0 1 1 1 1
1 0 1 1 0 0 1 1 1 1 0 1 1 1 1
1 0 1 1 0 0 1 1 1 1 0 1 1 1 1
1 0 1 1 0 0 1 1 1 1 0 1 1 1 1
1 0 1 1 0 0 1 1 1 1 0 0 1 1 1
1 0 1 1 0 0 1 1 1 1 0 0 1 1 1
1 0 1 1 0 0 1 1 1 1 0 0 1 1 1
1 0 1 1 0 0 1 1 1 1 0 0 1 1 1
TABLE II OPERATING COST AND CALCULATION TIME OF EACH SYSTEM IN CASE 1 Ind. Syst. Operating cost ($) Total calculation time (s) S1 338813 1.2 S2 383183 1.6
1 0 1 1 0 0 0 1 1 1 0 0 1 1 1
B. 118-Bus System The IEEE 118-bus test system is used to study the decentralized UC algorithm. The entire system has 54 generating units and 91 demand sides. As shown in [19], this power system has three areas each of which is an independent system with its own rules and policies. The first independent system (S1) includes 42 buses, 19 units, and 30 loads; the second system (S2) has 48 buses, 20 units, and 37 loads; and the third system (S3) consists of 28 buses, 15 units, and 24 loads. A local UC problem is formulated for each independent system. UC problem 1 has one pseudo generator supplied by S2; UC problem 2 includes one pseudo loads related to S1 and a pseudo generation supplied by S3; and UC problem three consists of a pseudo load associated to S2. The initial value for
PGt* , PDt* , αt0 , and βt0 (t=1:24) are set to 200, 200, 1, and 1, respectively; and the convergence thresholds ε1 and ε2 are set to 0.1 and 0.001. The SoS-based UC algorithm is applied to find the optimal operating point of these three independent systems. The algorithm converges after 6 iterations. The proposed decentralized algorithm takes 20 s in order to obtain an optimal solution with a total operating cost of $1342162 for the entire SoS-based power system. Table III shows the detail of calculation time and operating cost of each independent system. S2 has higher load compared with S1 and three, and thus its operating cost is higher than others. Moreover, the local UC problem of area two has more number of design variables, shared variables, and constraints in comparison with other two UC problems, and the solver needs more time to solve the optimization problem of S2.
5 TABLE III OPERATING COST AND CALCULATION TIME OF EACH SYSTEM IN CASE 2 Ind. Syst. Operating cost ($) Total calculation time (s) S1 373571 3 S2 625329 14 S3 343262 3
V. CONCLUSION This paper presented a decentralized decision-making framework in order to determine an economical hourly generation schedule for a multi-area power system. The entire power system was modeled as a system of systems in which each self-governing area was an independent system. Using the proposed algorithm, each area required to only share the amount of power exchange with its own neighbors. If fact, the areas do no need to share all of their own information, which might be commercially sensitive, to find the UC solution, and therefore their privacy of each independent system was taken into account. An iterative decentralized model was proposed to find the optimal operating point of all independent systems in the SoS-based power system architecture. The numerical results on the IEEE 48-bus and IEEE 118-bus test systems showed the efficiency of the proposed SoS framework and solution methodology. REFERENCES [1] [2] [3]
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environment,” Innovative Smart Grid Technologies - India (ISGT India), 2011. [15] F. N. Lee, and Q. Feng, “Multi-area unit commitment” IEEE Trans on Power Syst., vol. 7, no. 2, pp. 591-599, May 1992. [16] C. Wang, and M. Shahidehpour, “A Decomposition approach to nonlinear multi-area generation scheduling with the tie-line constraints using expert systems,” IEEE Trans on Power Syst., vol. 7, no. 4, pp. 1409-1418, Nov 1992. [17] R. Amgai, J. Shi, S. Abdelwahed, and Y. Fu, “Research trends in high performance computing application on large scale power system operation,” Grand Challenges in Modeling & Simulation, Italy, pp. 112-119, 2012. [18] S. Tosserams, L. F. P. Etman, P. Y. Papalambros, and J. E. Rooda, “An augmented lagrangian relaxation for analytical target cascading using the alternating directions method of multipliers,” Struct. Multidisc Optim., vol. 31, no. 3, pp. 176–189. 2006. [19] Y. Fu, M. Shahidehpour, and Z. Li, “AC contingency dispatch based on security-constrained unit commitment,” IEEE Trans on Power Syst., vol. 21, no. 2, pp. 897-908, May 2006. Amin Kargarian (S’10) received his BS and MS degrees in E.E. from the University of Isfahan and Shiraz University, Iran, in 2007 and 2010, respectively. He is currently working toward the Ph.D. degree in Electrical Engineering at Mississippi State University. His research interests include power system operation and optimization, system of systems engineering, and distributed optimization techniques. Yong Fu (SM’13, M’05) received his BS and MS in E.E. from Shanghai Jiaotong University, China, in 1997 and 2002, respectively and Ph.D degree in E.E. from Illinois Institute of Technology, USA, in 2006. Presently, he is an assistant professor in the Department of Electrical and Computer Engineering at Mississippi State University. Ping Liu (S’11) received her BS and MS in E.E. from Southwest Jiaotong University, China, in 2007 and 2010, respectively. She is currently working toward the Ph.D. degree in Electrical Engineering at Mississippi State University. Her research interests include power system operation and economics, and optimization with uncertainty in smart grid. Chunheng Wang (S’12) received his B.S. degree in Electrical Engineering from Dalian University of Technology, China in 2008. He also received his M.S. degree in Electrical and Computer Engineering from Illinois Institute of Technology in 2010. Currently, he is pursuing his Ph.D. degree in Electrical Engineering at Mississippi State University. His research interests include power systems optimization and economics.
