Robust n–k contingency constrained unit commitment ...

www.ietdl.org Published in IET Generation, Transmission & Distribution Received on 20th January 2014 Revised on 2nd May 2014 Accepted on 9th May 2014 doi: 10.1049/iet-gtd.2014.0052

ISSN 1751-8687

Robust n–k contingency constrained unit commitment with ancillary service demand response program Jamshid Aghaei, Mohammad-Iman Alizadeh Department of Electrical and Electronics Engineering, Shiraz University of Technology, Shiraz, Iran E-mail: [email protected]

Abstract: Today, many Independent System Operators (ISOs) establish programs to manipulate the reserve provided by Demand Response (DR) in ancillary service markets. In this study, Ancillary Service DR (ASDR) programs, recently introduced by Federal Energy Regulatory Commission, is integrated to an n–k Contingency Constrained Unit Commitment problem to investigate the capability of the DR Providers (DRPs), newly added ancillary market participants, in mitigating the impacts of simultaneous multiple contingencies in a power system. In the proposed model, an n–k security criterion by which power balance constraint is satisfied under any contingency state comprising simultaneous outages in generation units is investigated in the presence of the ASDR programs. Demand side reserve is supplied by DRPs, which have the responsibility of aggregating and managing customer responses to offer a bid-quantity to the ISO. The proposed formulation is a Mixed Integer Programming problem based on primal-dual optimisation, reported recently in literature. In addition, a detailed discussion about versatile effects of ASDR programs on n–k security criterion is presented to demonstrate the applicability of the proposed model.

Nomenclature

U0i V0i

t i n o T NQ Ngen k Ω x(.) DR(.) C min SC(.) NSC(.) D(.) UT(.), DT(.) S0i

index for time index for conventional unit number of components in contingency analysis index for discrete demand bid price segments set for time horizon number of discrete demand bid price segments number of generation units number of unavailable generators set for contingency indices binary contingency indicator, which is 0 if ith unit in hour t is on outage state and 1 otherwise elastic part of demand in an hour minimum demand side bid price spinning reserve offer cost of a generation unit non-spinning reserve offer cost of a generation unit initial demand in an hour minimum up/down time of a unit number of hours that unit i has been offline before starting the scheduling period

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P min(.), P max(.) Pi RUp-S(.) RU(.), RD(.) RUp-NS(.) SD(.) a(.), b(.) SU(.) DRmax x c(.) u(.) P(.) rS(.) r NS(.) u NS(.) v(.), w(.) CostGen(.) CostSpin(.) CostNspin(.) CostASDR(.)

number of hours that unit i has been online before starting the scheduling period initial commitment state of unit i at the beginning of the scheduling period minimum/maximum generating capacity of a unit decision variable of the subproblem upper bound for the up-spinning reserve contribution of unit i ramp up/down limit of a unit upper bound for the non-spinning reserve contribution of unit i shutdown ramp limit of generator i generation offer coefficients startup ramp limit of generator i maximum allowed demand curtailment incremental step between bid prices demand side bid price in an hour binary unit status indicator power generation in an hour spinning reserve in an hour non-spinning reserve in an hour binary non-spinning reserve indicator dual variables of the lower level optimisation problem offer cost spinning reserve cost non-spinning reserve cost demand side reserve cost

IET Gener. Transm. Distrib., 2014, Vol. 8, Iss. 12, pp. 1928–1936 doi: 10.1049/iet-gtd.2014.0052

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Introduction

In newly founded deregulated electricity markets, over than economic concerns, safety and security of the power system seem crucial. Hence, many security standards in power systems mainly concentrate on random failure of a single or at most two components, commonly known as n − 1 and n − 2 security criteria. In a large power system with lots of components, however, the adequacy of such criteria is questionable. Thus, a new approach is begun by many experts and researchers to assess the impacts of simultaneous multiple contingencies on power systems. Power balance constraint, however, may lead to infeasibilities in multiple contingency states. In order to prevent such infeasibilities, Demand Response (DR) programs are integrated to the power balance constraint. DR in its latest definition announced by Federal Energy Regulatory Commission is said to any changes in electric use by demand-side resources from their normal consumption patterns in response to changes in the price of electricity, or incentive payments designed to induce lower electricity use at times of high wholesale market prices or when system reliability is jeopardised [1]. Ancillary Service DR (ASDR) programs can provide demand side reserves to the upstream wholesale reserve markets. To make these programs implementable, new market participants designed as DR Providers (DRPs) are defined. A DRP is an interface between retail consumers and wholesale ISOs. In our approach, unlike the conventional reserve procurement, DRPs provide reserve by offering discrete bid prices to the reserve market in a way that power balance constraint is satisfied under both normal and contingency states. In this regard, competitive transactions between expensive spinning/non-spinning reserves of thermal generation units and demand side reserve resources are imaginable. Demand side reserves may have some outstanding features comparing to the conventional thermal units’ reserves such as low procurement cost, flexibility of capacity and availability within the least possible time. Within this context, this paper extends the consideration of multiple contingencies to the Contingency Constrained Unit Commitment (CCUC) in presence of ASDR programs.

2

Literature review

One of the first pioneer works about demand response and price elasticity of demand is [2]. In [2], price elasticity of demand in a pool-based electricity market has been taken into consideration when generation scheduling is done with responsive loads and different amount of incentives are paid to interruptible loads. Prior to [2], spot pricing concept was stated by Schweppe and his team in [3]. They showed customers can control their load based on the spot prices. The concept of demand responsiveness has been tracked with Federal Energy Regulatory Commission staff surveys issued every year since 2006 [4–6]. In [4], DRPs was classified into two major categories namely, time-based programs and incentive-based programs, for the first time. Modifications were done in [5] by announcing more detailed sub-branches of the incentive-based programs including voluntary-based programs, mandatory-based programs and market clearing programs. The recent issue of the surveys [6] declared fifteen versatile programs without any clustering because of combined programs. IET Gener. Transm. Distrib., 2014, Vol. 8, Iss. 12, pp. 1928–1936 doi: 10.1049/iet-gtd.2014.0052

In [7], a new demand response approach distinguished from demand elasticity platform has been presented. In [7], the amount of curtailed load is restricted with some intertemporal load characteristics such as minimum up/ down time, load pick up rates, minimum hourly curtailments and maximum daily curtailments to make load biddings more realistic. Security constrained unit commitment is considered as the base tool for implementing DRPs in [8], where it is suggested that DRP may pile up discrete retail customer responses and submit a bid-quantity offer to the Independent System Operators (ISOs). DRPs have been investigated in UC platform in [9, 10]. Carrión et al. proposed a real-time pricing scheme for DR in [9], where robust optimisation technique is employed to model price uncertainty in the time of use (TOU) programs. In the same field, Roscoe and Ault [10] investigated real-time DRP in highly penetrated renewable generation systems. In [10], it is recommended that instead of applying mandatory load curtailment programs, implementing real time pricing programs cannot only provide desired benefits from DRPs but also relive the discomfort caused by mandatory load curtailment programs. Recent tendencies towards robust optimisation are increasing in power system applications [11–15]. A day ahead unit commitment and real-time economic dispatch based on robust optimisation frame work are presented, in [11], to deal with the growing uncertainties in power systems because of the increasing penetration rate of renewables. The objective of the robust unit commitment problem is to minimise the total commitment cost and the worst-case dispatch cost. The objective of the robust dispatch problem, however, is to minimise the sum of the dispatch cost, the regulation capacity cost and the worst-case regulation performance cost. In [12], a robust counterpart of unit commitment (UC) is presented considering wind power generation as an uncertain parameter where the conventional stochastic scenario-based technique is replaced by uncertainty sets regarding the uncertainty parameter. In [13], net power injection is considered as the uncertain parameter and it is stated that unlike previous stochastic optimisation techniques that the scenario-based optimisations were highly dependent on the probability distribution function (PDF) of the uncertain parameters, in the current approach, it is not necessary to know the PDF of the uncertain parameter and uncertainty sets are defined instead. Recently, robust optimisation is also going to be applied in real ISOs’ decision making frameworks [14, 15]. In [14], Look-ahead unit Commitment in a two stage robust optimisation framework previously presented by PJM and Alstom Grids is solved with more computational tractable techniques such as linear decision rule and two stage decomposition approaches. A similar investigation this time on midcontinent independent system operator (MISO) is applied, in [15], where the commitment decisions are made in the first decision stage and the economic dispatch decisions are made in the second stage where the decision variables are a function of the uncertain load. Owing to moving towards extreme reliability for future power systems some papers started to investigate the vulnerability of the power systems to multiple contingencies [16–18]. According to such concerns stochastic optimisation techniques are applied in early papers [16]. Gradually, robust optimisation technique is detected as a proper technique to deal with multiple contingencies because two following reasons. First, there is no need to 1929

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www.ietdl.org know the probability distribution function of different scenarios and next is the acceptable computational tractability provided by robust optimisation [19, 20]. In [19], proposed an innovative application of robust optimisation in n–k contingency constrained unit commitment problem. They suggest that although power systems are not designed to operate safely in case of multiple contingencies occurred and n − 1 cannot guarantee the security of a large power system with lots of components. Hence, they proposed a primal-dual base optimisation procedure to provide the best feasible UC solution which immunises from the worst-case n–k contingency scenarios. In a more recent paper, both multiple generation and transmission contingencies are investigated in [20]. Authors proposed a two stage optimisation problem where the upper stage decides about commitment status of the generation portfolio and the lower stage discover the worst case contingency scenarios. Duality theory along with decomposition techniques are applied in [20] to not only convert the two stage problem to one stage but also improve the tractability of the solutions. The robust optimisation technique has been used in some more recent papers by considering energy and reserve scheduling under both generation and transmission contingencies [21]. Joint energy and reserve scheduling in a robust framework is implemented in a trilevel programming problem where the upper-level problem aims at minimising total costs of energy and reserves while ensuring that the system is able to withstand each contingency. The next lower level maximises the power imbalances occurred in contingency state and the last level models the operator’s best reaction for a given contingency by minimising the system power imbalance. According to one the premier duties of DRPs in providing reliability of the power system, these programs are accommodated to the current paper to enable investigating the real capabilities of such programs in mitigating the impacts of multiple contingencies. Implementing DRPs, however, requires infrastructures such as AMIs and two way communication bases. When ISO runs the optimisation process, a day ahead to the operation day, accepted demand reduction bids are recognised and sent to the customers through communication gateways. In the operation day, if any multiple contingencies occur, ISO notifies the accepted customers to curtail their loads in order to guarantee the desired reliability level. According to the mentioned reasons, the main contributions of this research work can be summarised as follows:

3 3.1

Problem formulation Demand response model

Ancillary service demand participation can be modelled as increasing bid-quantity offers to the ISO. Thus, the elastic part of demand comprises increasing discrete quantities, named as load curtailment bid blocks, as follows DR(t) = Dmin sinit (t) +

NQ


m(o, t)s(o, t)

(1)

o=1

where D min is the minimum amount of load curtailment and s init (t) are the binary indicators corresponding to the minimum demand curtailment block in hour t. μ(o, t) is continuous variable representing the difference between two subsequent curtailment blocks, as in (2), and s(o, t) is its corresponding binary indicator

m(o, t) ≤ q(o, t) − q(o − 1, t)

(2)

where q(o, t) is the quantity (load curtailment) offer in block o in hour t, as illustrated in Fig. 1. The amount of total accepted demand curtailment in an hour is a summation of all accepted individual blocks. To avoid obtaining infeasible results, the following constraint (3) on the binary variables is added s(1, t) ≤ sinit (t) s(o, t) ≤ s(o − 1, t),

o = 2, . . . , NQ

(3) (4)

m(o, t) ≤ q(o, t) − q(o − 1, t) Equations (3) and (4) say that a load curtailment block may be accepted when its previous less expensive blocks are accepted. With the same trend, demand-side reserve bids can be obtained as follows CostASDR (t) = C

min

D

s (t) +

min init

NQ


c(o, t)m(o, t)s(o, t)

o=1

(5) The monotonic increasing trend of demand-side bid prices is depicted in Fig. 1.

(i) Implementing ASDR programs for handling multiple contingencies in n–k contingency constrained UC framework. To the best of the authors’ knowledge, no previous work has investigated multiple contingency states in the presence of demand response programs and ancillary service markets. (ii) Robust approach is applied in the proposed model to find the best feasible UC solution which immunises from the worst-case n–k contingency scenarios. The proposed robust contingency constrained unit commitment along with demand response can be a potential tool for future ISOs/ RTOs to operate the next generations of power systems with extreme reliability and security. The rest of this paper is arranged as follows. Problem formulation is presented in Section 3. Section 4 is dedicated to the several case studies and discussions. Eventually, Section 5 is allocated to the conclusion. 1930 & The Institution of Engineering and Technology 2014

Fig. 1 Discrete demand bid prices and their corresponded costs [22] IET Gener. Transm. Distrib., 2014, Vol. 8, Iss. 12, pp. 1928–1936 doi: 10.1049/iet-gtd.2014.0052

www.ietdl.org 3.2

Pmin (i)u(i, t) ≤ P(i, t) ≤ Pmax (i)u(i, t),

Objective function

The contingency-constrained unit commitment problem determines the optimal generation schedule and reserve allocation under both normal and contingency states over a specific short term time span, Fig. 2. n–k security criterion by which all combinations of up to k unit outages are modeled in each period guarantees the power demand, in presence of ASDR program, is supplied under both normal and contingency states min

+ CostASDR (i, t)



(6)

where the objective function includes generation, spinning/ non-spinning reserve and demand side reserve costs, respectively. In order to present the formulation in a unique mixed integer linear programming (MILP) form, linear offer cost functions are considered for the generators CostGen (i, t) = a(i)u(i, t) + b(i)P(i, t)

P(i, t) + rS (i, t) ≤ Pmax (i)u(i, t), 0 ≤ rS (i, t) ≤ RUp−S (i)u(i, t),

CostNspin (i, t) = NSC(i)r (i, t)

(9)

NS

∀i [ Ngen , ∀t [ T

(13) (14) (15)

+ Pmax (i)[1 − u(i, t − 1)], ∀i [ Ngen , ∀t [ T (16) P(i, t) + rS (i, t) ≤ P(i, t − 1) + RU(i)u(i, t − 1) + SU(i)[u(i, t) − u(i, t − 1)] + Pmax (i)[1 − u(i, t)], ∀i [ Ngen , ∀t [ T (17)

where a(i) and b(i) are the linear offer cost coefficients, and u(i, t) is the binary variable which indicates the status of generation unit i in hour t. CostSpin(i, t) and CostNspin(i, t) are defined as follows (8)

∀i [ Ngen , ∀t [ T

P(i, t − 1) ≤ P(i, t) + RD(i)u(i, t) + SD(i)[u(i, t − 1) − u(i, t)]

(7)

CostSpin (i, t) = SC(i)rS (i, t)

(11)

∀i [ Ngen , ∀t [ T (12)

Pmin (i)uNS (i, t) ≤ rNS (i, t) ≤ RUp−NS (i, t)uNS (i, t), ∀i [ Ngen , ∀t [ T u(i, t) + uNS (i, t) ≤ 1,

Ngen T

 CostGen (i, t) + CostSpin (i, t) + CostNspin (i, t) t=1 i=1

∀i [ Ngen , ∀t [ T

(UTi
−U0i )V0i

[1 − u(i, t)] = 0

(18)

t=1

t+UT
i −1

u(i, n) ≥ UTi [u(i, t) − u(i, t − 1)]

n=t

∀i [ NGen , ∀t = (UTi − U0i )V0i + 1, . . . , T − UTi + 1 (19) 3.3

Constraints

T 


The constraints of the proposed model are as follows


n=t

Ngen

P(i, t) = D(t) − DR(t),

∀t [ T

 u(i, n) − [u(i, t) − u(i, t − 1)] ≥ 0

(20)

∀i [ NGen , ∀t = T − UTi + 2, . . . , T

(10)

(DTi −S 0i )(1−V0i )


i=i

u(i, t) = 0

(21)

t=1 t+DT
i −1



 1 − u(i, n) ≥ DTi [u(i, t − 1) − u(i, t)]

n=t

∀i [ NGen , ∀t = (DTi − S0i )(1 − V0i ) + 1, . . . , T − DTi + 1 (22) T 
 1 − u(i, n) − [u(i, t − 1) − u(i, t)] ≥ 0 n=t

(23)

∀i [ NGen , ∀t = T − DTi + 2, . . . , T 0 ≤ DR(t) ≤ DRmax , Ngen


∀t [ T

(24)

x(i, t, k)[P(i, t) + rS (i, t) + rNS (i, t)]

i=1

≥ D(t) − DR(t),

Fig. 2 Bilevel model IET Gener. Transm. Distrib., 2014, Vol. 8, Iss. 12, pp. 1928–1936 doi: 10.1049/iet-gtd.2014.0052

∀k [ C, ∀t [ T

(25)

The optimisation problem comprises regular unit commitment constraints as follows. Equation (10) states the power balance constraint including demand side reserve in 1931

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www.ietdl.org order to regulate the hourly demand in case of emergencies. Constraint (11) indicates the power generation limits and (12) is the spinning reserve constraint. Constraints (13) and (14) are spinning and non-spinning reserve limits, respectively, and (15) verifies that the non-spinning reserves are only provided by not-committed generation units. Constraints (16) and (17) are allocated to up/down ramp rate limits as well as startup and shutdown ramp. Relations (18)–(20) represent the minimum up time constraints and (21)–(23) are belonged to minimum down time constraints, respectively. Equation (18) is related to the initial status of the units. The term (UTi − U0i)V0i represents the number of initial periods during which unit i must be online. Set of constraints (19) is used for the periods following (UTi − U0i)V0i to satisfy the constraint for the minimum up time during all possible sets of consecutive periods with size UTi. Set of constraint (20) are needed for the last UTi − 1 periods, in other words, if a unit is started in one of these periods, it remains online during the remaining periods. Finally, constraints (21)–(23) are identical to constraint (18)–(20) by changing u(i, t) by 1 − u(i, t). The maximum allowable load curtailment of each hour is limited in (24). Equation (25) specifies that all available generation capabilities such as power generation, spinning and non-spinning reserves in all contingency states must satisfy the power demand minus curtailed load by implementing DRP. x(i, t, k) is applied to characterise contingency states, which is a constant equal to 0 if unit i is unavailable in period t under the contingency state k, and 1 otherwise. In order to have a robust scheduling against the worst situation with multiple contingencies (25) can be converted to an optimisation problem by defining a new vector of decision variables X (t) = [x1 (t), . . . ., xn (t)]T x(t) [ V

(26)

where xi (t) is a binary variable which is equal to 0 when generator i is unavailable and 1 otherwise. Consequently, the above single-stage optimisation problem can be formulated as a bilevel optimisation problem, where (6)– (24) represent the upper optimisation problem, and (25) is replaced by its corresponding lower optimisation problem, respectively, as follows d W (t) ≥ D(t) − DR(t),  d (t) = min W

x(i,t)

Ngen


∀t [ T NS

i=1 Ngen


x(i, t) ≥ n − k

(29)

i=1

0 ≤ e(i, t) ≤ 1,

3.4

Robust optimisation

Prior to finding the robust counterpart of the lower-level problem, it is worth to discuss briefly on robust optimisation. The robust counterpart of a linear optimisation problem is written, without loss of generality, as

∀i [ N , ∀t [ T



(30)

where dw(t) is an additional variable to specify the maximum power that can be supplied in period t under the worst-case contingency scenario for given amount of scheduled power output, spinning and non-spinning reserve. Equation (28) specifies that the lower level is a combination of out-of-service generators so that the available post-contingency power output in each period is minimised. For each period, security criterion is enforced to be restricted by the constraint (29), which considers all contingency states. Equation (30) states that e(i, t) is a 1932 & The Institution of Engineering and Technology 2014

min : cT x

(31)

∀a1 [ U1 , . . . , am [ Um

(32)

subject to Ax ≤ b,

where x [ Rn is the vector of decision variables, ai represents the ith row of the uncertain matrix A and takes values in the polyhedral uncertainty set Ui # Rn . In fact, when U is polyhedral, the subproblem becomes linear, and the robust counterpart is equivalent to a linear optimisation problem. Then, aTi x ≤ bi , ∀ai [ Ui , if and only if max aTi x ≤ bi ,

{ai [Ui }

∀i

(33)

which is called ‘subproblem’ and must be solved. Its structure determines the complexity of solving the robust optimisation problem [23]. According to dual optimisation theory, each constraint in the primal problem can be replaced by its corresponding dual variable and each primal variable has to be substituted with its corresponding dual constraint [23]. Accordingly, the lower optimisation problem, stated in (34) and (35), can be replaced by (36)–(38), where pTi represents the dual variables of the primal objective function. It is noted that x is not a variable of the lower optimisation problem, in (33) max aTi x

(34)

s.t. Di ai ≤ di

(35)

min pTi di

(36)

s.t. pTi Di = x

(37)

(27)

e(i, t)[P(i, t) + r (i, t) + r (i, t)] (28) S

binary variable. The resulted bilevel optimisation problem can be converted to a single stage MILP problem, which can easily be solved using off-the-shelf optimisation software, based on strong robust optimisation theory. The theory is briefly provided in the next section.

pi ≥ 0,

i = 1, . . . , m

(38)

Based on the strong duality theorem, each convex and continuous lower-level problem can be replaced by its constraints, the constraints of its dual problem and the strong duality condition [24]. In addition, unlike the stochastic optimisation approach, the uncertainty model in the robust optimisation formulation is not a probability distribution, but rather a deterministic set. Moreover, the lower-level problem of the current framework has a unimodular matrix structure which guarantees that for integer values of contingencies k there is an optimal solution. In other words, generator availability set X(t) = [x1(t), …, xn(t)]T is a polyhedral set defined by (29) and (30) where each vertex of the polyhedral indicates unavailability of one component and availability of others. This set, in robust optimisation nomenclature, is called polyhedral uncertainty set just like a generic set in (39). IET Gener. Transm. Distrib., 2014, Vol. 8, Iss. 12, pp. 1928–1936 doi: 10.1049/iet-gtd.2014.0052

www.ietdl.org Generally, the polyhedral uncertainty set can be described as    N 
 |u | ≤ G, |ui | ≤ 1 U = u  i=1 i

Units

ai − ai  ai

(40)

where ai and  ai are the nominal value and the deviation from the nominal value of the uncertain parameter ai, respectively. The scaled deviation of a parameter always belongs to [−1, 1]. Finally, the robust linear optimisation becomes min : cT x s.t.: pTi di ≤ bi , pTi Di = x,

(41)

i = 1, . . . , m i = 1, . . . , m

(42) (43)

pi ≥ 0

Ngen


w(i, t) ≥ D(t),

∀t [ T

(45)

i=1

v(t) − w(i, t) ≤ P(i, t) + rS (i, t) + rNS (i, t) ∀i [ Ngen , ∀t [ T w(i, t) ≥ 0,

∀i [ Ngen , ∀t [ T

v(t) ≥ 0, ∀t [ T

(46) (47) (48)

In addition, the budget of the uncertainty set is [0, 1]. It is worth noting that, the presented robust optimisation is not adaptive, so that the budget of uncertainty is fixed. Consequently, the result is an MILP framework with (6) as the objective function, (9)–(24) as constraints and (45)–(48) as the linear dual equivalents of (28)–(30). As a result, the MILP contingency constraint unit commitment can be solved in one stage with off-the-shelf linear optimisation softwares.

4

b(i), $/MWh

c(i), $/MW2h

Pmax(i), MW

Pmin(i), MW

1 2 3 4 5 6 7 8 9 10

1000 970 700 680 450 370 480 660 665 670

16.19 17.26 16.6 16.5 19.7 22.26 27.74 25.92 27.27 27.79

0.00048 0.00031 0.002 0.00211 0.00398 0.00712 0.00079 0.00413 0.00222 0.00173

455 455 130 130 162 80 85 55 55 55

150 150 20 20 25 20 25 10 10 10

Units

CSC(i), $/MWh 9000 10 000 1100 1120 1800 340 520 60 60 60

HSC(i), $/MWh 4500 5000 550 560 900 170 260 30 30 30

CST(i), h

MU(i), h

MD(i), h

1 2 3 4 5 6 7 8 9 10

5 5 4 4 4 2 2 0 0 0

8 8 5 5 6 3 3 1 1 1

8 8 5 5 6 3 3 1 1 1

(44)

Based on the stated primal-dual conversion rules, the bilevel optimisation problem, stated in (6)–(30), can be replaced by its mixed integer robust equivalent by introducing v(t) and w(i, t) as dual variables of (29), (30). Accordingly, dual problem of the lower part can be described as follows (n − k)v(t) −

a(i), $/h

(39)

where Γ is the budget of uncertainty, and ui is ‘scaled deviation’ of parameter ai from its normal value as follows [25] ui =

Table 1 Unit characteristic of conventional 10-unit test system

Case studies and discussion

Versatile case studies are implemented to demonstrate the applicability of the proposed model. To explicitly declare the efficiency of ASDR programs in handling multiple contingency states, each case study is solved with and without DR programs. Besides, 10, 20, 30 and 40 unit test systems are applied to analyse the performance of the proposed method in different power system scales. Including and excluding effects of ASDR programs are analysed for each of these four test systems, and this made eight versatile case studies. The optimisation solver used in the current paper is General Algebraic Modeling System IET Gener. Transm. Distrib., 2014, Vol. 8, Iss. 12, pp. 1928–1936 doi: 10.1049/iet-gtd.2014.0052

(GAMS). According to the linear feature of the framework CPLEX linear solver embedded in GAMS software is utilised. CPLEX solver is selected because of its high performance in solving linear optimisation problems and its capability to guarantee the optimal solution for linear problems. The base test system data is provided in Table 1, in which coefficients of the quadratic cost functions, minimum up/down time and ramp rates are provided along with their startup costs. This standard test system comprises ten generation units and larger test systems have been generated by replicating the base system. It is noticeable that the up spinning cost for all of the generation units are 80% of their linear coefficients, named as ‘b(i)’ in this paper. Moreover, the non-spinning cost is allocated to be the linear coefficient of each generation unit. Demand reduction bid prices for the base case are 3.5, 7 and 10.5 MW at 1, 1.3 and 1.6 $/MW while the maximum allowed demand reduction is set to 25% of the base demand. In order to increase the accuracy of the solutions, the optimality gap is set to 3% of the optimal solution.

4.1

10-unit case study

The aim of this case study is to observe the applicability of the proposed method on 10-unit test system. As previously mentioned, both including and excluding ASDR programs are considered in this case. The corresponded results of the current case study are tabulated in Table 2. In this Table 2 Results for the schedules of the 10-unit system without ASDR Costs

K

total cost, $ spin. res. cost non-spin. res. cost, $ demand red. cost, $

0

1

2

3

265 589 0 0 –

374 808 728 69 302 –

458 848 27 625 106 650 –

inf. inf. inf. –

1933

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www.ietdl.org Table 3 Results for the feasible schedules of the 10-unit system with ASDR Costs

K

total cost, $ spin. res. cost non-spin. res. cost, $ demand red. cost, $

0

1

2

3

246 854 0 0 3584

340 972 8340 51 536 3815

408 607 3565 113 677 4123

486 443 26 222 154 025 4123

framework, no demand side reserve is allowed and as can be seen in this table, three simultaneous contingencies is infeasible because of restrictions in power balance constraints and small scale of the problem. Moreover, the offer cost for no contingency state is $265 589 and this increases by 41% because of the augmentation in multiple contingencies’ number when k is equal to 1, and 71% increase when k is equal to 2. Spinning reserve costs also show the same trend from no required spinning reserve in no contingency state to $728 and $27 625 when k is equal to 1 and 2, respectively. In case of participating demand side reserves, presented in Table 3, not only generation cost decreased magnificently but also the utilisation of expensive grid side reserves has been diminished noticeably. It can be inferred from Table 3 that a 7% reduction occurred in total generation cost with no contingency when demand reserves are allowed. Moreover, in case of one contingency, although a 9% reduction occurred in total generation cost, spinning cost is increased magnificently. Non-spinning reserve, however, reduced up to 25% comparing to its counterpart in the base case. It should be noted that the opposite trend is observable for two simultaneous contingencies state with 87% increasing in spinning reserve cost and 6% decreasing in non-spinning reserve cost. It is observable that a reduction up to 10% is occurred in total generation cost in case of two simultaneous contingencies. The main contribution of integrating ASDR programs is clearly observable when k is equal to three. Although the generation cost in this state is extra high, the security of power system is guaranteed because of the feasibility of the generation scheduling. It is worth noting that although power systems are not designed to operate with multiple simultaneous contingencies, demonstrating the ability of demand side reserves to handle the maximum feasible contingencies is investigated in this subsection. The feasible solution when k is three can be justified by the flexibility of the power balance constraint that causes the infeasibilities in conventional generation frameworks. This flexibility will be more clear if the reduced demand is compared with the base demand, as depicted in Fig. 3. It is clearly observable that the reductions are occurred majorly based on the high demand periods, which make contingency handling more difficult because of the capacity lack. Moreover, the difference between the valley and peak in the curtailed demand profile is reduced significantly. In addition, the minimum curtailment is 16.2% and the maximum curtailment, occurred in hour 13, is 25%.

Fig. 3 Demand reductions for different K in 10-unit system

required because of the adequacy of cheap generation units to supply demand. n − 1 security criterion, imposed a magnificent 26% increase in objective function comparing to its counterpart in the base case. This upward trend along with allocating expensive spinning/non-spinning reserve costs continued to larger k factors for which extremely high generation scheduling costs are obtained. It is noticeable that the maximum elapsed time for the current case was not more than 37 min. Integrating demand side reserve contributions had significant effects on the final solution. As presented in Table 5, total cost reduction for each case, comparing to its base case counterpart, was at least 7% and at most 10%, respectively, for k factors 1 and 4. Spinning reserve comparison shows a meaningful reduction in any contingency state, as presented in Fig. 4. It can be inferred from Fig. 4 that because of capacity lack spinning reserve’s costs increase sharply by increasing k factor. Specifically, when k is equal to three, spinning reserve is at its most without ASDR, while this variable remains zero when ASDR is applied. Owing to this reduction in expensive spinning reserve contribution, the corresponded total generation cost reduced by 10%. Fig. 5, however, verifies that demand side reserve can be considered as a suitable substitution for spinning reserve and no significant effect on non-spinning reserve is expected. 4.3

Larger test systems

In the current case study, two larger test systems, 30 and 40 unit systems are employed to investigate the efficiency of the proposed method on large power systems. n − 1 security criterion is analysed in the presence of demand response Table 4 Results for the schedules of the 20-unit system without ASDR Costs

total cost, $

4.2

20-unit case study

In order to investigate the applicability of the proposed method on larger systems with more components, this case study is implemented. As can be seen in Table 4, just like the previous case, no spinning or non-spinning reserve is 1934 & The Institution of Engineering and Technology 2014

spin. res. cost non-spin. res. cost, $ demand red. cost, $

K 0

1

2

3

4

5

6

523 531 0

660 592 3200

745 517 9940

inf.

inf.

inf.

98 622 –

125 069 –

911 610 16 050 251 993 –

inf.

0

826 748 19 978 177 566 –

inf.

inf.

–

–

–

IET Gener. Transm. Distrib., 2014, Vol. 8, Iss. 12, pp. 1928–1936 doi: 10.1049/iet-gtd.2014.0052

www.ietdl.org Table 5 Results for the feasible schedules of the 20-unit system with ASDR Costs

total cost, $ spin. res. cost non-spin. res. cost, $ demand red. cost, $

K 0

1

2

3

4

5

6

486 424 0 0 2751

612 325 253 78 667 3661

676 118 6400 111 468 3738

738 525 384 175 195 3892

812 910 12 720 234 591 4123

89 484 23 800 278 297 4123

974 296 14 549 342 433 4123

Table 6 Results for the feasible schedules of the 30-unit system States

Without ASDR

With ASDR

K

0

2

0

2

769 968 0 0 –

1 040 803 13 328 128 952 –

721 568 0 0 3584

951 260 9585 109 817 3969

total cost, $ spin. res. cost non-spin. res. cost, $ demand red. cost, $

Table 7 Results for the feasible schedules of the 40-unit Fig. 4 Spinning reserve with and without utilising DR programs

system States

Without ASDR

K total cost, $ spin. res. cost non-spin. res. cost, $ demand red. cost, $

Fig. 5 Non-spinning reserve with and without utilising DR programs

programs in quite enough researches. Thus, in this section, ASDR programs are considered to handle n − 2 security criterion. The results for 30-unit test system are tabulated in Table 6. Total cost of the base case without DR programs is $769 968 while implementing DR programs caused a 7% reduction. n − 2 security criterion caused an additional cost up to 35% of the base case. This augmentation is declined by 12% when demand reserves are allowed. Spinning reserve cost also diminished, in case of n − 2 contingencies, down to 40%, indicated by grey coloured cells. The same trend is observed for the non-spinning reserve with 17% reduction, highlighted with orange coloured cells in Table 7. In case of 40-unit test system, the aforementioned discussion is valid. In this context, 28% increasing in total generation cost because of n − 2 security criterion, is declined by 8% in case of integrating ASDR. Although spinning cost increased by 48%, non-spinning reserve cost declined significantly from $190 230 for the base n − 2 case to $141 235 for the integrated DR program case. It is worth noting that the simulation elapsed time for large scale test systems exponentially rose. The elapsed time for the first case study without considering DRPs was less than a IET Gener. Transm. Distrib., 2014, Vol. 8, Iss. 12, pp. 1928–1936 doi: 10.1049/iet-gtd.2014.0052

With ASDR

0

2

0

2

1 027 911 0 0

1 323 370 2506 190 230

956 871 0 0

1 222 520 3720 141 235

–

–

3451

3892

second. The high speed computational calculation is because of the mixed integer linearity of the framework. The simulation time rose to 1.5 min when considering the DRPs. The increased duration is mainly because of additional integers introduced to the problem by demand bid packages. The case study B took 3 and 10 min (rounded numbers) to be solved for the case with and without implementing DRPs, respectively. The elapsed time for the last case, however, rose form 5 min for the case disregarding DRPs to 37 min for the worst-case scenario considering DRPs in a test system with 40 units. The next important sharp increasing in simulation time occurred between two consecutive contingencies in the last two case studies. The n − 1 security criterion for 30-unit test system lasts 20 min, whereas n − 2 criterion simulation lasts about 37 min in presence of DRPs.

5

Conclusions

In this current paper, ASDR programs are investigated in an n–k contingency constraint unit commitment. To implement an ASDR program, DRPs may offer their bid prices to ISOs in discrete quantities to compete with expensive supply side reserve offers. The main purpose to integrate demand side reserve into the framework is to make power balance constraint more flexible, which is the main cause of infeasibilities in power generation scheduling because of capacity lack. In our present work, finding the maximum number of simultaneous multiple contingencies is targeted, 1935

& The Institution of Engineering and Technology 2014

www.ietdl.org which is supported with the versatile case studies. The results indicated that integrating demand side reserves into generation scheduling not only caused a significant reduction in generation cost but also led to improve the number of maximum possible multiple contingencies. According to the mixed integer linearity nature of the framework, this structure can be applied in real world by ISOs, Regional Transmission Operators, day ahead market operators and any other short term power generation schedulers to make the extreme reliability concept of the future smart grids a reality. The current work is a base research for future works. Future research is needed to cover more realistic constraints such as multiple transmission line contingencies. Considering multiple generation and transmission line contingencies along with investigating many available demand response programs can reveal the real capabilities of DRPs in mitigating the impacts of multiple contingencies and improving the security of the future power systems.

6

References

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IET Gener. Transm. Distrib., 2014, Vol. 8, Iss. 12, pp. 1928–1936 doi: 10.1049/iet-gtd.2014.0052