... Hamidreza Zareipour, Senior Member, IEEE, Ali Reza Seifi, Member IEEE ...... [3] F. Barati, H. Seifi, M. S. Sepasian, A. Nateghi, M. Shafie-khah, and J. P. S..
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A Chance Constrained Programming Approach to the Integrated Planning of Electric Power Generation, Natural Gas Network and Storage Babatunde Odetayo, Student Member, IEEE, Mostafa Kazemi, John MacCormack, Member, IEEE, W. D. Rosehart, Senior Member, IEEE, Hamidreza Zareipour, Senior Member, IEEE, Ali Reza Seifi, Member IEEE
Abstract—Natural gas is increasingly preferred as a choice of fuel for electricity generation globally resulting in electricity systems whose reliability is progressively dependent on that of the natural gas transportation system. The cascaded relationship between the reliabilities of these system necessitates an integrated approach to planning both systems. This paper presents a chance constrained programming model that minimizes the investment cost of integrating new natural gas-fired generators, natural gas pipeline, compressor and natural gas storage required for meeting future stochastic natural gas and power demand. The proposed model also highlights the role of natural gas storage in managing short time uncertainties in developing a long-term expansion plan for both the electric and natural gas systems. A two-stage chance constrained solution algorithm is employed in solving the mix-integer non-linear programming optimization problem and illustrated on a standard IEEE 30 bus test system superimposes on the Belgian high-calorific gas network. Index Terms—Integrated planning, Chance constrained programming, Natural gas transmission system, Natural gas-fired electricity generators;
I. N OMENCLATURE A. Indices b, r Index of source and demand electricity buses c Index of NG compressor stations i, j Index of source and demand NG nodes p Index of NG pipeline types s Index of NG storage sizes t Index of planning time horizon, t = 0,1,...T % Index of NG-fired generator sizes B. Binary Variables yp,i,j,t 1 if a pipeline y is installed between NG nodes i, j in time t. 0 otherwise zi,j,t 1 when there is NG flows from node i to j in time t and 0 otherwise. λb,t 1 if a new NG-fired generator (NGG) is connected to b in time t. 0 otherwise ψi,j,t 1 when compressor is installed between nodes i to j in time t. 0 otherwise. φi,t 1 when the upper limit of NG sources is increased C. Variables (n) Pb,t Power generation from new(n) NGG connected to bus b in time t (e) Pb,t Generation from existing (e) NGG at bus b in t f lb,r,t Power flow from bus b to r in time t Θi,t , Θj,t Square of NG nodal pressure at source i and sink j NG nodes in time t
Φi,j,t NG flow rate between NG nodes i, j in time t Γi,t NG pressure at source NG nodes i in time t D. Random Variables ξ1 Electricity demand forecast uncertainty ξ2 NG demand uncertainty E. Chance constraint variable/parameters α Desired confidence-level for electricity system β Desired confidence-level for the NG system γ Desired confidence level for the entire system F. Parameters (I) Overnight investment (I) cost of NGG of size % C% (I) Overnight investment (I) cost of NG pipeline p Cp (I) The overnight investment (I) cost of compressor Cc (c) station (o) Operating cost (o) of NGG % C% (o) Operating cost (o) of NG pipeline p Cp (o) Operating cost (o) of NG compressor c Cc (e) di,t NG demand by existing (e) NGG at node i in time t (n) di,t NG demand by new (n) NGG at node i in time t (h) di,t NG demand for non-electric power generation at node i in time t ir Interest rate on capital Li,j Length of existing and candidate pipeline connecting NG nodes i and j M1 Large number equivalent to maximum possible NG flow in a pipeline in time t M2 Large number used to limit the square of the nodal pressure M3 A large value greater or equal to the maximum acceptable pressure ngsi,t NG supply at node i in time t ngsi,t Lower limit on NG supply at node i in time t ngs i,t Upper limit on NG supply at node i in time t (L) Pb,t Electricity demand at bus b in time t Xb,r Reactance of the transmission line connecting buses b, r. θb,t Voltage phasor angle at bus b in time t ∆i,j Diameter of existing and candidate pipeline connecting nodes i and j Lower limit on the NG nodal pressure in time t Γi,t Γi,t Upper limit on the NG nodal pressure in time t κ Maximum pressure increase multiplier at a compressor station
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σ G. Sets Ωb () Ωc Ωdn Ωn Ωp Ωs Ωsn Ω(tr) Ω%
discretized storage and inflow/outflow rate used to linearize the properties of the NG storage Set of electricity buses (indexed by (b,r)) Sets of NG compressor nodes Set of NG demand nodes Set of NG nodes Set of all pipe types Set of NG storage sizes Set of NG supply nodes Set of transhipment nodes Set of NGDG sizes II. I NTRODUCTION
The worldwide electric power generation capacity is expected to grow by 69% from 21.6 trillion kilowatt-hours (kWh) in 2012 to 36.5 trillion kWh by 2040. In the same period, electricity generation from natural gas (NG) is expected to grow by 110% from 4.8 trillion kWh to 10.1 trillion kWh. [1]. This trend is driven by the availability of relatively cheap NG supply, environmentally responsible regulations, and the improved efficiencies, competitive investment cost, relatively short ramp rates, modularity and scalability of NG-fuel generators (NGGs) [2]–[4]. Increasing electricity generation from NG, however, exposes the electricity grid to reliability concerns on the NG transportation system [5]. Potential reliability concerns on the NG transmission system includes outages resulting from component failures, malicious attacks, competing NG demand from other sectors such as transportation and extreme weather conditions as experienced in the northeast of the USA during 2013/1014 polar vortex [4], [6]. These risks make the long-term planning of both systems challenging as the planner is constantly faced with the dilemma of the high cost of building robust blackundancy consequently, overbuilding both systems and the cost of outages. To some extent, energy utilities can manage some of these reliability concerns by utilizing NG storage and employing an integrated approach to the long-term planning of the electricity and NG systems. NG storage can provide contingent NG supply in the advent of an outage on NG transmission network, a spike in power demand or a sudden drop in power generation from renewable energy resources. Integrated planning of the electricity and NG systems is challenging because of the traditional planning culture of the individual systems, non-collusion concerns between them, the overall reliability of the cascaded systems, the computational complexities of planning and managing both systems for uncertain power and NG demands. The integrated planning of electricity and NG system have been approached from an operational [5], [7]–[12] and long-term investment [3], [4], [13], [14] point of view. In the operation planning, the prevalent objective is ensuring adequate NG for daily variations in electric power generation. The integrated operation planning problem is modelled as a mixed-integer linear programming (MILP) security-constrained optimal power and NG flow in [10], while [11] employed steady-state analysis based on Newton-Raphson method to solve the coordinated operation of both systems. In [12], a multi-stage stochastic model for the integrated operation of both systems considering
the uncertainties of electric and NG supply was presented, while [15] proposes a single-stage linear optimization model for the coordinated operation of NG and power system In contrast to the short-term adequacy objective of the integrated operation planning problem, the chief goal of the longterm investment plan is the adequacy of electric power generation, NG transportation capacity for both electric power generation and non-electric power generation considering stochastic future power and NG demands. Recent studies have modelled the long-term planning problem as electricity Generation Expansion Planning (GEP) - NG planning problem [2], [16], [17], GEPNG-electric transmission planning problem [3], [18] and GEPNG planning problem considering resilience constraints enhancing grid resiliency [19]. NG storage is usually employed by utilities to augment NG supplies and improve supply reliability. The operational characteristic of NG storage for contingency management was introduced in [20], while its value for energy supply security was discussed in [9]. Models for the constraints associated with the effective operation of an NG storage is presented in [9], [16] while the inflow /outflow rates to and from the NG storage are modelled in [21]. The long-term integrated planning problem is often modelled as a cost minimization mixed integer non-linear problem (MINLP) which is usually difficult to solve [4]. A number of approaches have been employed to manage this challenge. For instance, [3], [22] employed a sequential approach where the electricity system is planned first and then the NG system. In [18], [23], the complexities of the integrated planning problem was managed by decomposing it into manageable units, while [24] employed an Elitistist non-dominated Sorting Genetic Algorithm to solve the integrated planning problem. The cost of implementing a very reliable integrated plan can become astronomical as the level of dependency increases because of the cascaded relationship between the reliabilities of both systems. This is because the reliability of the entire system is a function of the product of the reliabilities of the constituent systems. Probabilistic planning approach such as Chance Constrained programming (CCP) [25], [26], allows the planner to minimizes the investment and operation cost of the system within acceptable confidence levels of electric and NG power supplies. In this paper, the objective of the long-term integrated planning problem is the minimization of investment cost of NGfiblack generators, NG pipelines, and NG storages while ensuring the uncertain power and NG demands are met with desiblack levels of confidence. The proposed model extends previous model [3], [13], [16], [27] for the integrated long-term planning of electric and NG systems to include NG compressors, storage and the dynamics of NG flow in and out of the storage for managing stochastic power and NG demand. Furthermore, compablack to [3], [4], [13], [16], the proposed model accommodates the cascaded relationship between the reliabilities of the power and NG systems in developing of a long-term expansion plan. The CCP framework is employed in solving the proposed integrated planning model in two stages. First, a non-convex MINLP deterministic planning problemis solved, and then based on the solution of the deterministic problem a convex NLP op-
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eration problem is solved for optimality. The operation solution accommodates the stochasticity of the electric power and NG demand by testing the solution from the deterministic expansion plan for sufficient scenario sets of the uncertain demand pairs. The contribution of this paper is summarized as follows. • A CCP long-term integrated planning model that accommodates the interactions between the confidence levels of meeting the stochastic power and NG demands is presented. The proposed model allows the system operator or a planner in a vertically integrated utility accommodate the expansion of the NG system in the planning of a reliable power system. This planning approach that can be very valuable when planning under tight budgetary constraints. • Secondly, the proposed model incorporates the dynamics of NG compressors, NG inflow and outflow from an NG storage and its role in managing short-term demand uncertainties when developing a long-term integrated expansion plan. This can help blackuce investment cost, properly size NG storage and develop operation policies of NG storage. The rest of this paper is structublack as follows. The integrated planning problem is described in Section III, while the mathematical representation of the problem is presented in Section IV. The solution methodology is presented in Section V, in section VI, the solution methodology is implemented on a test system and Section VI concludes the paper.
Fig. 1.
Schematics of a couple NG and electric power system
development of a long-term expansion plan for the electric and NG systems.
III. T HE I NTEGRATED PLANNING PROBLEM WITH NATURAL G AS S TORAGE The power system planner is often tasked with ensuring adequate electric power supply from NGG, other power generators, and the interties. NGG as a power source is however dependent on the adequacy and reliability of the NG supply system as shown in Fig. 1. NG storage offers an option for substitute investment in new production and transportation capacity, provide high security of supply, and bridging the gap between peak NG demand and supply. For example, NG storage can provide prompt NG boost supply to NGGs in the advent of a spike in electricity demand or a dip in power output from renewable energy resources. This might be more effective in comparison to increasing NG supply at the source and transporting it at circa 20 miles/hour to the NGG. The prevalent NG storages are abandoned oil and gas, salt caverns, aquifers, LNG-storages and pipeline line-pack. Abandoned oil and gas, salt caverns, aquifers are less flexible because they are naturally occurring or base on naturally occurring geological formations. LNG-storages is not dependent on naturally occurring factors hence, it offers significant location and size flexibilities [21]. The operational characteristic of an NG storage includes the base (cushion) gas, working gas, injection and extraction range and cycling as illustrated in Fig. 2. The base gas is a permanent NG inventory requiblack to maintain adequate NG pressure and outflow rate. The amount base gas requiblack is dependent on the engineering of the storage, contractual obligations and regulatory requirements.The working gas i.e. the NG available for cushioning the effects of extensive competing NG demand is tied to the contractual obligation, injection and extraction rate. The proposed model considers the operation characteristic in the
Fig. 2.
A illustration of the operational characteristic of NG storage
IV. I NTEGRATED PLANNING MODEL The objective of the integrated planning problem is to minimize the investment and operational cost of electric and natural gas systems as modeled in (1). T X (n) (n/e) min z = CCt + OCt + CCt t (ς)
+ CCt
(ϕ)
+ OCtς + CCt
(ϕ)
+ OCt
(1)
(n) CCt
Where in (1) is the present value (PV) of the capital cost (n) of building new (n) NGG. CCt is further defined in (2). (n)
CCt
=
X X
δt λb,%,t C%(I)
(2)
b∈Ωb %∈Ω% ir δt = 1−(1+ir) −t . The operating cost of electric power generation from both existing and new generators i.e., the second term of (1) is modeled in (3). This cost constitute the fixed and variable operating cost of the electric power generator. X X (n/e) (n/e) OCt = δt Pb,t C%(o) (3) b∈Ωb %∈Ω%
The cost of building new or extending existing pipelines to new NG demand nodes, i.e., the third term of (1) is modelled in (4) X X CCt = δt Cp(I) Li,j (yp,i,j,t+1 − yp,i,j,t ) (4) (i,j)∈Ωn p∈Ωp
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The capital cost of the NG pipeline constitutes the cost of the pipeline material and installation per diameter per length. The operating cost of NG pipeline a small fraction of the cost of NG hence it is ignoblack. Compressors stations increase NG flow rate by increasing pressure differential along a pipeline segment. Its PV and operation costs are modeled in (5) and (6) respectively. X X (ς) CCt = δt Cc(I) (ψi,j,t+1 − ψi,j,t ) (5) (ς)
OCt
=
X
∀i ∈ Ωsn , t ∈ T (10a)
Γi,t ≤ Γi,t ≤ Γi,t
∀i ∈ Ωdn , t ∈ T (10b)
Φi,j,t ≤ M1 zi,j,t
∀(i, j) ∈ Ωn , t ∈ T (10c)
ngsi,t ≤ ngsi,t ≤ (ngs i,t + 5φi,t ) (e) di,t
di,t =
(n) di,t
+
ngsi,t +
(i,j)∈Ωn c∈Ωc
X
Γi,t = Γi
+ X
∀i ∈ Ωd ∪ Ωc , t ∈ T (10e) X Φi,j,t = di,t (ξ2 ) Φj,i,t − j∈Ωn
j∈Ωn
δt ψi,j,t Cc(o)
(6)
∀(i, j) ∈ Ωn , t ∈ T (10f)
(i,j)∈Ωn c∈Ωc
X
The capital and operation cost of NG storage is modeled in (7) and (8), where S is the capacity of the NG storage. The cost of refrigeration/liquefaction accounts for the majority of the operating cost NG storage [28]. X X (ϕ) δt φs,j,t Cs(I) CCt = (7)
X
Φi,j,t −
(ϕ)
OCt
=
δt Ss,t Cs(o)
− Ψi,j Φ2i,j,t ≥ M2 (zi,j,t − 1) ∀(i, j) ∈ Ωn , t ∈ T (10h)
(Γ2i,t Ψi,j =
(8)
s∈Ωs
−
Γ2j,t )
−
Ψi,j Φ2i,j,t
1 1.1494 × 10−3
p∈Ωp
∀(i, j) ∈ Ωc , t ∈ T (10o)
The physical and operating constraints on the electricity system presented in (9) [29] . +
=
+
X
f lb,r,t
r∈Ωb (e) Pb,t +
f lb,r,t = θb,t
∀(i, j) ∈ Ωc , i < j, t ∈ T (10n)
Γj,t − κΓi,t ≤ M3 (2 − zi,j,t − ψi,j,t )
A. Physical and operating constraint on the electricity network
L Pb,t (ξ1 )
∀(i, j) ∈ Ωn , t ∈ T (10i) 2 GTf Li,j Zf Pb (10j) ∆5i,j Tb
yp,i,j,t ≤ yp,i,j,t+1 ∀(i, j) ∈ Ωp , t = 1, ...|T | − 1 (10m) ψi,j,t + ψj,i,t ≤ 1
(n) Pb,t
2
≤ M2 (1 − zi,j,t )
zi,j,t + zj,i,t ≤ 1 ∀(i, j) ∈ Ωn , i < j, t ∈ T (10k) X zi,j,t + zj,i,t ≤ yp,i,j,t ∀(i, j) ∈ Ωn , i < j (10l)
The minimization of the investment and operating cost of the electric and NG system is constrained by the physical and operating constraints on the electric and NG system and the coupling constraint between them. These constraints are discussed next.
(e) Pb,t
∀i ∈ Ω(tr) , t ∈ T (10g)
Φj,i,t = 0
j∈Ωn 2 (Γi,t − Γ2j,t )
j∈Ωn
s∈Ωs j∈Ωn
X
∀i ∈ Ωsn , t ∈ T (10d)
(h) di,t
Γj,t − κΓi,t ≥ M3 (zi,j,t + ψi,j,t − 2) ∀(i, j) ∈ Ωc , t ∈ T (10p) Γj,t − Γi,t ≤ M3 (1 − zi,j,t + ψi,j,t )
∀b, r ∈ Ωb , t ∈ T (9a)
∀(i, j) ∈ Ωc , t ∈ T (10q) (n) Pb,t
Pb,t = 1 (θb,t − θr,t )
Xb,r,t =0 ∀b = slackbus 0 ≤ Pb,t ≤ λb,%,t P G%
Γj,t − Γi,t ≥ M3 (zi,j,t − ψi,j,t − 1)
∀b ∈ Ωb , t ∈ T (9b)
∀(i, j) ∈ Ωc , t ∈ T (10r)
∀(b, r) ∈ Ωb , t ∈ T (9c)
ςψi,j,t − Φi,j,t ≤ M1 (ψi,j,t − 1)∀(i, j) ∈ Ωc , t ∈ T (10s)
∀b ∈ Ωb , t ∈ T (9d) ∀b ∈ Ωb , t ∈ T (9e)
f l ≤ f lb,r,t ≤ f l ∀(b, r) ∈ Ωb (9f) Constraint (9a) is the kirchhoff’s flow conservation nodal balance constraint. Constraint (9b) ensures that total electric power generation is always the sum of up existing (e) and new (n) power generations. The approximate power flow between connected electricity buses is modeled in (9c). Constraint (9d) sets the angle of the voltage phasor of the slack bus to zero. Constraints (9e) and (9f) models the limits on the capacity of new and existing generators and power flow in the transmission lines. P G% is the name place capacity of NGG type %.
X
ψi,j,t ≤ ψi,j,t+1 ∀(i, j) ∈ Ωc , t = 1, ..|T | − 1 (10t) X (in) (in) Φj,s,t ≤ νσ,s,t Πσ ∀s ∈ Ωs ∪ Ωn , t ∈ T (10u)
j
σ
X
X
Φs,j,t ≤
(out)
νσ,s,t Π(out) σ
∀s ∈ Ωs ∪ Ωn , t ∈ T (10v)
σ
j
X
(in)
νσ,s,t +
σ (in)
(νσ,s,t +
X
(out)
νσ,s,t = 1
σ (out) νσ,s,t )χσ−1
xs,t
∀s ∈ Ωs ∪ Ωn , t ∈ T (10w) (in)
(out)
≤ xσ,s,t ≤ (νσ,s,t + νσ,s,t )χσ (10x)
∀s ∈ Ωs , t ∈ T (10y) Ss ≤ xs,t ≤ Ss X X = xs,t−1 + Φi,s,t − Φs,i,t i∈i(s)
i∈i(s)
∀s ∈ Ωs , i ∈ Ωp , t ∈ T (10z) B. Physical and operational constraint on the NG system NG flow through the pipeline system is constrained by a number of physical, operation and contractual constraints as modeled in (10). [3], [30]–[32].
Constraint (10a) sets the NG pressure at the source nodes to the maximum. Constraints (10b) sets the limits on the square of the NG pressure at the demand nodes. Constraint (10c) limits the NG mass flow rate on each segment of the pipeline, where
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M1 is large number that indicates the maximum possible flow. The contractual limits of NG supply is modelled in constraint (10d). This limit can be physical or contractual, therefore (10d) is modelled such that the contractual limit increases by multiples 5M m3 /day when such increase does not violate the physical limits of the system. A multiple of 5M m3 /day is selected randomly to ease the tracking of results. Constraint (10e) models the NG demand for electricity and non-electricity purposes. The NG flow conservation equation at NG demand and supply nodes i is modeled in constraint (10f). Constraint (10g) is conservation equation at the transhipment nodes i.e., nodes where where there are no local NG demans. Constraints (10h) and (10i) models the NG pressure drop between two nodes of pipeline segment assuming isothermal flow through a horizontal pipeline such that the effect of kinetic energy is negligible [3], [30]–[32]. (10h), (10i) and (10c) ensure that NG only flows from higher to lower pressure nodes. Constraint (10h) and (10i) is formulated such that it is only tight when there is NG flow between two nodes i.e. when za,j,i,t = 1. M2 is a large value close to Γi,t 2 . The pipeline resistant Ψi,j is modeled in (10j) - ∆i,j and Li,j (km) are the diameter and length of the pipeline segment connecting nodes i and j respectively. Where f , Pb , Tb , G, Z, and Tf are the frictional factor, base pressure, base temperature, NG gravity, gas compressibility factor and flow temperature respectively. Constraint (10k) ensures a unidirectional flow of NG per time [30] while constraint (10l) ensure that for NG flow to be possible, a pipeline must be present. yi,j,t is a unique identifier for installed pipeline. Constraint (10m) ensures that new pipelines remain installed for the entire planning period. Constraint (10n) ensure that the compressor only increases NG pressure in one direction. Constraints (10o) and (10p) ensure the NG pressure increase through a compressor is less or equal to κ. Constraint (10o) and (10p) are only tight when a compressor exists and there is a positive NG flow i.e. when ψi,j,t = 1 and zi,j,t = 1. Constraints (10q) and (10r) ensures the pressure at the inlet and outlet of the compressor are equal in the absence of compression i.e. when ψi,j,t = 0, consequently treating the pipeline as a passive pipeline. In addition, constraints (10q) and (10r) dissociates Γi,t and Γj,t when there are no NG flow between nodes i and j i.e. when zi,j,t = 0 in the presence or absence of a compressor [30]. In addition, constrains (10l), (10o) - (10r) ensures that a compressor can only be installed between two nodes connected via a pipeline. Constraint (10s) ensures that flow rate of NG through a pipeline connected to a compressor station is less or equal to the total flow capacity of the compressor station. In this constraint, when a compressor is present between nodes i, j i.e., ψi,j,t = 1 the difference between the flow capacity of the compressor ς and the flow rate in the connecting pipe is less than zero. Otherwise, the flow rate in a pipeline without a compressor is less than M1. Constraint (10t) ensures that an installed compressor remains installed for the planning period. The maximum rate of NG flow rate in and out of NG storage is limited by the capacity of the connecting pipes and NG level in the storage. The maximum inflow and outflow of NG into storage are strictly decreasing and increasing convex function of the NG storage level. The level of NG in the storage is discretized by a
(in)
(in)
set of constant x1 , ....xσ , corresponding inflow Π1 , ... Πσ (out) (out) and outflow Π1 , ... Πσ rates, and variables ν1 , .....νσ is a convex combination of discretized storage and NG injection (in)/extraction (out) flow rate employed in linearizing the NG storage characteristic as shown in Fig. 4 [21], [33].
Fig. 3. Illustration of the linearization of the inflow and outflow rate of a NG storage system
The characteristic of NG flow in and out of the NG storage s is constrained constraints (10u) - (10z). Constraints (10u) and (10v) limits the NG injection (in)/extraction (out) from the NG storage in time t. Constraint (10w) ensures a uni-direction (in)/(out) NG flow per time. Constraint (10x) ensures that the NG levels between two point of approximation is constrained by the sum of the convex combination of the net NG flow to or from the storage. On the flip side, constraint (10x) ensures the discrete representation of the inflow or outflow is constrained by the level of the storage at a given time. Constraint (10y) place a capacity limit on the NG storage. Constraint (10z) ensures that the difference between storage levels at two given times is a function of the net NG inflow and outflow. a) Non-linear constraint approximation: The flow equation i.e. constraints (10h) and (10i) introduces non-linearity into the integrated planning model because of Γ2 and Φ2 . To improve the linearity of the model Γ2i is replaced with Θi and (10h) and (10i) are reformulated as (11). (Θi,t − Θj,t ) − Ψi,j Φ2i,j,t ≥ M2 (zi,j,t − 1) ∀(i, j) ∈ Ωn , t ∈ T (Θi,t − Θj,t ) −
Ψi,j Φ2i,j,t
(11a)
≤ M2 (1 − zi,j,t )
∀(i, j) ∈ Ωn , t ∈ T 2
(11b)
2
Γi,t ≤ Θi,t ≤ Γi,t (11c) where (11c) is the limit on the square of the nodal pressure Θi . The planning remains non-linear because of Φ2i,j,t , however, recent improvement in computational methods have provided acceptable solution to similar MINLP problems [30], [34]. C. Coupling constraint The NG requiblack by the existing (e) and new (n) NGGs as a function of the electric power injected into the electricity network at time t is computed with the relaxed coupling constraint present in (12) [4], [27]. (n/e)
(n/e)
(n/e)
di,t ≥ ai +bi Pb,t +ci (Pb,t )2 ∀b ∈ Ωb , ∀i ∈ Ωdn (12) Constraint (12) is the relaxed non-convex equality coupling constraint [4]. ai ,bi ,bi are the gas fuel rates coefficients of the
6
NGG. For simplicity purposes all candidate NGGs are assumed to have similar fuel rate coefficients. D. Chance constrained model of the stochastic constraints The CCP modelling approach restricts the feasible region of a stochastic constraint so that the confidence level of the solution is greater or equal to a set value. The stochastic constraints (9a) and (10f ) modelled as chance constraints in (13). X \ (e) (n) (L) P Pb,t + Pb,t − f lb,r,t ≥ Pb,t ≥α
(13a)
r∈Ωb
b
X X \ P ngsi,t + Φj,i,t − Φi,j,t ≥ di,t ≥ β (13b) i
j∈Ωn
j∈Ωn
γ =α×β
(13c)
0 ≤ α ≤ 1, 0 ≤ β ≤ 1 (13d) Equations (13a) states that the joint probability that the power generation, less the net power flow is greater or equal to the stochastic power demand must be greater or equal to a desiblack confidence level α. Similarly, (13b) ensures NG supply matches the stochastic NG demand with a confidence level β. Due to the cascaded relationship between these systems, the confidence level of meeting the system energy demand γ is a product of α and β as modelled in (13c). Assuming the power and NG demand follows a normal distribution, (13a) and (13b) is approximated to (14a) and (14b) [35]–[37]. X (L) (L) (e) (n) Pb,t + Pb,t − f lb,r,t = µPb,t + σ Pb,t Zα (14a) r∈Ωb
ngsi,t +
X j∈Ωn
Φj,i,t −
X
Φi,j,t = µdi,t + σ di,t Zβ
(14b)
Fig. 4.
Flowchart of the implemented CCP
The deterministic approximation of the stochastic long-term integrated planning problem modelled in (14) is solved for a randomly selected sets of inverse cumulative distributions Zα and Zβ [35]–[37]. The output of interest includes the location and size of NGGs, NG pipelines, compressors and storages are then passed to feasibility check sub-model as shown in Fig. 4
j∈Ωn
Where Zα and Zβ are the inverse cumulative distribution of the (L) random power and NG demands. µPb,t and µdi,t are the means (L) of the power and NG demands respectively. While, σ Pb,t and σ di,t are their respective standard deviations. V. S OLUTION METHODOLOGY The MINLP long-term integrated planning of the electricity and NG systems is challenging to solve, a challenge further complicated by the stochasticity of the power and NG demands. The CCP solution algorithm accommodates these challenges by iteratively solving the planning in two main stages illustrated in Fig. 4. First, expected power and NG demands are assumed consequently, approximating the stochastic planning problem to a deterministic one which is easier to solve. Based on the solutions of the deterministic problem, a convex Non-Linear Programming (NLP) operation problem i.e. the Natural Gas constrained Optimal Power Flow (NGOPF) is then solved for Ntimes samples of NG and power demands. Should the confidence levels be less than the desiblack values, the expected demand pairs in the deterministic problem are updated using by updating Zα and Zβ in (14) and re-evaluated. The details of this solution algorithm is discussed in the following subsections. A. The deterministic integrated planning model
B. The feasibility check model
The solution from section V-A might be a local optimum since it only accommodates a set of expected power and NG demand scenario. To ensure the desiblack confidence levels is achieved in the presence of stochastic demand pairs, the solution is tested for feasibility by solving a NGOPF modelled in (15), for Ntimes scenario of NG and power demands sets in t ∈ T . Scenarios of (L) (h) power demand, Pb,t and NG demand for heat di,t are generated via Monte Carlos simulation. The uncertainties in NG demand (L) for power generation is determined from Pb,t . The accuracies of this feasibility check increases with the value of Ntimes .
7
min z ∗ =
T X X X t
X
ui,j,t Cc(o) +
β
requiblack for achieving a system confidence level γ.
(n)
s.t. Pb,t + Pb,t −
X
X
Ss,t Cs(o)
(15a)
∀b ∈ Ωb
(15b)
∀i, j ∈ Ωn
(15c)
s∈Ωs
(i,j)∈Ωp c∈Ωc (e)
C%(o)
b∈Ωb %∈Ω%
X
+
(n/e)
Pb,t
Z1 and Z2 are the inverse distribution of the Pf eas associated with Z Hi and Z Lo respectively. Z γ is the minimum Zα
(L)
f lb,r,t = Pb,t
r∈Ωb
ngsi,t +
X
Φj,i,t −
j∈Ωn (n)
X
Φi,j,t = di,t
j∈Ωn
di,t , Pb ∀b ∈ Ωb , i ∈ Ωd HR 0 ≤ Φi,j,t ≤ M 1 ∀i, j ∈ Ωp ,
Pb ≤ Pb,t ≤ min
ngsi,t ≤ ngsi,t ≤ ngsi,t
∀i ∈ Ωs ∪ Ωd ,
Θi,t − Θj,t = Ψi,j Φ2i,j,t zˆi,j,t Θi,t − Θj,t = −Ψi,j Φ2i,j,t zˆi,j,t Γj,t − κΓi,t = 0
(15d) (15e) (15f)
∀i, j ∈ Ωp
(15g)
∀i, j ∈ Ωc
(15h)
∀(i, j) ∈ Ωc ,
(15i)
(9b − 9f ), (10a, 10b, 10e, 10j, 10u − 10z), (12) (15j) Constraint (15a) minimize the operation cost of generators, compressors, and storages. ui,j,t indicates the presence of a compressor. Constraints (15b) and (15c) ensures a nodal balance of electric power and NG flow. Constraint (15d) set the capacity limits on electricity generation based on the nameplate capacity and the Heat-rate (HR) of the generators. Constraints (15e) and (15f) sets the capacity limits NG flow rates and supply. Constraints (15g) and (15h) models the NG flow conservation equation in the passive and active pipelines, while (15i) models the flow conservation at a compressor station. zˆi,j,t indicates the direction of Φi,j,t . zˆi,j,t is the known variable from the deterministic problem sent to the lower level problem and treated as a constant. The components of (15j) are defined above in section IV. With the direction of Φi,j,t determined from the solution to the deterministic model, the NGOPF model is convex [38]. The average feasible cases i.e. pf eas of the Ntimes set of scenarios indicates the confidence level of the solution computed in the deterministic model. If pf eas ≥ γ, the solution to the deterministic model guarantees the desiblack confidence levels i.e. optimality, else the approximations in the deterministic model are updated using the Z-update algorithm [36], [39].
VI. T EST SYSTEM AND RESULT The proposed model is illustrated by developing a 20 years integrated expansion plan based on a ieee 30 bus test system superimposed on Belgium calorific gas network. The role of the NG storage in accommodating short-time uncertainties when developing the long-term expansion plan is investigated by considering the following scenarios: 1) Scenario 1: No NG storage - In this scenario, it is assumed that there is no NG storage in the system 2) Scenario 2: Low NG supply constraint - In this scenario, the long-term plan is expected to accommodate a maximum of 1 day of complete NG storage depletion annually 3) Scenario 3: Extensive NG supply constraint - this scenario requires a plan capable of accommodating a maximum of 3 days of continuous depletion of the NG storages. For examples an outage on a major NG transmission pipeline. A. The test system and basic assumptions A summary of the modified test system is presented Table II. The NG nodes and electric buses are label G1- G29 and E1-E30 respectively for easy trackability. TABLE I S UMMARY OF THE TEST SYSTEM [38], [40], [41] IEEE 30-bus system
Belgian NG system
Buses Generators Base lines Candidate transmission lines Existing gas generators Demand bus Bus voltages
30 6 41 41 2 20 132/33kV
Total nodes Existing compressors Base pipelines Candidate pipes Candidate nodes Existing NG Storage Norminal nodal pressure NG source (ngs)
20 3 24 30 29 4 80 bar 2 ( ngsG1 , ngsG8 )
The following assumption were made in our illustration 1) The overnight investment cost of candidate generators, compressors, storage, and pipelines are shown in Table II TABLE II C HARACTERISTIC AND CAPITAL COST OF CANDIDATE GENERATORS AND P IPELINE [42]–[44]
C. The Z-update Algorithm Generator
The approximation of the stochastic demand pairs modelled with a mean, µ, and standard deviation, σ , in (14a) and (14b), can be updated by updating the inverse cumulative distribution of the demands i.e. Zα or Zβ . The Z-update algorithm provides a framework to update the Zα or Zβ in terms of the initial, computed (pf eas ) and expected (γ) sets of confidence levels. If pf eas ≤ γ, γ is updated by either updating α or β or both based on (13c). The approximate power demand is updated by updating Zα in (14a) using (16) [35]–[37]. Z βγ − Z2 Hi Lo Zα = Z + Z − Z Lo (16) Z1 − Z2 where, Z Hi and Z Lo are the inverse distribution of the lower (P1 ) and upper (P2 ) limits on the confidence levels search space.
Compressor
LNG storage
MW
100
200
4974kW
1M m3 /day
∆(mm)
NG pipeline 890
590
315
Capital Cost $M
90
170
11.42
2.17
$M/km
2.18
1.45
0.96
$2, 450 per millimeter of diameter per kilometer [42]
2) The desiblack system reliability γ is set to 96% while the minimum acceptable reliability of the constituent system i.e. α and β are set to 98% for illustrative purpose 3) The cost associated with increasing the upper limit of NG supply is assumed to be negligible 4) f , Pb , Tb , G, Z, and Tf are taken to be 0.012, 1 bar, 288o K, 0.66, 0.805, and 283o K respectively [30] 5) The long-term integrated planning model is exploblack for a planning horizon of 20 years
8
6) The locations and capacities of existing NG storages are: G2 -12.6M m3 ; G5 - 7.2M m3 ; G1 3 -1.8M m3 ; G1 4 1.44M m3 [38], [40], [41]. 7) The power and NG demand growth is assumed to be 3% and 2% respectively 8) For the sake of simplicity, candidate power generators are assumed to be NG fiblack. The proposed model could be extended to include other types of generations such as renewable energy resources by treating them as negative loads [45] 9) In this illustration, we assumed a utility with large NG generators where the system operator or planner must accommodate future constraint in the NG network in other to ensure reliable power supply in the future 10) The non-convex MINLP deterministic integrated planning model is solved using the Branch-And-blackuce Optimization Navigator (BARON) [46] solver available in GAMS 24.2 [47]. The solution of the deterministic model is tested for optimality by solving the convex NGOPF using 1000 samples i.e., Ntimes = 1000 of power and NG demand pairs using the DIscrete and Continuous OPTimizer (DICOPT) [48] solver also available GAMS 24.2. B. Results and explanation First, the role of the NG storage in managing the shortterm stochasticity of NG demand is investigated. Secondly, the reliability and cost of the CPP long-term planning model is compablack to a deterministic model. Finally, the sensitivities of the expansion plan to the confidence levels is illustrated. 1) NG storage and the long-term planning model: The proposed expansion plans for the three storage scenarios indentified in Section VI are presented in Table III. The computed plans illustrate the role of NG storage in the long-term integrated planning of electric power and NG system while considering extensive constraint on NG supply. The result shows that a combination of investments in both NGG, NG compressors, NG pipeline and a possible increase in NG supply contract is requiblack to accommodate extensive constraint on the NG transportation system. The time and location of the proposed NGG’s remain the same for most of the three scenarios as shown in Table III, this is because the electrical demand remains the same and the NG system allows for cheaper means of mitigating against the constraints on NG supply. Increasing supply from Norwegian gas was preferblack over Algerian gas because the price of the NG from Norwegian gas was 26% cheaper. It should be pointed out that these solutions are unique to the test system’s ability to accommodate increased NG flow in the existing NG transportation system, existing NG storages and an assumption that the NG supply from Algerian gas and Norwegian gas can be increased. The difference between the plans for each of the scenarios is discussed in the following sub-sections. a) Scenario 1: When there are no NG storages in the test system, the confidence level of 96% is unachievable/infeasible from year 4 as shown in Table III. This is due to the inability of the NG network to transport enough NG to meet both power and non-power NG demands with the desiblack confidence level.
This highlight the ease with which reliability concerns can propagate between the two cascaded systems. b) Scenario 2: As shown in Table III, the expansion plan assuming scenario 2 constitute a combination of NGG, NG compressor, NG pipeline, and an increase in the upper limits of potential NG import from Algerian and Norwegian gas. The presence of daily depletable NG storage that can support complete daily depletion ensures that the power and NG demand are always meet the desiblack confidence levels. This is because the NG storage is able to provide NG boost that is requiblack to compensate for short time increase in NG demand. c) Scenario 3: In comparison to scenario 2, there a larger increase of the upper limits of NG supply is proposed. This is to accommodate the expected extensive utilization of the NG storages for NG supply compensation in the presence of extensive NG supply constraints. Another significant difference between the proposed plan in scenario 2 and 3 is the location and integration timing of NG compressors. The integration of a compressor was recommended earlier in the scenario 3 focused expansion plan. A compressor ψ(G14,G15) between NG demand nodes G14 and G15 is recommended mainly to boost NG flow rate to France. The compressor recommended in scenario 2 i.e. ψ(G11,G12) is mainly requiblack to increase NG pressure to allow increased NG supply from G8. This helps to compensate for extensive demand from the storage connected to G13 and G14. The extensive constraint on NG supply influenced the location of NGG i.e. 100E16 in year 20 in comparison to the previous scenario. The NGG is sited closer to the NG sources i.e. E16 to avoid potential pressure drop that might result from the depletion of the NG storages connected to G13 and G14. The minimal difference between scenario 2 and 3 is attributed to the ability of the existing test system to accommodate the increase in NG supply and negligible cost associated this increase. 2) Deterministic model vs chance constrained programming model: A comparison of the solution to the deterministic and stochastic long-term integrated planning model for scenario 2 is presented in Table IV. As expected the proposed expansion plan resulting from the deterministic model is cheaper than that of the CCP model. Although the investment cost is lower, the reliability cost is higher as captublack by the frequency with which the electric power nodal balance constraint is violated. This aligns with intuitive expectations i.e. investment cost typically increases with increase in the reliability power supply. The computational time for the deterministic and CCP solution is presented in Table V. Although there exists significant difference in the computation time of both solution algorithm, both times are sufficient for long-term planning. 3) Sensitivity of the integrated plan to the confidence levels of cascaded constituent systems: Thus far, α and β are set to 98% each in order to meet the system wide energy demand with a confidence level, γ of 96%. However, an 0.96 γ can be achieved with other combinations of different α’s and β’s. In consequence to this, a sensitivity of the outputted expansion plan to the values of α, β is investigated and the result is shown in Table VI. As shown in Table VI, the selected α and β combination can affect the cost of the expansion plan. In this specific case, higher confidence level on the NG system, β results in an overall cheaper expansion plan. However, there is no cost
9
TABLE IV C OMPUTED ELECTRICITY GENERATION AND NG EXPANSION TARGETS Expectation integrated (n)
TABLE VI S ENSITIVITY OF INTEGRATED PLAN TO CONFIDENCE LEVELS TARGETS α = 97%, β = 99%
CCP model - Scenario 2 (n)
(n)
α = 98%, β = 98% (n)
α = 99%, β = 97% (n)
Year (t)
Pb (MW)
yi,j / ψi,j
Violation (per 1000)
Pb (MW)
yi,j / ψi,j
Violations (per 1000)
Year t
Pb (MW)
yi,j (km) / ψi,j
Pb MW
yi,j (km)/ ψi,j
Pb (MW)
yi,j (km) / ψi,j
1 2 3 4 5 6 7 8 9 10 11 12 13 14
100E24 100E21
y(G8 →E24) -
23 3 8 24 42 73 113 154 203 263 312 366 423 40
100E24 100E10 100E21
2 8 9 9 13 19 3 4 8 13 10 9 18 2
1 2 3 4 7 8 9
100E24 -
y(G8 →E24) ngsG1 ↑ 5∗
100E10
-
100E24 100E10 -
y(G8→E24) ngsG1 :↑ 5∗ y(G18 →E10) ngsG1 ↑ 5∗ -
100E24 100E10 -
y(G8→E24) ngsG1 :↑ 5∗ y(G18 →E10) ngsG1 ↑ 5∗ -
10
-
-
-
-
-
11 12 13 14
-
100E21
15 16 17 18 19 20
-
91 169 231 312 398 488
100E17 100E12
y(G8→E24) ngsG1 :↑ 5∗ y(G18 →E10) ngsG1 ↑ 5∗ ψ(G14,G15) ngsG8 ↑ 5∗ y(G8 →21) ngsG8 ↑ 5∗ ngsG8 ↑ 5∗ y(G17 →E17) y(G15 →E12) ngsG1 :↑ 5∗
15 16
100E21
17 18 19 20
100E17 -
ψ(G14,G15) ngsG8 ↑ 5∗ y(G8 →21) ngsG8 ↑ 5∗ ngsG8 ↑ 5∗ y(G17 →E17) y(G15 →E12) ngsG1 :↑ 5∗
100E21 100E17 100E12
- ψ(G14,G15) ngsG8 ↑ 5∗ y(G8 →21) ngsG8 ↑ 5∗ ngsG8 ↑ 5∗ y(G17 →E17) y(G15 →E12) ngsG1 :↑ 5∗
Summary
200
186.8
500
Summary
400
ngsG1 ↑ 5∗ ngsG1 ↑ 5∗ y(G8 →E21) ψ(G14,G15) ngsG8 ↑ 5∗ ngsG1 ↑ 5∗ 21km ngsG8 ↑ 10∗ ngsG1 ↑ 10∗ ψ(G14,G15)
57km ngsG8 ↑ 15∗ ngsG1 ↑ 15∗ ψ(G14,G15)
8 16 14 5 16 2
9.4
* M m3 /day
-
y(G18 →E10) ngsG1 ↑ 5∗ ψ(G14,G15) ngsG1 ↑ 5∗ ngsG8 ↑ 5∗ y(G8 →21) ngsG8 ↑ 5∗ ngsG8 ↑ 5∗ y(G17 →E17) ngsG1 ↑ 5∗
100E17 100E12 -
55km ngsG8 ↑ 15∗ ngsG1 ↑ 20∗ 1 × ψ(G14,G15)
500
57km ngsG8 ↑ 15∗ ngsG1 ↑ 15∗ 1 × ψ(G14,G15)
500
57km ngsG8 ↑ 15∗ ngsG1 ↑ 15∗ 1 × ψ(G14,G15)
* M m3 /day
TABLE V C OMPUTATIONAL TIME OF DETERMINISTIC AND STOCHASTIC MODEL Solution algorithm
Deterministic model
Chance constrained programing model
Computational time
33sec
93.3min
difference when confidence level of the electric system, α is set higher. The influence of the individual confidence levels on the expansion plan is dependent on the level of integration between the systems, the ratio of the NG demand used for electricity generation and time horizon under consideration. A higher β, for example, results in a cheaper plan because NG demand for electricity is a small part of the total NG demand, therefore the main contributor to the system-wide reliability violation is the non-electric NG demand. This collaborates earlier results where the expansion plans were infeasibility because of the violation of adequacy constraint on the NG system. In addition, since the electric system in the test system is highly dependent on the NG system considering that all candidate generation is expected to be NG fiblack, increasing the reliability of the NG system significantly influence the increase in the system-wide reliability. Finally, as shown in Table VI, if the planning horizon had been 9 years, the three combinations of confidence levels consideblack would have resulted in similar solution from a cost perspective. VII. CONCLUSION In conclusion, the increasing dependency of the electric system on NG systems has necessitated an integrated planning approach towards planning both systems. Due to the cascaded relationship between the power and NG system, the expansion cost of ensuring reliable energy supply can become financially unattainable. A planner can manage this potential challenge by modelling the integrated planning problem in probabilistic
terms that allows manageable violations of the nodal balance constraints. In this paper, we proposed a chance constrained programming model of the integrated planning problem and show that NG storages are efficient in managing short-time stochasticity of the demand pairs in the long-term integrated planning of the electric and NG system and in achieving the desiblack confidence levels of energy supply. In addition, we show how the CPP model results in increased reliability in comparison to a deterministic mode. Finally, we show that the sensitivity of the long-term expansion plan to the selected confidence levels is dependent on the level of integration between the systems, the ratio of the NG demand used for electricity generation and time horizon under consideration R EFERENCES [1] US energy information administration, “International energy outlook 2016,” 2016, accessed 4 July 2017. [Online]. Available: https://www.eia.gov/outlooks/ieo/electricity.php [2] C. Unsihuay-Vila, J. Marangon-Lima, A. de Souza, I. J. Perez-Arriaga, and P. P. Balestrassi, “A model to long-term, multiarea, multistage, and integrated expansion planning of electricity and natural gas systems,” IEEE Transactions on Power Systems, vol. 25, no. 2, pp. 1154–1168, 2010. [3] F. Barati, H. Seifi, M. S. Sepasian, A. Nateghi, M. Shafie-khah, and J. P. S. Catalao, “Multi-period integrated framework of generation, transmission, and natural gas grid expansion planning for large-scale systems,” Power Systems, IEEE Transactions on, vol. PP, no. 99, pp. 1–11, October 2014. [4] C. Borraz, R. Bent, S. Backhaus, S. Blumsack, H. Hijazi, P. van Hentenryck et al., “Convex optimization for joint expansion planning of natural gas and power systems,” in 2016 49th Hawaii International Conference on System Sciences (HICSS). IEEE, 2016, pp. 2536–2545. [5] I. G. Sardou, M. E. Khodayar, and M. Ameli, “Coordinated operation of natural gas and electricity networks with microgrid aggregators,” IEEE Transactions on Smart Grid, 2016. [6] M. Babula and K. Petak, “The cold truth: Managing gas-electric integration: The iso new england experience,” IEEE power and energy magazine, vol. 12, no. 6, pp. 20–28, 2014.
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11
TABLE III C OMPUTED ELECTRICITY GENERATION AND NG SYSTEM EXPANSION PLANS Scenario 1 Year (t)
(n)
Pb
(MW)
1
100E24
3 4 6 7 8 11 12 14
-
17 18 19 20
Summary
* M m3 /day
100
yi,j / ψi,j
Scenario 2 (n)
Pb
(MW)
yi,j / ψi,j
Scenario 3 (n)
Pb
(MW)
yi,j / ψi,j
y(G8→E24) ngsG8 :↑ 15∗ ngsG1 :↑ 10∗ ψ(G11,G12) ngsG1 :↑ 10∗ infeasible infeasible infeasible infeasible infeasible infeasible infeasible
100E24
y(G8→E24)
100E24
y(G8→E24) ngsG8 :↑ 15∗ ψ(G11,G12)
-
ngsG1 :↑ 5∗
100E10
ngsG1 :↑ 5∗ y(G18→E10) ngsG1 :↑ 5∗ ngsG8 :↑ 5∗ y(G8→21)
infeasible infeasible infeasible infeasible
100E17
y(G18→E10) ngsG1 :↑ 5∗ ψ(G14,G15) ngsG8 :↑ 5∗ y(G8→21) ngsG8 :↑ 5∗ ngsG8 :↑ 5∗ y(G17→E17) y(G15→E12) ngsG1 :↑ 5∗
100E10 100E21 100E17 100E16
ngsG1 :↑ 5∗ y(G17→E17) ngsG8 :↑ 5∗ y(G13→E16) ngsG1 :↑ 5∗
27km ngsG8 :↑ 15∗ ngsG1 :↑ 20∗ 1 × ψ(G11,G12)
100E21
100E12
500
57km ngsG8 :↑ 15∗ ngsG1 :↑ 15∗ 1 × ψ(G14,G15)
500
52km ngsG8 :↑ 25∗ ngsG1 :↑ 20∗ 1 × ψ(G11,G12)
