A Hybrid Stochastic/Robust Planning Model for Integrated Energy System Considering Multiple Uncertainties

Complex energy flows exist in integrated energy systems (IESs), and the optimal assignment of different energy sources has become a primary challenge in the initial stage of its construction. In addition, multiple uncertainties, such as wind power fluctuations and multiple energy load fluctuations, are not well-respected in IES planning. In order to solve the above problems, the generation expansion planning model in IES is proposed. In the model the uncertainties of renewable energy resources (RESs) are described by scenarios, while the power price and fuel price are described by intervals

Operation unit prices of WT w, ESS e and equipment ch.
The performance coefficient of CS and AC.

CS cop cop
Active and reactive power demands at node i in typical day d during period t.
Heat and cold power demands at node j in typical day d during period t.
Minimum and maximum storage energy of ESS e.
Conductance of line ij.
Maximum output heat or cold power of equipment ch.
Upper limit of apparent heat power of hot pipe h.
Probability of wind fluctuation scenario s.
Lower and upper limit of electricity purchasing power of substation st.
Forecast output power of WT w under scenario s in typical day d during period t.
Minimum and maximum output power of DG g.
Maximum output power of ESS e.
Maximum output power of equipment ch.
Upper limit of electricity purchasing reactive power of substation st.
Lower and upper limit of static var compensator svc.
Upper limit of apparent power of line ij and DG g.
Base voltage of reference node i.

VOLL
Value of lost load.
A random parameter to represent the electricity purchasing price under scenario s in typical day d during period t.
A random parameter to represent the fuel purchasing price under scenario s in typical day d.
Heat to electric ratio of GE and GT.
Fuel transfer efficiency of GE, GT and GB.
Discharging efficiency of ESS e.

Decision Variables
Binary variables to represent the absorbing and releasing state of candidate HP, HS and CS.
Fuel consumption of equipment ful under scenario s in typical day d during period t.
Active heat output power of equipment GE, GT and GB under scenario s in typical day d during period t.
Active cold output power of equipment HP and AC under scenario s in typical day d during period t.
Active heat output power of equipment HP and AC under scenario s in typical day d during period t.
Active absorbing power of equipment HS and CS under scenario s in typical day d during period t.
Active releasing power of equipment HS and CS under scenario s in typical day d during period t.
Active heat power flow of hot pipe h under scenario s in typical day d during period t.
Load curtailment at node i under scenario s in typical day d during period t.
Interger variable to represent the construction number of equipment ch.
Charging and discharging output power of candidate e under scenario s in typical day d during period t.
Active and reactive output power of WT w under scenario s in typical day d during period t.
Active and reactive power flows from node i to node j under scenario s in typical day d during period t.
Active output power of equipment ch under scenario s in typical day d during period t.
Reactive output power of static var compensator svc under scenario s in typical day d during period t.
, , , , , ; Wind curtailment of WT w under scenario s in typical day d during period t.

Introduction
With the massive increase in renewable energy sources (RESs), such as wind power and photovoltaics, the optimal assignment of different energy sources (such as electricity, heat and cold) becomes the primary challenge at the initial stage of the integrated energy system (IES) [1][2][3].Especially, the planning of generating units [4] has become one of the most significant challenges in IES.
Many scholars have analyzed the planning problem of IES.Reference [5] proposes a stochastic optimization (SO) model for IES planning considering RES and energy storage system (ESS), and only the uncertainty of wind turbine (WT) output is considered and it is described by scenarios.
Reference [6] proposes a SO planning method for IES based on Energy Hub model, where the load uncertainty is described by scenarios.Reference [7] proposes a robust optimization (RO) planning model for Energy Hub, where the energy price uncertainty is described by intervals.Reference [8] proposed an adaptive RO planning model.The load, RES output and number of vehicles are all described by intervals.References [9][10] propose robust planning models to considering uncertainties of WT output, load and electricity price, and these uncertainties are all described by intervals.
Reference [11] proposes a multi-scenario based stochastic programming model of IES, which described the uncertainties of WT and photovoltaic (PV) by scenarios by k-means clustering.
For the researches mentioned above, it can be found that the uncertainties in the model are described by either discretized scenarios of probability distribution function (PDF) or intervals, and the corresponding RO or SO models are established.
However, for a specific uncertainty is concerned, if enough PDF information can be obtained, this uncertainty can be described by PDF and it can be further discretized by scenarios.If it does not have enough PDF information, this uncertainty can be simply described by intervals.Therefore, the best description measure of an uncertainty depends on the corresponding information that can be obtained.If multiple uncertainties are considered in an optimization model and they described by both Nowadays, a hybrid S/R optimization is proposed where multiple uncertainties are described in terms of both scenarios and intervals.However, the hybrid S/R optimization has not attracted much attention.
Reference [12] proposes a hybrid S/R scheduling model for grid-connected microgrid, where the availability of equipment is described by scenarios, and active and reactive load, energy price, active power of RES are all described by intervals.Reference [13] proposes a hybrid S/R optimization method to minimize the overall cost of all energy management units, where market electricity price uncertainty is described by scenarios, WT and load uncertainties is described by intervals.
It can be found that the S/R optimization mentioned above are all applied in scheduling problems.The S/R optimization is not applied in planning problems, especially in IES planning.
In this paper, a hybrid S/R optimization is proposed for IES planning.Multiple uncertainties are respected.The uncertainty of WT output is expressed by scenarios, while the uncertainties of energy price which includes the electricity price and fuel price is expressed by intervals.T The proposed multi-level S/R optimization model is based on the duality theory.The main contributions of this article are as follows: 1) A hybrid S/R optimization is proposed for IES planning, where multiple uncertainties are respected in terms of both scenarios and intervals.
2) By strong dual theory, the proposed multi-level S/R optimization model is reconstructed into a MILP model, and can be efficiently solved by commercial solvers.
The later parts of this article are arranged below.The second section briefly introduces mathematical models and box uncertainty sets.The third part describes the linearization and resolution of the model.Section IV provides case studies and Section V summarizes conclusions.

Mathematical model
The objective function is the minimum solution for the total cost, including annual investment cost, operating cost, and reliability cost.The objective function model is as follows. ( (2) where CINV, C DIS OPE , C HC OPE and CR are the annual investment cost, and reliability cost, respectively.In (2), the capital recovery factor Rn can be expressed as below.(6) where u is the discount rate, and yn is the economic life.The operation cost function of unit g in ( 3) can be easily expressed as below. (7)

Distribution system constraints
The constraints of the distribution system are listed as below.

1) Electric power balance constraints
g s g d t g s g d t g s g d t a P b P c

= ( ) , , ,, s st d t s g d t s w d t st bus i g bus i w bus i s e d t s ch d t e bus i ch bus i P s ij d t s ji d t i d t s i d t i start ij
i end ij P P P Equation ( 8) and ( 9) represent the active and reactive power balance at node i of IES, respectively; (10) represents the total power consumption of cold-heat equipment.

2) Line flow and voltage magnitude constraints
The traditional AC power calculation equations are difficult to be solved.So for simplicity, the linear AC power flow calculation equations with voltage magnitude and phase angle [14] are applied and it is shown as below. ( Constraint ( 13) gives the transmission line flow limit, and ( 14) represents the voltage magnitude limits.
3) Power output constraints s g d t s g d t g x P P Equation (15) gives the active and reactive power output limits of upstream grid.Similarly, ( 16) and (17) give the power output limits of DG g and WTs w, (18) gives the power and energy limits related to ESS e, (19) is the upper and lower output limits of the static reactive power compensator svc.4) Wind spillage and loss of load constraints (20) Constraint (20) shows that the wind spillage and loss of load are less than the forecast values of wind and load, respectively.

Cold-heat system constraints
The constraints of the cold-heat system are listed as below.
1) Heat and cold power balance constraints The constraints of ( 21) and ( 22) represent the heat power balance and cold power balance at node j, respectively; (23) gives the upper limits of hot pipe transmission power flow.
2) Heat and cold equipment constraints , , , , , , , , , , , , , , The equipment in our proposed cold-heat system includes GE, GT, GB, HP, HS, CS, and AC, and all the physical constraints of cold-heat equipment are as below. ( (29) Constraints (24) gives the output limits, fuel conversion electrical power efficiency limits and electrical power conversion heat power efficiency limits of GE and GT, respectively; (25) gives the output limits and fuel conversion heat power efficiency limits of GB; (26) describes the output limits of AC; (26), ( 27) and ( 28) are similar constraints that characterize the heat and cold conversion limits of equipment HP, HS and CS, respectively.

Ambiguity uncertain sets
In the objective function, the electricity price and fuel price are uncertain values, and this section will specifically introduce the construction process of theirs uncertain sets.
The uncertainties of electricity and fuel price can be described by an intervals [15].By introducing auxiliary variables, random variables are represented by midpoints of interval and auxiliary variables [16].With the introduction of auxiliary variables, the random variable  " !,#,$ %&'# can be expressed as below. (30) Here, z+ s,d,t and zs,d,t are auxiliary variables and they satisfy that z+ s,d,t and zs,d,t [0, 1]. # #,$ %&'# is the midpoint of the interval and  % #,$ %&'# is the radius of the interval.
In order to limit the conservativeness, the number of scenarios that reach the worst-case at the same time is constrained by corresponding budget of constraint, and the budget of uncertainty is shown as below. (31) In (31), Γd is the conservativeness parameter representing the number of periods that reaches the worst case in typical day d.
After that, the entire ambiguity set of electricity price can be written as below. (32) Similarly, with the introduction of auxiliary variables v+ s,d and v-s,d [0, 1],  # !"#$% and  $ !"#$% , the random variable  % &,! "#$% can be expressed as below. (33) After that, the entire ambiguity set of fuel price can be written as below. (34) In (34), Γ is the conservativeness parameter represent the number of days that reaches the worst case.

Solution method
In the proposed model, the uncertianties caused by electricity price and fuel price are descirbed by intervals.Besides, the uncertainties caused by RES output in the model are descrived by scenarios.The model minminzes the expected value with respect to the scenarios, and it also belongs to a SO model.Therefore, the proposed S/R optimizaiton model possesses the characteristcs of RO and SO, and it combines the advantages of RO and SO.
After the random energy prices are expressed by the deterministic forms, the entire optimization will be recast as a min-max robust optimization problem.The objective function of (1) can be reformulated as below. (37) where CINV and CR are the same as ( 2) and ( 5), C DIS OPE and C HC OPE can be reformulated as below. ( where obj 1 and obj 2 represent the electricity purchasing cost and fuel purchasing cost, and they can be expressed by ( 40) and ( 41). (40) (41) ( ) where the inner maximization part of obj1 and obj2 tries to simulate the electricity price and fuel price, and it cannot solved directly [16], the bi-level objective function transformed into the singlelevel optimization problems as below.
4 Case studies

Data of the case studies
The proposed model was tested in an IEEE 33-node power distribution system [18].The test system structure is shown in Fig 1 .Four typical days are chosen to validate the cases [20].The line capacity is 0.5MW.The original power load is expanded by a factor of 2.0.WT4-WT6 is 1MW respectively.The capacity factor of each WT is 0.9.WT uncertainties follow Gaussian distribution.The } , , , distribution is discretized into five scenarios and the probability of each scenario is 0.062, 0.242, 0.392, 0.242 and 0.062, respectively.Two static var compensators (SVC) is added on node 3 and 28, respectively.The VOLL is set as 5000 $/MWh.The discount rate d is set as 0.06, and the economic life yn is set as 20.
Five schedulable DGs exist in the system, and the specific data of DGs are shown in Table 1.Ten candidate DGs and three candidate ESSs can be added in the system and their parameters are shown in Table 2 and 3, respectively.
The cold-heat system [19] consists of 11 nodes and 12 branches.The hot pipe capacity is 0.3 MW.The specific data of existing GE and GT are shown in Table 4.The specific data of candidate GB, HP, HS, CS and AC are shown in Table 5.
The interval radius  % #,$ %&'# and  % # ()*+ are set as 0.2 times of interval midpoint  # #,$ %&'# and  # # ()*+ , respectively.The conservativeness parameter Γd is set as 12, which means 12 periods of electricity price in one day that reach the worst-case.And the conservativeness parameter Γ is set as 2, which means two days of fuel price in four typical days that reach the worst-case.The case study is encoded on a General Algebra Modeling System (GAMS) [21] platform and solved using the MILP solver of CPLEX [22].

Effect of various uncertainties
To analyze the effect of uncertainties and different optimization methods, the following subsections will be studied through case studies.
1) Case A: no uncertainty is considered in the model.
2) Case B: only the uncertainty of WT is considered, which is described as five scenarios.
3) Case C: both the uncertainty of WT and electricity price are considered.
4) Case D: the uncertainties about WT, electricity price and fuel price are all considered.
5) Case E: the uncertainty about WT is not considered, but both the uncertainties of electricity price and fuel price are considered.
Different costs of Cases A, B, C, D and E are shown in Table 6.The construction plans of DGs and ESSs of Cases A, B, C, D and E are shown in Table 7, and the construction plans of cold-heat equipment of Cases A, B, C, D and E are shown in Table 8.From Table 6, compared Case A with B, it can be found that when the uncertainty of WT is considered, the total cost, investment cost, electricity purchasing cost and reliability cost all increase in order to resist the power fluctuation caused by WT fluctuation.Compared Case A with E, it can be found that when the interval uncertainties are considered, all of the cost increase in order to cope the market price fluctuation.Compared Case B, C and D, it can be found that the market price fluctuation has a great impact on reliability cost, because the IES adopts wind and load curtailment to dealing with the market price fluctuation.
From Table 7, it can be found that when more uncertainties are considered, more DGs and ESSs with large capacity are constructed to address the uncertainties caused by WT, electricity price and fuel price.It is worth mentioning that the hybrid stochastic/robust optimization (Case D) makes a compromise construction plan between optimism of stochastic optimization (Case B) and conservativeness of robust optimization (Case E).The effectiveness of S/R optimization in our IES model is proved.
From Table 8, compared Case A with B, it can be seen that the WT uncertainty has a great impact on the construction plan of cold-heat system, because the WT penetration is relatively high in the IES system.Compared Case A with E, the uncertainty of market price has little impact on the construction plan of cold-heat system, the construction plans about Case B, C and D are nearly the same.

Effect of different Γd and Γ
The total cost with respect to different Γd and Γ in Case E are shown in Fig. 2. The unit of the total cost in Figure 1 is 10 4 $.When Γ is set as zero, the electricity purchasing cost with respect to different Γd is shown in Fig. 3.When Γd is set as zero, the fuel purchasing cost with respect to different Γ is shown in Fig. 4.  From Figure 2, it can be found that when Γd increases, more serious situation is considered.The total cost increases but finally the increase becomes saturate.And when Γ increases, the total cost also increases but the total cost value about Γ =3 and 4 is basically the same.
From Figure 3 and Figure 4, it can be found that when Γ and Γd increase, the purchasing cost of electricity and fuel grows at the beginning, but tend to saturation.It illustrates that the amount of electricity and fuel purchasing cost is gradually decreasing in order to resist the price fluctuation by robust optimization.

Analysis of computational efficiency
To analysis the computation efficiency of stochastic, robust and stochastic/robust optimization method, the CPU times about Case A, B, C, D and E are shown in Table 9.From Table 9, it can be seen that the CPU time of Case B is 29 times larger than Case A, which reveals that the SO needs significant computation resource.It is worth mentioning that the CPU time of Case E is a little bit longer than that of Case A, and the CPU times of Case C and D are also a little bit longer than that Case B, which illustrates that RO shows desirable computation efficiency.The proposed S/R optimization possesses the characteristics of RO and SO in terms of computation efficiency.

Conclusion
In this paper, a hybrid S/R optimization planning model for IES considering multiple uncertainties is proposed.The uncertainty of WT is described by scenario while the uncertainties of energy price is described by interval.The proposed model involves random parameters and it is finally recast as a single-level optimization problem by strong dual theory.The case studies show that the cost and construction plan are significantly influenced by multiple uncertainties.

and β 3 d
are duality variables of constrains , and y 1 d , y 2 d and χ 3 are duality variables of constrains .By doing so, C DIS OPE and C HC OPE can be reformulated as (48) and (49).

Figure 1 .
Figure 1.The specific structure of the test system.

Table 1 .
Specific data of existing DGs.

Table 2 .
Specific data of candidate DGs.

Table 3 .
Specific data of candidate ESSs.

Table 4 .
Data of GE and GT

Table 5 .
Data of GB, HP, HS, CS and AC

Table 6 .
Cost of the Five Cases

Table 7 .
Construction DGs and ESSs of Five Cases

Table 8 .
Construction Cold-Heat equipment of Five Cases

Table 9 .
Computation Time