Ordinal Optimization-based Multi-Energy System Scheduling for Building Energy Saving Zhong-Hua Su 1, Qing-Shan Jia 1,*,1 , Chen Song 2 1
Center for Intelligent and Networked Systems, Department of Automation, Tsinghua University, Beijing 100084, China 2 Ubiquitous Energy Research Center, ENN, Langfang, Hebei Province, China * corresponding author
[email protected],
[email protected],
[email protected]
Abstract. Buildings contribute a significant part in the energy consumption and CO2 emission in many developed and developing countries. Building energy saving has thus become a hot research topic recently. The technology advances in power co-generation, on-site generation, and storage devices bring us the opportunity to reduce the cost and CO2 emission while meeting the demand in buildings. A fundamental difficulty to schedule this multi-energy system, besides the other difficulties, is the discrete and large search space. In this paper, the multi-energy scheduling problem is modeled as a nonlinear programming problem with integer variables. A method is developed to solve this problem in two steps, which uses ordinal optimization to address the discrete and large search space and uses linear programming to solve the remaining sub-problems. The performance of this method is theoretically quantified, and compared with enumeration and a priority-and-rule-based scheduling policy. Numerical results show that our method provides a good tradeoff between the solution quality and the computational time comparing with the other two methods. We hope this work brings more insight on multi-energy scheduling problem in general. Keywords: Multi-energy system, ordinal optimization, linear programming, building energy saving, renewable energy.
1
Introduction
Buildings consume about 30% of the primary energy and 70% of the electricity in many developed and developing countries [1]. The technology advances in power co-generation, on-site generation, and storage devices bring us the opportunity to reduce the cost and CO2 emission while meeting the demand in buildings. For example, the combined cooling, heating, and power generation (CCHP) [2] consumes natural gas to satisfy the cooling, heating, and electricity loads in the same time, 1
This work was supported in part by the National Natural Science Foundation of China under grants (Nos. 60704008, 60736027, and 90924001), the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20070003110), and the Programme of Introducing Talents of Discipline to Universities (the National 111 International Collaboration Project, No. B06002).
which could increase the overall efficiency for energy generation. The building integrated photovoltaic (BIPV) [3] is a photovoltaic system integrated on building surfaces, which consumes solar energy and output larger electricity and hot water than traditional PVs. The technology advances in heat and electricity reservoirs [4] have significantly increased the capacity of the storage and reduced the conversion loss in the same time. Thus it is of great practical interest to schedule such a multi-energy system in order to satisfy the demand in buildings. However, such a multi-energy system scheduling problem is nontrivial due to at least the following difficulties. First, the nonlinear system dynamics. For example, the power generation of CCHP requires a minimal consumption of natural gas, which means when the input gas is smaller than the threshold value, CCHP does not output any cooling, heat, nor electricity. Second, the large search space that is caused by discrete integers. One approach to handle the nonlinearity of the aforementioned system dynamics is to approximate by piecewise linear functions. However, this approximation usually introduces indicator variables which take discrete and integer values. The size of the search space increases exponentially fast when the number of such indicator variables increases. This is also known as the curse of dimensionality. There have been abundant existing literatures to address the above difficulties, some of which will be briefly reviewed in section 2. On one hand, when there are integer variables, many of the existing works use approximate solution methodologies. Then it is usually difficult to theoretically quantify the performance of the resulting solutions. On the other hand, the scheduling of multi-energy system in practice is usually based on heuristics or experience. These methods may work well. But the performance degradation from the global optimum becomes even less clear. The key issue here is how to address the integer variables while theoretically quantifying the performance loss of the resulting solutions. We focus on this important fundamental difficulty in this paper. In this paper, the multi-energy scheduling problem on a daily basis is considered. This problem is modeled as a mixed integer programming problem, where the nonlinear dynamics of CCHP is approximated by piecewise-linear functions. In order to handle the resulting large search space of the indicator variables, a method is developed which combines ordinal optimization and linear programming. A unique advantage of this method is that the global performance of the resulting solution can be theoretically quantified. This helps us to understand the performance loss from the global optimum. This method is compared with the well-known enumeration method and a priority-and-rule-based method. Numerical results show that our method provides a good tradeoff between the solution quality and the computational time. The rest of this paper is organized as follows. In section 2, we briefly review related literatures. In section 3, the scheduling problem of multi-energy systems is mathematically formulated. In section 4, the method that combined ordinal optimization and linear programming (COOLP) is introduced and theoretically analyzed. In section 5, the numerical comparison among COOLP, enumeration, and priority-and-rule-based method is presented. We briefly conclude in section 6.
2
Literature Review
There have been abundant existing literatures on the scheduling of individual systems for energy saving. For example, the discussions in [5] on the schedule of CCHP, the discussions on the schedule of distributed renewable energy generation and distributed storage in [6], and the discussions on the schedule of solar powers in [7], just to name a few. The problem becomes more interesting and more practical when multiple energy systems are scheduled jointly. Li et al. [8] considered a joint scheduling problem of solar power, wind power, and power grid, and considered a linear system model. Genetic algorithm was then used to solve the problem. Derewonko and Pearce [9] jointly scheduled the solar PV panels and CHP to provide a constant voltage. Shen [10] addressed the daily scheduling problem in a sequential way. In each step, the scheduling problem of a single stage is solved using linear programming. Gupta et al. [11] considered the unit commitment problem in renewable energy generations, and scheduled the units according to the costs from small to large. Guan et al. [12] considered the joint schedule of multi-energy systems and solved the problem using the mixed integer programming package in CPLEX. Sun et al. [13] considered the joint schedule of multiple terminal devices for energy saving, and combined Lagrangian relaxation and dynamic programming to find an approximate optimal solution. Some of the aforementioned joint scheduling problems do not consider integer variables. When there are integer variables, the above works usually use approximate solution methods. Though numerical results were usually used to demonstrate the performance of these approximate methods, the performance loss from the global optimum is usually lack of theoretical analysis in existing works. This important problem is considered in this paper.
Fig. 1.
A multi-energy system.
3
Problem Formulation
In this section, we model the daily scheduling problem of a multi-energy system as a mixed integer programming problem. The multi-energy system in Fig. 1 is considered, which includes CCHP, BIPV, electric power grid, and heat grid as energy supplies; electricity and heat reservoirs as storages; and the intelligent building (IB) as demands. There are two types of demand from the IB, namely electricity and heat. The question is how to schedule the multiple energy system in a day to meet the demands from the IB while minimizing the cost of buying commercial electricity, heat and natural gas. Consider a discrete-time version of the problem, where a day is discretized into 24 stages with each stage representing one hour. The detailed explanations of all modules are given in the following subsection. 3.1
Models of Devices
1) Model of CCHP: We use a typical CCHP model as mentioned in [10]. This CCHP consumes natural gas V(t) (Unit: cubic meter) and electricity E b2 (t) (Unit: kWh) and outputs heat H g (t) (Unit: kWh) and electricity E g (t) (Unit: kWh). Due to the system architecture, assume that the electricity that is consumed by CCHP is always bought from the grid. The dynamics of CCHP can be described as follows. k (t ) 2 − k (t ) = 0 , (1) E g (t ) − 70k (t ) ≤ 0 , (2) 28k (t ) − Eg (t ) ≤ 0 , (3) 1.1696E g (t ) k (t ) + 82.7070k (t ) − H g (t ) = 0 ,
(4)
0.2741 E g (t ) k (t ) + 7.3034 k (t ) − V (t ) = 0 ,
(5)
0.0178Eg (t )k (t ) + 0.5457k (t ) − Eb 2 (t ) = 0 .
(6) where k(t)∈{0,1} depicts the off/on state of CCHP in stage t, t=1,…,24. The above equations approximate the CCHP dynamics by piecewise linear functions. 2) Model of BIPV: The BIPV consumes solar radiation S(t) (Unit: kWh) and outputs heat H s (t) (Unit: kWh) and electricity E s (t) (Unit: kWh) simultaneously. The system dynamics are as follows. Es (t ) = 0.1S (t ) , (7) H s (t ) = 0.6S (t ) . (8) 3) Model of electricity reservoir: The electricity reservoir receives (part of) the electricity from CCHP E g1 (t) (Unit: kWh) and (part of) of the electricity from BIPV E s1 (t) (Unit: kWh), and outputs electricity with amount E r (t) (Unit: kWh) to the IB. Denote the state-of-charge of the battery by R E (t) (Unit: kWh), which represents the battery level. In addition, we assume that the energy loss of the reservoir is 0.042%R E (t+1) during stage t, i.e., RE (t + 1) = 0.9996[ RE (t ) + Eg1 (t ) + Es1 (t ) − Er (t )], t = 1,..., 23 . (9) 4) Model of heat reservoir: This heat reservoir receives (part of) the heat from CCHP H g1 (t) (Unit: kWh) and (part of) the heat from BIPV H s1 (t) (Unit: kWh), and outputs heat with amount H r (t) (Unit: kWh) to the IB. Denote the heat level in the
reservoir as R H (t) (Unit: kWh). We assume that the energy loss of the reservoir is 0.01% R H (t+1) during stage t, i.e., (10) RH (t + 1) = 0.9999[ RH (t ) + H g 1 (t ) + H s1 (t ) − H r (t )], t = 1,..., 23 .
3.2 E b (t): E b1 (t): E g2 (t): H g2 (t): E s2 (t): H s2 (t): E x (t): H x (t): H b (t): C(t):
3.3
Other Nomenclatures electricity bought from power grid in stage t the portion of E b (t) that is directly dispatched to IB in stage t the portion of E g (t) that is directly dispatched to IB in stage t the portion of H g (t) that is directly dispatched to IB in stage t the portion of E s (t) that is directly dispatched to IB in stage t the portion of H s (t) that is directly dispatched to IB in stage t electricity demand of IB in stage t heat demand of IB in stage t heat that is bought from heat grid in stage t price of electricity on power grid in stage t (Unit: Yuan). C(t)=0.30 when t=1,…6,23,24. C(t)=0.4883 when t=7,…24. Interconnection among Devices
The interconnection among devices can be described as follows. Eg (t ) = Eg1 (t ) + Eg 2 (t ) ,
(11)
H g (t ) = H g1 (t ) + H g 2 (t ) ,
(12)
Es (t ) = Es1 (t ) + Es 2 (t ) , H s (t ) = H s1 (t ) + H s 2 (t ) ,
(13) (14)
(15) Eb (t ) = Eb1 (t ) + Eb 2 (t ) . The balances between the supply and demand on electricity and heat are described as (16) EX (t ) = Es 2 (t ) + Er (t ) + Eg 2 (t ) + Eb1 (t ) , H X (t ) = H s 2 (t ) + H r (t ) + H g 2 (t ) + H b1 (t ) .
3.4
(17)
Objective Function
At stage t the control variables in our model include the decisions on CCHP, BIPV, heat and electricity reservoirs, electric power grid, and heat grid, i.e., X(t)= [E b (t), E b1 (t), E b2 (t), E g (t), E g1 (t), E g2 (t), H g (t), H g1 (t), H g2 (t), E s1 (t), E s2 (t), H s1 (t), H s2 (t), H b (t), V(t), k(t), E r (t), H r (t), R E (t), R H (t)]. Thus the control variables are X=(X(1),…,X(24)). It is trivial and straightforward to verify that there are 480 variables in total. Our objective is to minimize the cost of electricity, heat, and natural gas in a day, i.e., 24 (18) f (X ) = [ 0.18 H (t ) + 1.5 V (t ) + C (t ) E ( t ) ] .
∑ t =1
b
b
where 0.18 and 1.5 are the price. Now the multi-energy system scheduling problem can be mathematically described as min X f ( X )
subject to the constraints in Eqs. (1)-(17). In the following discussion, we refer this problem as problem P. This is a mixed integer programming problem, where k(t), t=1,…,24 are integer variables.
4
Solution Methodology
As aforementioned the integer variables in this problem make it difficult to solve problem P. We now introduce our solution methodology combining ordinal optimization and linear programming (COOLP). Note that the decision variables in X can be divided into two groups, namely X 1 and X 2 , which contain the integral and continuous variables, respectively. Let Θ be the set of values that X 1 can take, which is usually called the design space. Note that Θ is a discrete set but could have a large size. Each element θ∈Θ is a vector with dimensionality of |X 1 |. The idea of COOLP contains two steps. In the first step, we use ordinal optimization [14] to select a set of θ’s from Θ. Denote this set as S. In the second step, for each θ∈S, we solve a linear programming to determine the values of variables in X 2 , denoted as X 2 (θ). Then we pick the best θ in S such that f(θ,X 2 (θ)) is minimized, i.e., argminθ ∈S f θ , X 2 (θ ) .
(
)
Let us sort the designs θ’s in Θ from small to large according to f(θ,X 2 (θ)). Define the good enough set G as the top-g designs in Θ. Then if we blindly pick s designs in the above first step, it can be shown that [14] ⎛ g ⎞⎛ N − g ⎞ ⎟⎜ ⎟ min ( s , g ) ⎜ i s−i ⎠ Pr {| G ∩ S |≥ k } = ∑ ⎝ ⎠ ⎝ , (19) ⎛N⎞ i=k ⎜ ⎟ ⎝ s ⎠ where g=|G|, s=|S|, and N=|Θ|. This probability is also known as the alignment probability. In other words, by restricting to the selected set S, we can still find a good enough design with high probability. If s
