Comparison of Procedures to Combine Operating Zones of Multiple Units into Plant-Based Operating Zones

Abstract

The hydropower unit commitment (HUC) is often indirectly considered by combining individual units in a hydropower plant into a plant-based generator to alleviate the dimensional difficulty in short-term hydropower scheduling of cascaded reservoirs. This work presents three procedures to combine operating zones of units in a hydropower plant into the plant-based operating zones, including an exhaustive method (M01) that enumerates all the possible combinations, a discrete method (M03) that investigates discrete values to see if they are in an operating zone, and a bound oriented method (M04) that explores new operating zones with an optimization solver. The procedures are compared with a previous one (M02) that merges the operating zones of a unit one by one in order. The experiments of the methods in 11 case studies involving a variety of hydropower plants reveal that the second method (M02) should always be recommended due to its strength, which is even more prominent than any other methods in dealing with large-scale problems, and errors occur when using interpolation to estimate the plant-based operating zones at a water head between two sampling water heads.

1. Introduction

People have made great use of moving water to produce electricity for centuries, and hydropower is now one of the crucial energy sources in modern industry, especially in recent years when the diversity of power suppliers and demanders is complicating the power systems [1,2], which require for more units to be flexible in their power outputs to integrate more renewable energy, which, as well known for wind and solar powers, is volatile, intermittent, and growing fast on a large scale [3,4]. A hydropower reservoir is naturally as a massive battery that can be recharged by refilling and discharged by drawing down its storage, providing flexible power outputs when appropriately scheduled to ensure the security and stability of the power systems.

The hydropower unit commitment (HUC) is an essential task to securely, accurately, and economically implement the hydropower schedule of a hydro-plant. The HUC is typically a nonlinear and discrete optimization problem [5]. The nonlinearity comes from the power output, a nonlinear function of the generating discharge of the unit, and the water head of its hydropower plant, which, again, has a nonlinear relationship with the outflow and storage of the reservoir. The discreteness is incurred by isolated operating zones and must be imposed by introducing integer variables to avoid the vibration for security concerns. The significant challenges in solving the problem have attracted a variety of optimization methods to be applied, including dynamic programming (DP) [6,7], mixed integer linear programming (MILP) [5,7], as well as many heuristic algorithms, such as the artificial neural networks (ANN) [8], the genetic algorithm (GA) [9], and so on.

When detailed with comprehensive constraints involving the up/down duration, startup frequency, and vibration zones of a turbine unit, for instance, the HUC can be integrated into the hydropower scheduling of cascaded reservoirs, but on a small scale, and often limited to a single hydropower reservoir. For instance, Zhou et al. [10] proposed a multi-objective mixed-integer nonlinear programming model considering feasible operating zones of hydropower plants, which was applied to four plants on the Beipanjiang River in China to regulate peak load and enhance power efficiency. Zhang et al. [11] developed an ultra-short-term hydropower scheduling model for four cascade hydropower plants on the Beipanjiang River in China based on MILP with multiple vibration zones constraint included. Yu et al. [12] examined an information-gap decision theory-based model to solve the short-term hydro scheduling with four reservoirs in series, taking forbidden operation zones of hydropower plants into account.

When integrated into the operation of cascaded hydropower reservoirs, the HUC is often indirectly considered by combining individual units in a hydropower plant into a plant-based generator to alleviate the dimensional difficulty in solving the problem. Cheng et al. [13], for instance, proposed a method to combine multiple vibration zones and combined the vibration zone avoidance strategy with a progressive optimality algorithm for an optimization framework, which was applied to 176 hydropower plants to solve the short-term hydro scheduling problem, obtaining a good and time-acceptable result. Wang et al. [14] took the operating zone constraints into account for short-term hydro scheduling and investigated how and how much the cooperation between reservoir-based operation and unit commitment can contribute to spillage reduction in a hydropower system of 45 reservoirs.

The conditions when a hydropower unit vibrates in operation are usually observed and recorded to define the operating zones favorable to the unit. Zhao et al. [15], for instance, identified the vertexes of the vibration zones and approximated them as polygons to define the operating zones by subtracting the union of all the vibration zones. The unit operating zones, usually in tabular form, are given at discrete water heads, where the plant-based operating zones can be accurately determined. However, the plant-based operating zones at a water head between two discrete water heads, when estimated by interpolation between the plant-based operating zones at the discrete water heads, will very likely have nonnegligible errors. Even the number of operating zones at a water head can differ from that at its neighboring water head, resulting in the plant-based operating zones being inconsistent with the unit operating zones.

At any water head between two discrete ones, an accurate way is to combine the unit into plant-based operating zones after the unit operating zones are determined by interpolation between those at two discrete water heads. In other words, the unit operating zones at any water head must be determined first and then combined into the plant-based operating zones, rather than the plant-based operating zones at the discrete water heads are determined first and then used to determine those at any water head by interpolation.

The plant-based operating zones, however, change over time during a scheduling horizon since they are dependent on the water head that changes over time. The quarterly water heads during the day may also change during the solution procedure and can be updated iteratively to achieve convergence. Li et al. [16], for instance, presented a procedure that initiated the hourly tailwater levels during the study horizon, which were then updated until the convergence was achieved. Cheng et al. [17] employed an iterative strategy to update the water heads until the convergence, and in each iteration during the solution procedure of the scheduling problem of cascaded hydropower reservoirs, the unit operating zones at each quarter during a day must be updated frequently and then combined into the plant-based zones that match the water head more accurately. Apparently, the efficiency of a procedure to combine the unit into plant-based operating zones will significantly impact the efficiency in solving the hydropower scheduling problem.

The plant-based operating zones can be identified by a procedure, denoted as M01, which enumerates all the possible combinations of the operating zones of all the units in the hydropower plant. Our previous work [18] presented a procedure, denoted as M02, to combine the unit operating zones in a hydropower plant into plant-based operating zones by merging the operating zones of the units one by one. It was unclear, however, if changing the merging order of the units in the procedure would lead to different results. Additionally, the number of merging trials can likely be reduced to improve the solution efficiency of the procedure.

This work will present two more procedures to be compared with the M01 and M02 in combining the unit into plant-based operating zones denoted as M03 employs an optimization solver to reduce the number of zone-merging trials, and M04 checks discrete points to identify the possible operating zones of a combined generator. The experiments will also be performed to investigate the precision and efficiency of the procedures used to combine the unit into plant-based operating zones, as well as the errors if interpolated between the plant-based operating zones at two discrete water heads.

2. Problem Formulation

An individual hydropower unit is generally required to operate in certain isolated zones to avoid vibration with its generating discharge enforced in only one of the operating zones,

$${\displaystyle \sum _{k=0}^{K(i)}{\mathrm{L}}_i^{(k)}(h)\cdot z_i^{(k)}}\le q_i\le {\displaystyle \sum _{k=0}^{K(i)}{\mathrm{U}}_i^{(k)}(h)\cdot z_i^{(k)}}$$

(1)

$${\displaystyle \sum _{k=0}^{K(i)}{z}_i^{(k)}}=1$$

(2)

where i and k = indices for units and operating zones of a unit, respectively, with k = 0 to indicate the shutdown status; K(i) = number of operating zones of unit i; ${\mathrm{L}}_i^{(k)}(h)$ and ${\mathrm{U}}_i^{(k)}(h)$ = lower and upper bounds in m³ in the k^th zone of unit i, functions of the water head (h) of a hydropower reservoir; $z_i^{(k)}$ = binary variable to indicate whether it is in the k^th zone of unit i; q_i = generating discharge in m³ of unit i.

The operation of multiple hydropower reservoirs, when detailed into the operating zones of individual units, will often encounter dimensional difficulties in solving the problem. A more practical strategy is to combine the operating zones of the individual units in a hydropower plant into plant-based operating zones, in which the generating discharge from a hydropower plant is

$$Q={\displaystyle \sum _{i=1}^I{q}_i}$$

(3)

$$\{\begin{array}{l}{\displaystyle \sum _{n=0}^N{\mathrm{LB}}^{(n)}(h)\cdot z{n}^{(n)}}\le Q\le {\displaystyle \sum _{n=0}^N{\mathrm{UB}}^{(n)}(h)\cdot z{n}^{(n)}}\\ {\displaystyle \sum _{n=0}^{N}z{n}^{(n)}}=1\end{array}$$

(4)

where Q = generating discharge in m³ from the hydropower plant; I = number of units in the hydropower plant; N = number of plant-based operating zones of the hydropower plant; ${\mathrm{LB}}^{(n)}(h)$ and ${\mathrm{UB}}^{(n)}(h)$ = lower and upper bounds in m³/s of the n^th operating zone of the hydropower plant, respectively; functions of water head (h); zn⁽ⁿ⁾ = binary variable to indicate whether the hydropower plant operates in its n^th zone.

With the water head (h) fixed to a specific value, the task is to identify the plant-based lower (LB) and upper bounds (UB) of operating zones so that for any generating discharges from individual units (q_i) that satisfy the constraints (1) and (2) we can always find the binary variables [zn⁽ⁿ⁾] and Q that meet the constraints (3) and (4); and vice versa; for any generating discharge (Q) that meet constraints (4), we can also find binary [$z_i^{(k)}$] and generating discharges (q_i) that satisfy the constraints (1), (2) and (3). This is a very challenging task since it is difficult to express “for any generating discharges mathematically.”

3. Solution Procedures

This work will experiment with four procedures to identify the plant-based operating zones. The first enumerates all the possible combinations of the operating zones of all the units, the second merges the operating zones sequentially unit by unit into the plant-based operating zones, the third uses discrete values to represent “any generating discharge” of the hydropower plant, and the fourth one, in a more accurate way, will identify one by one the lower and upper bounds of the plant-based operating zones.

3.1. M01: A Way to Enumerate All the Possible Combinations

This exhaustive method enumerates all possible combinations of unit-operating zones to determine all the plant-based operating zones that may overlap with each other but will be merged to obtain their union that defines the plant-based operating zones without any overlap.

All the possible plant-based operating zones that may overlap with each other can be determined as follows:

$$\{\begin{array}{l}LB(m)={\displaystyle \sum _{i=1}^I{L}_i^{(k(i))}}\\ UB(m)={\displaystyle \sum _{i=1}^I{U}_i^{(k(i))}}\end{array} \mathrm{for} 1\le k(i)\le K(i)$$

(5)

by enumerating all the possible k(i), which represents the index of one of the operating zones of the i^th unit, and the set of plant-based operating zones without overlaps can be obtained via

$$A={\displaystyle \underset{m=1}{\overset{M}{\cup }}[LB(m),UB(m)]}$$

(6)

where $m=1,2,\cdots ,M$ with M being the number of all the possible combinations by selecting one [= k(i)] of the operating zones of each unit, determined as

$$M={\displaystyle \prod _{i=1}^{I}K(i)}$$

(7)

which will be 59,049 = 3¹⁰ for ten units and three operating zones for each unit.

3.2. M02: A Way to Merge Zones Sequentially

The second method (M02) was proposed by Xianliang Cheng [18], who improved the exhaustive method (M01) by sequentially merging the operating zones unit by unit into the plant-based operating zones, leading to much fewer trials of combination.

The procedure starts by initiating the plant-based operating zones to be those of the 1st unit,

$$\{\begin{array}{l}L{B}_1^{(m_0)}=L_1^{(k)}\\ U{B}_1^{(m_0)}=U_1^{(k)}\end{array} \mathrm{for} 1\le m_0=k\le M_0(1)=K(1)$$

(8)

which is used to generate the plant-based operating zones with overlaps by including one more unit,

$$\{\begin{array}{l}L{X}_i^{(m_1)}=L{B}_{i-1}^{(m_0)}+L_i^{(k)}\\ U{X}_i^{(m_1)}=U{B}_{i-1}^{(m_0)}+U_i^{(k)}\end{array}\forall [(m_0,k)|1\le m_0\le M_0(i-1);1\le k\le K(i)]$$

(9)

where $m_1=1,2,\cdots ,M_1(i)$, which is the number of all the possible combinations of (m₀, k).

The set of plant-based operating zones without overlaps can be updated with,

$$A_i={\displaystyle \underset{m_1=1}{\overset{M_1(i)}{\cup }}[L{X}_i^{(m_1)},U{X}_i^{(m_1)}]}$$

(10)

which determines the lower and upper bounds of the plant-based operating zones,

$$\{\begin{array}{l}L{B}_i^{(m_0)}=L(A_i,m_0)\\ U{B}_i^{(m_0)}=U(A_i,m_0)\end{array} \mathrm{for} 1\le m_0\le M_0(i)$$

(11)

where $L(A_i,m_0)$ and $U(A_i,m_0)$ = lower and upper bounds, respectively, of the m₀^th isolated zone of set A_i; M₀(i) = the number of isolated zones of A_i.

The final plant-based operating zones are determined as

$$\{\begin{array}{l}LB(m)=L{B}_I^{(m_0)}\\ UB(m)=U{B}_I^{(m_0)}\end{array} \mathrm{for} 1\le m=m_0\le M_0(I)$$

(12)

after all the units have been included.

3.3. M03: A Way to Use Discrete Values

From (1), when all the units operate at their upper bounds of the last operating zone, the generating discharge of the hydropower plant will reach its maximum with,

$$0\le Q\le {\displaystyle \sum _{i=1}^I{U}_i^{(K(i))}(h)}=\mathrm{QMAX}$$

(13)

which will be represented with discrete values,

$$\widehat{Q}(j)=j\cdot \Delta \mathrm{for} j=0,1,\dots ,J$$

(14)

With

$$J\cdot \Delta =\mathrm{QMAX}$$

(15)

where QMAX = maximum of generating discharge of the hydropower plant; $\Delta$ = precision tolerance.

For any $j=0,1,\cdots ,J$, let $\lambda (j)=1$ if there exists a feasible solution [$q_i$ and $z_i^{(k)}$] that satisfies (1), (2) and

$$\widehat{Q}(j)={\displaystyle \sum _{i=1}^I{q}_i}$$

(16)

Otherwise, let $\lambda (j)=0$. The procedure may be time-consuming since every discrete value of the generating discharge has to be checked to see if there is a feasible solution.

Then, the plant-based operating zones can be determined by following the flowchart shown in Figure 1. Obviously, the first operating zone is special,

$$L{B}^{(0)}=U{B}^{(0)}=0$$

(17)

and a lower bound occurs when

$$\{\begin{array}{l}\lambda (j)=1\\ \lambda (j-1)=0\end{array} \mathrm{for} j=1,2,\cdots ,J$$

(18)

while an upper bound comes after when j = J or

$$\{\begin{array}{l}\lambda (j)=1\\ \lambda (j+1)=0\end{array} \mathrm{for} j=0,1,\cdots ,J-1$$

(19)

3.4. M04: A Way to Identify Zone Bounds Explicitly

Started from the first plant-based operating zone with all individual units shut down,

$$L{B}^{(n)}=U{B}^{(n)}=0$$

(20)

with n = 0, the procedure extends the current or explores a new operating zone by minimizing a potential lower bound,

$$\min LB$$

(21)

subject to the definitions of lower and upper bounds of a plant-based operating zone,

$$LB={\displaystyle \sum _{i=1}^I{\displaystyle \sum _{k=0}^{K(i)}{L}_i^{(k)}\cdot z_i^{(k)}}}$$

(22)

$$UB={\displaystyle \sum _{i=1}^I{\displaystyle \sum _{k=0}^{K(i)}{U}_i^{(k)}\cdot z_i^{(k)}}}$$

(23)

a lower bound on the upper bound to ensure more operating space to be explored,

$$UB\ge U{B}^{(n)}+\zeta$$

(24)

with a unit to be operating in only one of its operating zones,

$${\displaystyle \sum _{k=0}^{K(i)}{z}_i^{(k)}}=1$$

(25)

where $\zeta$ = a small positive for precision tolerance.

Solving the problems (12)–(16) gives the optimums: $L{B}^{*}$ and $U{B}^{*}$, then let

$$U{B}^{(n)}:=\max [U{B}^{(n)},U{B}^{*}]$$

(26)

to extend the current plant-based operating zone if the lower bound explored has an overlap on the current operating zone, that is

$$L{B}^{*}<U{B}^{(n)}$$

(27)

Otherwise, let n: = n + 1 to start a new plant-based operating zone with

$$\{\begin{array}{l}L{B}^{(n)}=L{B}^{*}\\ U{B}^{(n)}=U{B}^{*}\end{array}$$

(28)

The procedure repeats solving the optimization problem until there is no feasible solution to it, then the number of plant-based operating zones can be determined as:

$$N=n+1$$

(29)

4. Case Studies

4.1. Data Preparation

The models and procedures will be applied to six real-world hydropower plants on the Lancang River, along with three virtual hydropower plants (VH01-VH03) based at the Nuozhadu Hydropower Plant, to investigate the efficiency of the procedures in dealing with problems on various scales. 3, 4, 5, 6, 7, 9, 12, 15 and 18 units, as summarized in

Table 1. Profile of installed units in hydropower plants.

Unit Type	Rated Capacity (MW)	Hydropower Plant	Number of Units
1#	140	Lidi	3
2#	350	Miaowei	4
3#	700	Xiaowan	6
4#	300	Manwan	1
5#	250	Manwan	5
6#	120	Manwan	1
7#	650	Nuozhadu	9
8#	350	Jinghong	5
U09	650	VH01	12
U10	650	VH02	15
U11	650	VH03	18

, are installed in hydropower plants: Lidi, Miaowei, Jinghong, Xiaowan, Manwan, Nuozhadu, VH01, VH02, and VH03, respectively. The locations of the selected hydropower plants are illustrated in Figure 2. Except for the Manwan, which has three types of units, the other hydropower plants have identical units installed.

Table 2. Operating zones of the units.

Unit Type	Water Head (m)	Shutdown (MW)		Zone 1# (MW)		Zone 2# (MW)
		LB(0)	UB(0)	LB(1)	UB(1)	LB(2)	UB(2)
1#	36	0	0	60	140	--	--
1#	37	0	0	60	140	--	--
2#	81.6	0	0	120	170	230	292.4
2#	93	0	0	120	170	230	350
3#	212.33	0	0	120	240	480	676
3#	222	0	0	120	240	480	700
4#	89	0	0	--	--	--	--
4#	90	0	0	168	300	--	--
5#	89	0	0	140	250	--	--
5#	90	0	0	140	250	--	--
6#	89	0	0	90	120	--	--
6#	90	0	0	90	120	--	--
7#	152	0	0	211	220	420	467
7#	156	0	0	420	490	--	--
7#	162	0	0	420	523	--	--
8#	58	0	0	200	340	--	--
8#	62	0	0	200	350	--	--
U09	152	0	0	211	220	420	467
U10	152	0	0	211	220	420	467
U11	152	0	0	211	220	420	467

gives the operating zones of units at some typical water heads, where the operating zones are different from one water head to another, and even the number of them can be different at two adjacent water heads, with the 4# unit, for instance, having one operating zone at 89 m but two zones at 90 m. Here, the shutdown state [0, 0] is regarded as one of the operating zones, and the operating zones at a water head between two neighboring water heads of a unit are determined by interpolation, which is not an easy task when the number of operating zones is different between two adjacent water heads. To simplify, the operating zones at a water head (h₀) between two sampling water heads will be determined as those at one of the sampling water heads closer to the water head (h₀).

4.2. Comparison of Procedures

The models and procedures, run under AMD Ryzen 7 5800 H computer environment, are coded in C++ on Microsoft Visual Studio 2019, with the Gurobi 9.5.0. serving as the MILP solver in M03 and M04. With 10 cases studied,

Table 3. Solution efficiency of the procedures at a precision of 0.1 MW.

Hydro Plant	Water Head (m)	Scale (Zone×Unit)	Installed Capacity (MW)	Solution Time (Milliseconds)
				M01	M02	M03	M04
Lidi	36.4	2 × 3	420	<1	<1	106	12
Miaowei	85	3 × 4	1238	<1	<1	377	15
Manwan	89.4	1 × 1 + 2 × 5 + 2 × 1	1370	<1	<1	218	21
Jinghong	60	2 × 5	1724	<1	<1	181	24
Xiaowan	215	3 × 6	4095	<1	<1	528	66
Nuozhadu	158	2 × 9	4509	<1	<1	1624	73
Nuozhadu	152	3 × 9	4203	3	<1	1558	120
VP01	152	3 × 12	5604	73	<1	1602	234
VP02	152	3 × 15	7005	2411	1	1860	446
VP03	152	3 × 18	8406	71,480	1	2112	518

summarizes the experiments on a variety of hydropower plants to investigate how the scale of a problem will impact the solution efficiency of the procedures. The first six case studies, which may represent most of the hydropower plants in China, have the water heads chosen between two adjacent water heads that are given in Table 2, and the last four cases are designed to reveal the limit of a procedure in solving large-scale problems. The scale of a problem, when solved with the first three procedures (M01, M02, and M03) that do not have any numerical errors, is basically determined by the product of the number of units and the number of operating zones of a unit, while it is primarily decided by the installed capacity and the precision of discretization when using the last method (M04) that do have numerical errors in identifying the plant-based operating zones.

At the precision of 0.1 MW, all four procedures can identify the same plant-based operating zones but in different solution efficiency, as shown in the last four columns in Table 3. Impressively, the exhaustive method (M01) takes negligible time in solving problems in scales smaller than 18 (units times zones), which in fact, can cover most of the hydropower plants in China. The second method (M02), however, demonstrates more prominent strength than any other method in dealing with problems in scales larger than 36, taking negligible time even if the scale reaches 54. Apparently, the second method (M02) should always be recommended to solve problems on any scale.

All four methods identify the plant-based operating zones by trials. The first two methods (M01 and M02), for instance, trial possible combinations, the third one (M03) checks on discrete values, and the last one (M04) investigates the possible bounds of plant-based operating zones.

Table 4. The information behind the procedures that have different solution efficiency.

Hydro Plant	CPU Time Per Trial (Milliseconds)				Number of Trials
	M01	M02	M03	M04	M01	M02	M03	M04
Lidi	--	--	0.03	4	8	8	4200	3
Miaowei	--	--	0.03	3	81	33	12,380	5
Manwan	--	--	0.02	2.33	64	30	13,700	9
Jinghong	--	--	0.01	2.18	32	22	17,240	11
Xiaowan	--	--	0.01	4.71	729	39	40,950	14
Nuozhadu (158 m)	--	--	0.04	4.29	512	82	45,090	17
Nuozhadu(152 m)	--	--	0.04	6.67	19,683	204	42,030	18
VP01	--	--	0.03	9.75	312	303	56,040	24
VP02	--	--	0.03	14.87	315	402	70,050	30
VP03	--	--	0.03	14.39	318	501	84,060	36

gives the information that is related to the procedures having different solution efficiency. As shown by the information, trials in M01 and M02 take almost no time, but a huge number of trials will make the exhaustive method (M01) inapplicable. A trial in M03, which uses the MILP solver to check on the feasibility of a discrete power output, takes less time than the M04, which seeks new plant-based operating zones with the MILP solver. The M03, however, takes much longer time to secure the plant-based operating zones since it needs many more trials than the M04, which requires the least trials among all the four procedures. Apparently, almost no time is taken in a trial, and the second least number of trials makes the M02 the most favorable method in all the case studies.

4.3. Detailed Results

The M01, M02, and M04 can accurately identify the plant-based operating zones, which the M03 can also do at certain precision. The results of the plant-based operating zones in seven case studies are summarized in

Table 5. Results of plant-based operating zones at a water head.

Zone #	Lidi (h = 36.4 m)	Miaowei (85 m)	Manwan (89.4 m)	Jinghong (60 m)	Xiaowan (215 m)	Nuozhadu
						(158 m)	152 m
0	[0, 0]	[0, 0]	[0, 0]	[0, 0]	[0, 0]	[0, 0]	[0, 0]
1	[60, 420]	[120, 170]	[90, 120]	[200, 345]	[120, 4096]	[420, 501]	[211, 220]
2	--	[230, 340]	[140, 1370]	[400, 1725]	--	[840, 1002]	[420, 467]
3	--	[350, 1238]	--	--	--	[1260, 1503]	[631, 687]
4	--	--	--	--	--	[1680, 2004]	[840, 934]
5	--	--	--	--	--	[2100, 2505]	[1051, 1154]
6	--	--	--	--	--	[2520, 4509]	[1260, 1401]
7	--	--	--	--	--	--	[1471, 1621]
8	--	--	--	--	--	--	[1680, 1868]
9	--	--	--	--	--	--	[1891, 2088]
10	--	--	--	--	--	--	[2100, 4203]

, with a water head selected for the hydropower plant in each case study. Generally, more fragmented operating zones of the units in a hydropower plant will lead to more fragmented plant-based operating zones. The results tell that Nuozhadu has the most fragmented plant-based operating zone at 152 m of water head, while the Xiaowan has a very broad operating zone.

Figure 3 illustrates how the operating zones (green blocks) of six units installed in the Manwan can be combined into plant-based operating zones. At the water head of 89.4 m, the first unit (type #4) can only remain shut down, while the other units have one more zone to operate in. Here, the shutdown state is regarded as the first operating zone for both an individual unit and a hydropower plant. Careful observation reveals that the second plant-based operating zone is the same as the second operating zone of the seventh unit (type #6). The third plant-based operating zone covers an extensive range, making the hydropower plant more flexible in scheduling its power output.

4.4. Errors from Interpolation

As shown in Table 2, the operating zones of a unit are provided for a sample of water heads, where the plant-based operating zones can be identified directly, as given in the second and sixth columns of

Table 6. Errors in interpolation with Nuozhadu as an example.

Zone #	156 m	158 m			162 m
		Unit-Based	Plant-Based	Error
0	[0, 0]	[0, 0]	[0, 0]	0.0	[0, 0]
1	[420, 490]	[420, 501]	[420, 490]	[0, 11]	[420, 523]
2	[840, 980]	[840, 1002]	[840, 980]	[0, 22]	[840, 1046]
3	[1260, 1470]	[1260, 1503]	[1260, 1470]	[0, 33]	[1260, 1569]
4	[1680, 1960]	[1680, 2004]	[1680, 1960]	[0, 44]	[1680, 2092]
5	[2100, 2450]	[2100, 2505]	[2100, 2450]	[0, 55]	[2100, 4707]
6	[2520, 4410]	[2520, 4509]	[2520, 4410]	[0, 99]

for the Nuozhadu at the sampling water heads: 156 m and 162 m. At a water head of 158 m between two sampling water heads: 156 m and 162 m; however, the plant-based operating zones can either be estimated as those in the fourth column with interpolation between the plant-based operating zones at two sampling water heads or determined as those in the third column by estimating the unit operating zones at the middle water head with interpolation between two sampling water heads and then combining them into the plant-based operating zones. As shown in the fifth column in Table 6, there are gaps between two ways of identifying the plant-based operating zones, which must be updated with the operating zones of units whenever the water head has changed, especially frequently in an iterative solution process in hydropower scheduling.

5. Conclusions

This work presents three procedures (M01, M03, and M04) to compare with a previous one (M02) in combining operating zones of multiple units into plant-based operating zones. The first one (M01) is an exhaustive method that enumerates all the possible combinations of the operating zones of all the units in a hydropower plant. The M02 improves on the M01 by merging the operating zones of the units one by one in order. The M03 employs an optimization solver to reduce the number of zone-merging trials, and the M04 checks discrete points to identify the possible operating zones of a combined generator. The experiments of the procedures on a variety of hydropower plants reveal that, impressively, the exhaustive method (M01) takes little time in solving problems in scales smaller than 18 (measured as the product of the number of units and the number of zones per unit), which in fact can cover most of the hydropower plants in China.

The second method (M02) demonstrates more prominent strength than any other method in dealing with problems in scales larger than 36 and, therefore, should always be recommended to solve problems on any scale.

The M03 takes much longer to secure the plant-based operating zones since it needs many more trials than the M04, which requires the least trials among all four procedures.

Errors occur when using interpolation to estimate the plant-based operating zones at a water head between two sampling water heads.

For the first time, this work discussed different procedures to combine operating zones of multiple units into plant-based operating zones and compared their efficiency. M02 is recommended when obtaining plant-based operating zones according to the results of this work, through which errors are reduced. Thus, the consistency of the plant-based and unit-based operating zones is enhanced. Moreover, the efficiency in solving the hydropower scheduling problem is improved compared to the other ways discussed.

Subjecting to the research advance in combining operating zones of units in a hydropower plant into plant-based operating zones, no more study cases are found in the other works, and M02 needs to be compared with more procedures. More research is expected to be conducted under the conditions of more potential combinations of different unit types in other regions of the world.